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Publishers’ page

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Publishers’ page

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Publishers’ page

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To our families

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vi Molecular Electronics: An Introduction to Theory and Experiment

January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Preface

The trend in the miniaturization of electronic devices has naturally led

to the question of whether or not it is possible to use single molecules
as active elements in nanocircuits for a variety of applications. The re-
cent developments in nanofabrication techniques have made possible the
old dream of contacting individual molecules and exploring their electronic
transport properties. Moreover, it has been shown that molecules can in-
deed mimic the behavior of some of today’s microelectronic components,
and even strategies to interconnect molecular devices have already been
developed. These achievements have given rise to what is nowadays known
as Molecular Electronics. There are still many problems and challenges
to be faced to make this novel electronics a viable technology, but the
exploration of molecular-scale circuits has already led to the discovery of
many fundamental effects. In this sense, molecular electronics has become
a new interdisciplinary field of science, in which knowledge from traditional
disciplines like physics, chemistry, engineering and biology is combined to
understand the electrical and thermal conduction at the molecular scale.
This book provides a comprehensive overview of the rapidly developing
field of molecular electronics. It focuses on our present understanding of
the electrical conduction in single-molecule circuits and presents a thorough
introduction to the experimental techniques and the theoretical concepts.
To be precise, our goal in this monograph is two-fold. On the one hand, we
want to provide a true textbook for advanced undergraduate and graduate
students both in physics and chemistry who are interested in the field of
molecular electronics or nanoelectronics in general. Our idea is to take
a student with a good background in quantum mechanics all the way to
be able to follow the specialized literature in molecular electronics or to
start working in this field. On the other hand, we also want provide a

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viii Molecular Electronics: An Introduction to Theory and Experiment

thorough review of the recent activities in molecular electronics from which

newcomers and specialists in the field can benefit.
Bearing these goals in mind, this book has been written in a self-
contained and unified way. It contains four parts that can be read indepen-
dently. In the first two ones we review the basic experimental techniques
and the main theoretical concepts concerning the electronic transport in
atomic-scale junctions. These two parts are meant to be textbook material
for an advanced course in molecular electronics. In particular, we have in-
cluded a collection of exercises at the end of most chapters, which in many
cases are motivated by recent experiments in the field. On the other hand,
Part 3 contains two chapters in which we describe at an introductory level
the physics of metallic atomic-size contacts and we also point out some of
the remaining challenges and open problems in this context. Finally, Part
4 is devoted to the electrical and thermal transport in molecular circuits,
with special emphasis on single-molecule junctions. Here, we do not only
review the recent activities in the field of molecular electronics, but we also
introduce the addressed topics at a basic level. In this sense, we have often
included unpublished material and additional exercises to help the reader
to gain a deeper insight into the fundamental concepts involved in the field
of molecular electronics.1
We have tried to cover in this monograph as many aspects of molecular
electronics as possible, but obviously the selection is limited for space rea-
sons and it reflects unavoidably our own research interests. We also want
to apologize with those authors that feel that their contribution was not
properly highlighted in the review part of this monograph, but it is by now
impossible to include all the huge amount of work done in this field. Fi-
nally, we just hope to have achieved, at least partially, the goal that truly
motivated the writing of this book, namely the sincere will to provide a
useful book for the new generation of researchers that should consolidate
molecular electronics as a solid pillar of the emerging nanoscience.

1 See section 1.3 for a more detailed description of the structure and scope of the book.
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Acknowledgments

It would not have been possible to write the book without the help of many
coworkers and colleagues. First of all, we want to thank Edith Goldberg
for encouraging one of us (JCC) to give a postgraduate course on molecular
electronics in the fall of 2008 in Santa Fe (Argentina). The excellent stu-
dents who attended that course demonstrated that, after a 50-hours course
and without any previous knowledge about this field, one can master the
basic concepts and techniques that now form the body of this monograph.
This fact provided the final boost that we needed to collect all our notes
and turn them into this book.
Similarly, for the experimental point of view of this book, the students
in the graduate course at Konstanz served as test candidates. Some of them
even got contaminated by this exciting field and went on asking questions
what finally resulted in contributions to this book. Very valuable input
came from my colleague Artur Erbe who was the real expert in molecular
electronics in our Department until he left to Dresden.
We also want to express our gratitude to Alvaro Martı́n Rodero, who
not only introduced one of us (JCC) to the exciting field of nanoelectronics,
but also contributed decisively to this manuscript with his personal notes,
which are the basis of several chapters of the theoretical background. The
same holds for Hilbert von Löhneysen and Cristián Urbina who sent the
other one of us (ES) to perform experiments with nanoelectronic circuits.
We would especially like to thank our coworkers Fabian Pauly, Janne K.
Viljas, Michael Häfner, Sören Wohlthat, Stefan Bilan, Linda A. Zotti, Cécile
Bacca, Stefan Bächle, Tobias Böhler, Uta Eberlein, Stefan Egle, Daniel
Guhr, Ning Kang, Thomas Kirchner, Christian Kreuter, Shou-Peng Liu,
Youngsang Kim, Hans-Fridtjof Pernau, Olivier Schecker, Christian Schirm,
Dima Sysoiev, Simon Verleger, and Reimar Waitz. They have contributed

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x Molecular Electronics: An Introduction to Theory and Experiment

to this manuscript with many results, special figures and very important
suggestions and critical comments about the text.
Thanks go also sincerely to our colleagues who have read different parts
of the manuscript and have provided helpful comments: Douglas Natelson,
Abraham Nitzan, Wilson Ho, Latha Venkataraman, and Arunava Majum-
dar.
This monograph reflects our view of this field, which has emerged thanks
to the collaboration and exchange of ideas with many colleagues over the
years. So in this respect, we want to thank Alfredo Levy Yeyati, Gerd
Schön, Jan Heurich, Wolfgang Wenzel, Jan M. van Ruitenbeek, Nicolás
Agraı̈t, Gabino Rubio, Roel Smit, Oren Tal, Markus Dreher, Peter Nielaba,
Christoph Sürgers, Maya Lukas, Christoph Strunk, Sophie Géron, Richard
Berndt, Paul Leiderer, Wolfgang Belzig, Marcel Mayor, Thomas Huhn,
Andreas Marx, Ulrich Steiner, and Ulrich Groth.
We also want acknowledge the contribution of all the authors who have
kindly granted us the permission to reprint their work in this monograph.
Finally, I (JCC) want to thank my parents and brothers for being always
by my side. I also want to thank Ana for being so patient and share my
time with this book for too many nights and weekends. ES thanks her
family for continuous support and reminding me steadily of what is really
important in life.
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Contents

Preface vii

Acknowledgments ix

Brief history of the field and experimental tech-

niques 1

1. The birth of molecular electronics 3

1.1 Why molecular electronics? . . . . . . . . . . . . . . . . . 5
1.2 A brief history of molecular electronics . . . . . . . . . . . 6
1.3 Scope and structure of the book . . . . . . . . . . . . . . 14

2. Fabrication of metallic atomic-size contacts 19

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Techniques involving the scanning electron microscope
(STM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Methods using atomic force microscopes (AFM) . . . . . 21
2.4 Contacts between macroscopic wires . . . . . . . . . . . . 22
2.5 Transmission electron microscope . . . . . . . . . . . . . . 23
2.6 Mechanically controllable break junctions (MCBJ) . . . . 24
2.7 Electromigration technique . . . . . . . . . . . . . . . . . 31
2.8 Electrochemical methods . . . . . . . . . . . . . . . . . . . 35
2.9 Recent developments . . . . . . . . . . . . . . . . . . . . . 37
2.10 Electronic transport measurements . . . . . . . . . . . . . 38
2.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

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xii Molecular Electronics: An Introduction to Theory and Experiment

3. Contacting single molecules: Experimental techniques 45

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2 Molecules for Molecular Electronics . . . . . . . . . . . . . 46
3.2.1 Hydrocarbons . . . . . . . . . . . . . . . . . . . . 47
3.2.2 All carbon materials . . . . . . . . . . . . . . . . . 50
3.2.3 DNA and DNA derivatives . . . . . . . . . . . . . 51
3.2.4 Metal-molecule contacts: anchoring groups . . . . 52
3.2.5 Conclusions: molecular functionalities . . . . . . . 52
3.3 Deposition of molecules . . . . . . . . . . . . . . . . . . . 53
3.4 Contacting single molecules . . . . . . . . . . . . . . . . . 55
3.4.1 Electromigration technique . . . . . . . . . . . . . 56
3.4.2 Molecular contacts using the transmission electron
microscope . . . . . . . . . . . . . . . . . . . . . . 58
3.4.3 Gold nanoparticle dumbbells . . . . . . . . . . . . 59
3.4.4 Scanning probe techniques . . . . . . . . . . . . . 60
3.4.5 Mechanically controllable break-junctions (MCBJs) 64
3.5 Contacting molecular ensembles . . . . . . . . . . . . . . . 66
3.5.1 Nanopores . . . . . . . . . . . . . . . . . . . . . . 66
3.5.2 Shadow masks . . . . . . . . . . . . . . . . . . . . 68
3.5.3 Conductive polymer electrodes . . . . . . . . . . . 69
3.5.4 Microtransfer printing . . . . . . . . . . . . . . . . 70
3.5.5 Gold nanoparticle arrays . . . . . . . . . . . . . . 71
3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Theoretical background 75

4. The scattering approach 77

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2 From mesoscopic conductors to atomic-scale junctions . . 79
4.3 Conductance is transmission: Heuristic derivation of the
Landauer formula . . . . . . . . . . . . . . . . . . . . . . . 81
4.4 Penetration of a potential barrier: Tunnel effect . . . . . . 83
4.5 The scattering matrix . . . . . . . . . . . . . . . . . . . . 88
4.5.1 Definition and properties of the scattering matrix 88
4.5.2 Combining scattering matrices . . . . . . . . . . . 91
4.6 Multichannel Landauer formula . . . . . . . . . . . . . . . 92
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Contents xiii

4.6.1 Conductance quantization in 2DEG: Landauer for-

mula at work . . . . . . . . . . . . . . . . . . . . . 97
4.7 Shot Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.8 Thermal transport and thermoelectric phenomena . . . . 104
4.9 Limitations of the scattering approach . . . . . . . . . . . 106
4.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5. Introduction to equilibrium Green’s function techniques 111

5.1 The Schrödinger and Heisenberg pictures . . . . . . . . . 112
5.2 Green’s functions of a noninteracting electron system . . . 113
5.3 Application to tight-binding Hamiltonians . . . . . . . . . 118
5.3.1 Example 1: A hydrogen molecule . . . . . . . . . 118
5.3.2 Example 2: Semi-infinite linear chain . . . . . . . 122
5.3.3 Example 3: A single level coupled to electrodes . 124
5.4 Green’s functions in time domain . . . . . . . . . . . . . . 128
5.4.1 The Lehmann representation . . . . . . . . . . . . 131
5.4.2 Relation to observables . . . . . . . . . . . . . . . 134
5.4.3 Equation of motion method . . . . . . . . . . . . 136
5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6. Green’s functions and Feynman diagrams 143

6.1 The interaction picture . . . . . . . . . . . . . . . . . . . . 144
6.2 The time-evolution operator . . . . . . . . . . . . . . . . . 146
6.3 Perturbative expansion of causal Green’s functions . . . . 148
6.4 Wick’s theorem . . . . . . . . . . . . . . . . . . . . . . . . 149
6.5 Feynman diagrams . . . . . . . . . . . . . . . . . . . . . . 151
6.5.1 Feynman diagrams for the electron-electron inter-
action . . . . . . . . . . . . . . . . . . . . . . . . . 152
6.5.2 Feynman diagrams for an external potential . . . 157
6.6 Feynman diagrams in energy space . . . . . . . . . . . . . 158
6.7 Electronic self-energy and Dyson’s equation . . . . . . . . 162
6.8 Self-consistent diagrammatic theory: The Hartree-Fock
approximation . . . . . . . . . . . . . . . . . . . . . . . . 167
6.9 The Anderson model and the Kondo effect . . . . . . . . . 170
6.9.1 Friedel sum rule . . . . . . . . . . . . . . . . . . . 171
6.9.2 Perturbative analysis . . . . . . . . . . . . . . . . 173
6.10 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . 175
6.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
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xiv Molecular Electronics: An Introduction to Theory and Experiment

7. Nonequilibrium Green’s functions formalism 179

7.1 The Keldysh formalism . . . . . . . . . . . . . . . . . . . 180
7.2 Diagrammatic expansion in the Keldysh formalism . . . . 184
7.3 Basic relations and equations in the Keldysh formalism . 186
7.3.1 Relations between the Green’s functions . . . . . 186
7.3.2 The triangular representation . . . . . . . . . . . 187
7.3.3 Unperturbed Keldysh-Green’s functions . . . . . . 189
7.3.4 Some comments on the notation . . . . . . . . . . 191
7.4 Application of Keldysh formalism to simple transport
problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
7.4.1 Electrical current through a metallic atomic contact 193
7.4.2 Shot noise in an atomic contact . . . . . . . . . . 199
7.4.3 Current through a resonant level . . . . . . . . . . 200
7.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

8. Formulas of the electrical current 205

8.1 Elastic current: Microscopic derivation of the Landauer
formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
8.1.1 An example: back to the resonant tunneling model 211
8.1.2 Nonorthogonal basis sets . . . . . . . . . . . . . . 212
8.1.3 Spin-dependent elastic transport . . . . . . . . . . 213
8.2 Current through an interacting atomic-scale junction . . . 215
8.2.1 Electron-phonon interaction in the resonant tun-
neling model . . . . . . . . . . . . . . . . . . . . . 217
8.2.2 The Meir-Wingreen formula . . . . . . . . . . . . 222
8.3 Time-dependent transport in nanoscale junctions . . . . . 224
8.3.1 Photon-assisted resonant tunneling . . . . . . . . 231
8.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

9. Electronic structure I: Tight-binding approach 237

9.1 Basics of the tight-binding approach . . . . . . . . . . . . 237
9.2 The extended Hückel method . . . . . . . . . . . . . . . . 241
9.3 Matrix elements in solid state approaches . . . . . . . . . 242
9.3.1 Two-center matrix elements . . . . . . . . . . . . 244
9.4 Slater-Koster two-center approximation . . . . . . . . . . 246
9.5 Some illustrative examples . . . . . . . . . . . . . . . . . . 247
9.5.1 Example 1: A benzene molecule . . . . . . . . . . 248
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Contents xv

9.5.2 Example 2: Energy bands in line, square and cubic

Bravais lattices . . . . . . . . . . . . . . . . . . . . 250
9.5.3 Example 3: Energy bands of graphene . . . . . . 252
9.6 The NRL tight-binding method . . . . . . . . . . . . . . . 253
9.7 The tight-binding approach in molecular electronics . . . 257
9.7.1 Some comments on the practical implementation
of the tight-binding approach . . . . . . . . . . . . 258
9.7.2 Tight-binding simulations of atomic-scale trans-
port junctions . . . . . . . . . . . . . . . . . . . . 259
9.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

10. Electronic structure II: Density functional theory 263

10.1 Elementary quantum mechanics . . . . . . . . . . . . . . . 264
10.1.1 The Schrödinger equation . . . . . . . . . . . . . . 264
10.1.2 The variational principle for the ground state . . 265
10.1.3 The Hartree-Fock approximation . . . . . . . . . . 266
10.2 Early density functional theories . . . . . . . . . . . . . . 268
10.3 The Hohenberg-Kohn theorems . . . . . . . . . . . . . . . 269
10.4 The Kohn-Sham approach . . . . . . . . . . . . . . . . . . 271
10.5 The exchange-correlation functionals . . . . . . . . . . . . 273
10.5.1 LDA approximation . . . . . . . . . . . . . . . . . 273
10.5.2 The generalized gradient approximation . . . . . . 275
10.5.3 Hybrid functionals . . . . . . . . . . . . . . . . . . 277
10.6 The basic machinery of DFT . . . . . . . . . . . . . . . . 277
10.6.1 The LCAO Ansatz in the Kohn-Sham equations . 278
10.6.2 Basis sets . . . . . . . . . . . . . . . . . . . . . . . 280
10.7 DFT performance . . . . . . . . . . . . . . . . . . . . . . 282
10.8 DFT in molecular electronics . . . . . . . . . . . . . . . . 284
10.8.1 Combining DFT with NEGF techniques . . . . . 285
10.8.2 Pluses and minuses of DFT-NEGF-based methods 291
10.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

Metallic atomic-size contacts 293

11. The conductance of a single atom 295
11.1 Landauer approach to conductance: brief reminder . . . . 296
11.2 Conductance of atomic-scale contacts . . . . . . . . . . . 297
11.3 Conductance histograms . . . . . . . . . . . . . . . . . . . 300
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xvi Molecular Electronics: An Introduction to Theory and Experiment

11.4 Determining the conduction channels . . . . . . . . . . . . 304

11.5 The chemical nature of the conduction channels of one-
atom contacts . . . . . . . . . . . . . . . . . . . . . . . . . 308
11.6 Some further issues . . . . . . . . . . . . . . . . . . . . . . 316
11.7 Conductance fluctuations . . . . . . . . . . . . . . . . . . 319
11.8 Atomic chains: Parity oscillations in the conductance . . . 322
11.9 Concluding remarks . . . . . . . . . . . . . . . . . . . . . 331
11.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 332

12. Spin-dependent transport in ferromagnetic atomic con-

tacts 335
12.1 Conductance of ferromagnetic atomic contacts . . . . . . 336
12.2 Magnetoresistance of ferromagnetic atomic contacts . . . 343
12.3 Anisotropic magnetoresistance in atomic contacts . . . . . 347
12.4 Concluding remarks and open problems . . . . . . . . . . 353

Transport through molecular junctions 355

13. Coherent transport through molecular junctions I: Basic
concepts 357
13.1 Identifying the transport mechanism in single-molecule
junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
13.2 Some lessons from the resonant tunneling model . . . . . 364
13.2.1 Shape of the I-V curves . . . . . . . . . . . . . . . 366
13.2.2 Molecular contacts as tunnel junctions . . . . . . 368
13.2.3 Temperature dependence of the current . . . . . . 369
13.2.4 Symmetry of the I-V curves . . . . . . . . . . . . 371
13.2.5 The resonant tunneling model at work . . . . . . 373
13.3 A two-level model . . . . . . . . . . . . . . . . . . . . . . 374
13.4 Length dependence of the conductance . . . . . . . . . . . 377
13.5 Role of conjugation in π-electron systems . . . . . . . . . 381
13.6 Fano resonances . . . . . . . . . . . . . . . . . . . . . . . 382
13.7 Negative differential resistance . . . . . . . . . . . . . . . 385
13.8 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . 388
13.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

14. Coherent transport through molecular junctions II:

Test-bed molecules 391
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Contents xvii

14.1 Coherent transport through some test-bed molecules . . . 392

14.1.1 Benzenedithiol: how everything started . . . . . . 392
14.1.2 Conductance of alkanedithiol molecular junctions:
a reference system . . . . . . . . . . . . . . . . . . 395
14.1.3 The smallest molecular junction: hydrogen bridges 401
14.1.4 Highly conductive benzene junctions . . . . . . . . 405
14.2 Metal-molecule contact: The role of anchoring groups . . 408
14.3 Tuning chemically the conductance: The role of side-groups 412
14.4 Controlled STM-based single-molecule experiments . . . . 416
14.5 Conclusions and open problems . . . . . . . . . . . . . . . 420

15. Single-molecule transistors: Coulomb blockade and

Kondo physics 423
15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 423
15.2 Charging effects in transport through nanoscale devices . 425
15.3 Single-molecule three-terminal devices . . . . . . . . . . . 429
15.4 Coulomb blockade theory: Constant interaction model . . 432
15.4.1 Formulation of the problem . . . . . . . . . . . . . 432
15.4.2 Periodicity of the Coulomb blockade oscillations . 435
15.4.3 Qualitative discussion of the transport characteristics436
15.4.4 Amplitudes and line shapes: Rate equations . . . 439
15.5 Towards a theory of Coulomb blockade in molecular tran-
sistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
15.5.1 Many-body master equations . . . . . . . . . . . . 447
15.5.2 A simple example: The Anderson model . . . . . 449
15.6 Intermediate coupling: cotunneling and Kondo effect . . . 451
15.6.1 Elastic and inelastic cotunneling . . . . . . . . . . 451
15.6.2 Kondo effect . . . . . . . . . . . . . . . . . . . . . 453
15.7 Single-molecule transistors: Experimental results . . . . . 456
15.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 468

16. Vibrationally-induced inelastic current I: Experiment 473

16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 473
16.2 Inelastic electron tunneling spectroscopy (IETS) . . . . . 475
16.3 Highly conductive junctions: Point-contact spectroscopy
(PCS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483
16.4 Crossover between PCS and IETS . . . . . . . . . . . . . 490
16.5 Resonant inelastic electron tunneling spectroscopy (RIETS) 493
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xviii Molecular Electronics: An Introduction to Theory and Experiment

16.6 Summary of vibrational signatures . . . . . . . . . . . . . 499

17. Vibrationally-induced inelastic current II: Theory 501

17.1 Weak electron-phonon coupling regime . . . . . . . . . . . 501
17.1.1 Single-phonon model . . . . . . . . . . . . . . . . 502
17.1.2 Ab initio description of inelastic currents . . . . . 512
17.2 Intermediate electron-phonon coupling regime . . . . . . . 520
17.3 Strong electron-phonon coupling regime . . . . . . . . . . 524
17.3.1 Coulomb blockade regime . . . . . . . . . . . . . . 524
17.3.2 Interplay of Kondo physics and vibronic effects . . 532
17.4 Concluding remarks and open problems . . . . . . . . . . 534
17.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 535

18. The hopping regime and transport through DNA

molecules 537
18.1 Signatures of the hopping regime . . . . . . . . . . . . . . 538
18.2 Hopping transport in molecular junctions: Experimental
examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 541
18.3 DNA-based molecular junctions . . . . . . . . . . . . . . . 546
18.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 552

19. Beyond electrical conductance: shot noise and thermal

transport 553
19.1 Shot noise in atomic and molecular junctions . . . . . . . 554
19.2 Heating and heat conduction . . . . . . . . . . . . . . . . 560
19.2.1 General considerations . . . . . . . . . . . . . . . 561
19.2.2 Thermal conductance . . . . . . . . . . . . . . . . 562
19.2.3 Heating and junction temperature . . . . . . . . . 565
19.3 Thermoelectricity in molecular junctions . . . . . . . . . . 569

20. Optical properties of current-carrying molecular junc-

tions 579
20.1 Surface-enhanced Raman spectroscopy of molecular junc-
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580
20.2 Transport mechanisms in irradiated molecular junctions . 583
20.3 Theory of photon-assisted tunneling . . . . . . . . . . . . 585
20.3.1 Basic theory . . . . . . . . . . . . . . . . . . . . . 586
20.3.2 Theory of PAT in atomic contacts . . . . . . . . . 590
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Contents xix

20.3.3 Theory of PAT in molecular junctions . . . . . . . 592

20.4 Experiments on radiation-induced transport in atomic and
molecular junctions . . . . . . . . . . . . . . . . . . . . . . 594
20.5 Resonant current amplification and other transport phe-
nomena in ac driven molecular junctions . . . . . . . . . . 601
20.6 Fluorescence from current-carrying molecular junctions . 604
20.7 Molecular optoelectronic devices . . . . . . . . . . . . . . 608
20.8 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . 613
20.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 614

21. What is missing in this book? 617

Appendixes 621
Appendix A Second Quantization 623
A.1 Harmonic oscillator and phonons . . . . . . . . . . . . . . 624
A.1.1 Review of simple harmonic oscillator quantization 624
A.1.2 1D harmonic chain . . . . . . . . . . . . . . . . . 626
A.2 Second quantization for fermions . . . . . . . . . . . . . . 628
A.2.1 Many-body wave function in second quantization 628
A.2.2 Creation and annihilation operators . . . . . . . . 630
A.2.3 Operators in second quantization . . . . . . . . . 632
A.2.4 Some special Hamiltonians . . . . . . . . . . . . . 634
A.3 Second quantization for bosons . . . . . . . . . . . . . . . 637
A.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 638

Bibliography 639

Index 697
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

PART 1

Brief history of the field and

experimental techniques

1
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

2
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Chapter 1

The birth of molecular electronics

How does the electrical current flow through a single molecule? Can a
molecule mimic the behavior of an ordinary microelectronics component or
maybe provide a new electronic functionality? How can a single molecule be
addressed and incorporated into an electrical circuit? How to interconnect
molecular devices and integrate them into complex architectures? These
questions and related ones are by no means new and, as we shall see later in
this chapter, they were already posed many decades ago. The difference is
that we are now in position to at least address them in the usual scientific
manner, i.e. by providing quantitative experimental and theoretical results.
The advances in the last two or three decades, both in nanofabrication
techniques and in the quantum theory of electronic transport, allow us now
to explore and to understand the basic properties of rudimentary electrical
circuits in which molecules are used as basic building blocks. It is worth
stressing right from the start that we do not yet have definitive answers for
the questions posed above. However, a tremendous progress has been made
in recent years and some concepts and techniques have already been firmly
established. In this sense, one of main goals of this book is to review such
progress, but more importantly, this monograph is intended to provide a
solid basis for the new generation of researchers that should take the field
of molecular electronics to the next level.
Molecular electronics, as used in this book, is defined as the field of
science that investigates the electronic and thermal transport properties of
circuits in which individual molecules (or an assembly of them) are used as
basic building blocks.1 Obviously, some of the feature dimensions of such

1 Molecular electronics, in the sense used here, should not be confused with organic

electronics, the field in which molecular materials are investigated as possible constituents
of a variety of macroscopic electronic devices.

3
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4 Molecular Electronics: An Introduction to Theory and Experiment

MESOSCOPIC QUANTUM
PHYSICS CHEMISTRY

ELECTRICAL INORGANIC
ENGINEERING CHEMISTRY

MATERIAL ORGANIC
SCIENCE CHEMISTRY
BIOLOGY

Fig. 1.1 Molecular electronics: An interdisciplinary field.

molecular circuits are of the order of nanometers (or even less) and there-
fore, molecular electronics should be viewed as a subfield of nanoscience
or nanotechnology in which traditional disciplines like physics, chemistry,
material science, electrical engineering and biology play a fundamental role
(see Fig. 1.1). Molecular electronics, in the sense of a potential technology,
is based on the bottom-up approach where the idea is to assemble elemen-
tary pieces to form more complex structures, as opposed to the top-down
approach where the idea is to shrink macroscopic systems and components.
Molecular electronics has emerged from the constant quest for new tech-
nologies that could complement the silicon-based electronics, which in the
meantime it has become a true nanotechnology. It seems very unlikely
that molecular electronics will ever replace the silicon-based electronics,
but there are good reasons to believe that it can complement it by provid-
ing, for instance, novel functionalities out of the scope of traditional solid
state devices. More importantly, molecular electronics has become in recent
years a true field of science where many basic questions and quantum phe-
nomena are being investigated. In this sense, the importance of molecular
electronics is unquestionable and we are convinced that different traditional
disciplines will benefit from advances in this new field.
In the rest of this introductory chapter, we shall first try to answer the
questions of why it is worth pursuing molecular electronics research and
why it is interesting to work in a field like this. Then, in section 1.2 we
shall briefly review the complex history of this field to set the stage for
this book. Finally, in section 1.3 we shall clearly define the scope of this
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

The birth of molecular electronics 5

monograph and explain its structure.

1.1 Why molecular electronics?

Every researcher is sooner or later confronted with natural questions like

“why do you work in your field?” or “what is your research good for?”
Of course, the answers are always personal, but in the case of molecular
electronics they also depend on whether one’s interests are closer to fun-
damental science or to technological applications. From the point of view
of basic science, molecular electronics offers, for instance, the possibility to
investigate electronic and thermal conduction at the smallest imaginable
scale, where the physics is completely dominated by quantum mechanical
effects. The small feature dimensions of molecular circuits together with the
great variety of electrical, mechanical and optical properties of molecules
can give rise to countless new physical phenomena. Molecular junctions are
also ideal systems where to investigate and shed new light into the funda-
mental electron transfer mechanisms that play a key role both in chemistry
and biology. These reasons and many others make molecular electronics a
very attractive field of basic research. Moreover, we should never forget that
the history of science proves that the exploration of new territories and the
subsequent discovery of novel phenomena often lead to unexpected tech-
nological applications. History also teaches us that there is no technology
without basic understanding and thus, the future of molecular electronics
as an emerging technology depends on our ability to understand the funda-
mental mechanisms that govern the electronic conduction at the molecular
scale.
From a technological point of view, there are also good reasons to inves-
tigate the use of molecules as electronically active elements for a variety of
applications. In comparison with the silicon-based technology, which is al-
ready a nanotechnology in the sense that the structure sizes are in the range
of nanometers,2 molecular electronics could in principle offer the following
major advantages [2]:

• Size. The reduce size of small molecules (between 1 and 10 nm)

could lead to a higher packing density of devices with the subse-
quent advantages in cost, efficiency, and power dissipation.
2 The next generation of transistors for advanced microprocessors will have gate lengths
of 22 nm and a SiO2 gate oxide thickness of less than 1.2 nm [1].
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6 Molecular Electronics: An Introduction to Theory and Experiment

• Speed. Although most molecules are poorly conductive, good

molecular wires could reduce the transit time of typical transis-
tors (∼ 10−14 s), reducing so the time needed for an operation.
• Assembly and recognition. One can exploit specific intermolecular
interactions to form structures by nanoscale self-assembly. Molec-
ular recognition can be used to modify electronic behavior, provid-
ing both switching and sensing capabilities on the single-molecule
scale.
• New functionalities. Special properties of molecules, like the exis-
tence of distinct stable geometric structures or isomers, could lead
to new electronic functions that are not possible to implement in
conventional solid state devices.
• Synthetic tailorability. By choice of composition and geometry, one
can extensively vary a molecule’s transport, binding, optical, and
structural properties. The tools of molecular synthesis are highly
developed.

Molecules have also obvious disadvantages such as instabilities at high

temperatures. Moreover, the fabrication of reliable molecular junctions
requires sometimes to control matter at an unprecedented level, which can
be not only difficult, but also slow and costly. Anyway, the advantages
described above are sufficient to motivate the exploration of a molecule-
based electronics.

1.2 A brief history of molecular electronics

It is always difficult to trace back the history of an emerging field and to

summarize it in a few pages. Anyway, even at the risk of being unfair leaving
out some important contributors, we find necessary to say a few words about
the history of molecular electronics as a tribute to those visionary scientists
that made possible that we are now working in this fascinating field. Our
brief account here is partially based on a delightful (non-scientific) article
by Choi and Mody [3], which reviews the history of molecular electronics
paying special attention to its social aspects.
We start this historical review in 1950’s, after the revolution in electron-
ics due to the invention of the transistor and the subsequent introduction
of integrated circuits. In that context and in view of the difficulties to rad-
ically miniaturize the existent electronic components, Arthur von Hippel,
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The birth of molecular electronics 7

a German physicist working at the MIT, formulated in 1956 the basis of a

bottom-up approach that he called molecular engineering [4]. He argued:
Instead of taking prefabricated materials and trying to de-
vise engineering applications consistent with their macroscopic
properties, one builds materials from their atoms and molecules
for the purpose at hand ...

The concept of molecular engineering introduced by von Hippel [5] led

to the first notion of “molecular electronics”, which crystallized in a col-
laboration between the company Westinghouse and the US Air Force at
the end of the 1950’s. Westinghouse had begun a program to implement
von Hippel’s ideas and it applied for the financial support of the US Air
Force, which at that time was receptive to new ideas and alternatives to the
recently introduced integrated circuits. The Air Force organized a confer-
ence on “Molecular Electronics” and invited scientists and engineers from
military and private research labs. In this conference, colonel C.H. Lewis,
director of Electronics at the Air Research and Development Command,
expressed the need for a breakthrough in electronics in the following way:
Instead of taking known materials which will perform explicit
electronic functions, and reducing them in size, we should build
materials which due to their inherent molecular structure will
exhibit certain electronic property phenomena. We should syn-
thesize, that is, tailor materials with predetermined electronic
characteristic. Once we can correlate electronic property phe-
nomena with the chemical, physical, structural, and molecu-
lar properties of matter, we should be able to tailor materials
with predetermined characteristics. We could design and create
materials to perform desired functions. Inherent dependability
might eventually result. We call this more exact process of con-
structing materials with predetermined electrical characteristics
MOLECULAR ELECTRONICS.

This is probably the first time that the term molecular electronics was
used publicly, although it originally referred to a new strategy for the fab-
rication of electronic components, and it had yet little to do with the vision
of using individual molecules as electronically active elements. Fig. 1.2
summarizes the vision of colonel Lewis, where molecular electronics should
constitute be the next breakthrough in electronics, although it was not yet
clear what molecular electronics was supposed to mean.
The collaboration between Westinghouse and the US Air Force, which
started after the mentioned conference, lasted a few years and certain
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8 Molecular Electronics: An Introduction to Theory and Experiment

Electronic
Equipment Conventioal Miniaturization Microminiaturization
Trend Component
Size Reduction (Transistor) (Integrated Circuit)

Size Breakthrough!!!
Weight
and
Space Asymptote for Present Technologies
Reduction
Molecular
Electronics
Asymptote for Molecular Technologies

1940 1950 1954 1958

Time

Fig. 1.2 Graph presented by colonel Lewis of the US Air Force in the first conference
on molecular electronics held in November 1958. Here, one can see the trend in the
miniaturization of the electronic components during the 1940’s and 1950’s. According to
Lewis, molecular electronics should have constituted the next breakthrough in electronics
by the end of the 1950’s. Adapted from [3].

progress was indeed made in the development of new fabrication strategies.

However, these initiatives were not able to compete with the steady minia-
turization of the semiconductor-based electronic devices and they were soon
abandoned.
From a more scientific point of view, one can consider that molecular
electronics, as we understand it today, started at the end of the 1960’s and
the beginning of 1970’s. At that time, different groups started to investi-
gate experimentally the electronic transport through molecular monolay-
ers. For instance, Hans Kuhn, a Swiss chemist working at the University of
Göttingen, and his coworkers studied at that time new ways of fabricating
the so-called Langmuir-Blodgett films.3 They were able to not only master
the fabrication of these molecular films, but also to sandwich them between
metal electrodes and to measure the electrical conductivity of the resulting
junctions. In Fig. 1.3 we reproduce the experimental results of Ref. [6] for
the low-bias conductivity of Al/S(n)/Hg junctions, where S(n) stands for
a monolayer of Cd salt of fatty acid CH3 (CH2 )n−2 COOH of different chain
lengths. There one can see the exponential decay of the conductivity with
the length of the molecules, which is still a very important issue in today’s
3A
Langmuir-Blodgett film contains one or more monolayers of an organic material,
deposited from the surface of a liquid onto a solid by immersing the solid substrate into
the liquid. A monolayer is adsorbed homogeneously with each immersion or emersion
step, thus films with very accurate thickness can be formed.
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

The birth of molecular electronics 9

Fig. 1.3 Measurements of the low-bias tunneling conductivity (σt ) vs. the distance (d)
between the electrodes in Al/S(n)/Hg junctions. Here, S(n) stands for monolayers of
Cd salt of fatty acid CH3 (CH2 )n−2 COOH with different lengths (n ranges between 18
and 21). The solid line is a linear fit to the experiment data. The measurements were
performed at two different temperatures: 20 and -35 o C. Reprinted with permission from
[6]. Copyright 1971, American Institute of Physics.

molecular electronics (see Chapter 13). This type of experimental results

can be considered as the starting point of molecular electronics as a modern
field of science.
The idea of molecular electronics reappeared in the States at the be-
ginning of the 1970’s at IBM and thanks to the enthusiasm of Ari Aviram,
a synthetic chemist. Aviram was working at that time on charge-transfer
salts, which had recently been discovered to be reasonably good conduc-
tors in their solid form. Although Aviram’s task at IBM was to synthesize
new types of charge-transfer salts, he started working on the theory of elec-
tron transfer through single organic molecules in collaboration with Mark
Ratner,4 at that time at New York University. In the course of their inves-
tigations, Aviram and Ratner saw a clear analogy between charge-transfer
salts like TTF-TCNQ (tetrathiafulvalene-tetracyanoquinodimethane), with
a functional unit (TTF) rich in electrons and another unit (TCNQ) poor
in electrons, and traditional semiconductor diodes. In 1974 they published
a now-famous paper on “molecular rectifiers” [8] in which they described
4 Indeed Ratner was officially Aviram’s thesis advisor during that time.
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10 Molecular Electronics: An Introduction to Theory and Experiment

how a modified charge-transfer salt could operate as a traditional diode

in an electrical circuit. This is probably the first proposal to use a sin-
gle molecule as an electronic component, which is something that lies at
the heart of the modern molecular electronics. Aviram and Ratner’s idea
was considered during a long time a theoretical curiosity that could not be
tested experimentally and in this sense, it did not have much impact in the
scientific community at that time.
In the late 1970’s and early 1980’s other scientists started to work on
ideas similar to Aviram-Ratner’s unimolecular concept. Let us mention for
instance the name of Forrest Carter, a chemist at the Naval Research Lab-
oratory, who was certainly influenced by Feynman’s (1960) famous “Room
at the Bottom” speech [9]. Carter introduced concepts such as molecular
computing or cellular automata, where the essence was to use individual
molecules as the ultimate electronic components or as elementary units
where to store bits of information in a hypothetical molecular computer.
These ideas were to a large extend purely theoretical and they were no sup-
ported by real experiments. However, Carter was able to nucleate a first
molecular electronics community around him and, in particular, the orga-
nization of a series of conferences on molecular electronics in the 1980’s
played an important role in the history of this field. People like Robert
Metzger, Mark Reed and others, who played later an important role in
molecular electronics, attended those conferences and they were inspired
by the discussions held there.
As for many other fields in nanoscience, the invention of the scanning
tunneling microscope (STM) by Gerd Binnig and Heinrich Rohrer (at IBM
Zurich) in 1981 [10, 11] changed the panorama for molecular electronics.
The STM was the first tool that provided a practical way to “see”, “touch”,
and manipulate matter at the atomic scale (see Fig. 1.4). Soon after its
invention, it became clear to the STM could provide a realistic way to
address single molecules and to study their electronic transport properties.
Since the original experiments of Kuhn and coworkers [7], many different
groups studied the electrical conductivity through Langmuir-Blodgett (LB)
multilayers and even monolayers. For instance, Fujihira and co-workers
demonstrated an LB monolayer photodiode already back in 1985 [13], which
is probably the first unimolecular electronic device. In the 1990’s one of
the main goals in this context was to confirm the ideas of Aviram and
Ratner about unimolecular rectification. The Aviram-Ratner mechanism,
slightly modified, was confirmed by Robert Metzger’s group in both macro-
scopic and nanoscopic conductivity measurements through a monolayer of
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The birth of molecular electronics 11

Fig. 1.4 Principle of a local probe like the scanning tunneling microscope: The gentle
touch of a nanofinger. If the interaction between tip and sample decays sufficiently
rapidly on the atomic scale, only the two atoms that are closest to each other are able to
“feel” each other. Reprinted with permission from [12]. Copyright 1999 by the American
Physical Society.

γ-hexadecyl-quinolinium tricyanoquinomethanide in 1997 [14].

At the end of the 1980’s and the beginning of the 1990’s the appear-
ance of the metallic atomic-sized contacts had an important impact in the
nanoscience community. Different groups showed that the STM and the
recently introduced mechanically controllable break junction (MCBJ) tech-
nique5 could be used to fabricate metallic wires of atomic dimensions (for a
review, see Ref. [15]). Since then these nanowires have become an endless
source of new physical phenomena and have played a crucial role in the fields
of mesoscopic physics and nanoelectronics. The relevance of these systems
for molecular electronics is two-fold. On the one hand, they provide the
basis to contact individual molecules with dimensions on the range of a few
nanometers, which is out of the scope of conventional lithographies. On
the other hand, the atomic contacts have allowed establishing the connec-
tion between the quantum properties of single atoms and the macroscopic
electrical properties of the circuits in which they are embedded, which is
an important lesson for molecular electronics.6
In 1997 the collaboration between the groups of Mark Reed (a physicist
at Yale University) and James Tour (a synthetic chemist at the University
of South Carolina) led to the publication of the results of what is often
considered as the first transport experiment in single-molecule junctions
[16].7 These authors used the MCBJ technique to contact benzenedithiol
5 This technique will be described in the next chapter.
6 The physics of these metallic nanowires will be described in the third part of this
monograph.
7 Let us clarify that the first transport measurements involving single molecules were

indeed performed with the STM, but the experiment of Reed et al. is the first one realized
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12 Molecular Electronics: An Introduction to Theory and Experiment

Wire stretched until breakage,

resulting in tip formation

(a) (c)
Gold Gold
Gold wire electrode electrode

Add THF and benzene−1,4−dithiol Solvent evaporates, then tips

brought together until the
SAM onset of conductance

(b) (d)
Gold Gold
Gold wire
electrode electrode

SAM

Fig. 1.5 Schematics of the first transport measurements through single-molecule junc-
tions performed with the MCBJ technique [16]. (a) The gold wire of the break junc-
tion before breaking and tip formation. (b) After addition of benzene-1,4-dithiol, self-
assembled monolayers (SAMs) form on the gold wire surfaces. (c) Mechanical breakage
of the wire in solution produces two opposing gold contacts that are SAM-covered. (d)
After the solvent is evaporated, the gold contacts are slowly moved together until the
onset of conductance is achieved.

molecules with gold electrodes (the principle of this experiment is schemat-

ically illustrated in Fig. 1.5).8 The importance of this experiment is that
it triggered off the realization of many others in the same spirit. Indeed,
our review on single-molecule conduction in the last part of this book will
cover the activities from the appearance of this experiment on.
At the end of the 1990’s new experimental techniques were intro-
duced and additional results were reported showing that molecules can
indeed mimic the behavior of ordinary microelectronics components. Thus
for instance, Reed’s group adapted the so-called nanopore technique (see
Chapter 3) to form metal-self-assembled monolayer-metal heterojunctions.
With this technique it was shown that junctions based on certain organic
molecules can exhibit, for instance, rectifying behavior [17] or a very pro-
nounced negative differential resistance [18]. On the other hand, James
Heath and Fraser Stoddart groups joined efforts to show that junctions
based on rotaxanes and catenanes could act as reconfigurable switches
[19, 20].
in a symmetric structure that could in principle be integrated in more complex circuits.
8 This experiment will be described in detail in section 14.1.1.
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The birth of molecular electronics 13

Techniques like electromigration [21], which were specially designed to

contact single molecules, were developed at the turn of the century. These
methods made possible to incorporate a gate electrode in single-molecule
junctions and thus, to mimic the measurements performed in solid state
devices like transistors or in nanostructures like quantum dots. With the use
of these techniques it was possible to show that single-molecule junctions
can behave as a new kind of single-electron transistors [22] or that they
can exhibit basic physical phenomena like Coulomb blockade or the Kondo
effect [23, 24], which are well-known in the context of other nanoscopic
structures.
These results obtained in academic institutions and research laborato-
ries attracted the attention of global players in information technology like
HP, IBM and others that decided to set up small molecular electronics
research groups. This gave a new impulse to the field by providing very
important missing ingredients like, for instance, strategies to link molecular
devices with each other and with external systems. As an example we can
mention the nanoscale circuits based on a configurable crossbar architec-
ture introduced by Stanley Williams and coworkers at the HP Laboratories
in Palo Alto [25], see Fig. 1.6(a-d). This strategy was used, for instance,
to show that molecular crossbar circuits fabricated from a molecular mono-
layer of [2]rotaxanes can function as an ultra-high-density memory [26],
see Fig. 1.6(e-f). The working principle of these molecular memories is
supposed to be based on the ability of molecules like rotaxanes to switch
between two metastable states upon the application of an external bias
voltage. The actual origin of the switching behavior in these molecular
junctions has been heavily debated and, in some cases, it has shown that
the metal electrodes or the metal-molecule interface are responsible for
the switching mechanism rather than the molecules themselves (see e.g.
Ref. [27]). The controversy about these results, and also about some of
the original experiments mentioned above, led to the extended belief that
molecular electronics was going through a midlife crisis [28], although it was
no more than a teenager. In the meantime, the situation concerning the
molecular memories has been clarified to a large extend and more recently
the densest memory circuit ever made (1011 bits cm−2 ) was fabricated using
a monolayer of bistable [2]rotaxane molecules as the data storage elements
[29]. Although many scientific and engineering challenges, such as device
robustness, remain to be addressed before these devices can be practical,
these results show clearly the potential of a molecule-based electronics.
On the other hand, the efforts in recent years of numerous research
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14 Molecular Electronics: An Introduction to Theory and Experiment

(a) (b)

(e)

(c) (d)

(f)

Fig. 1.6 Nanoscale molecular-switch crossbar circuits. (a) An optical microscope image
of an array of four test circuits, showing that each has 16 contact pads with micron-
scale connections leading to nanoscale circuits in the center. (b) An image taken with
a scanning electron microscope (SEM) showing two mutually perpendicular arrays of
nanowires connected to their micron-scale connections. (c) A SEM image showing that
the two sets of nanowires cross each other in the central area. (d) A 3D image of the
crossbar taken with an atomic force microscope. (e) Schematic representation of the
crossbar circuit structure in which monolayer of the [2]rotaxane is sandwiched between
an array of Pt/Ti nanowires on the bottom and an array of Pt/Ti nanowires on the top.
(f) Molecular structure of the bistable [2]rotaxane R. Reprinted with permission from
[26]. Copyright 2003 IOP Publishing Ltd.

groups world-wide have established molecular electronics as a true field of

science, where there is a lot of new physics and chemistry to be learned.
Although it is still difficult to fabricate reliable molecular junctions, in par-
ticular at the single-molecule level, and there are other basic problems to
be solved, many concepts and techniques are by now well established and
they are precisely the subject of this book. For us, it is clear that molec-
ular electronics has reappeared this time to stay forever with us. In the
next years we shall surely contemplate many basic discoveries in this field
and some of them will hopefully lead to new and unforeseen technological
applications.

1.3 Scope and structure of the book

By now molecular electronics is a very broad field with many different inter-
esting aspects and special topics. These topics can be divided in a natural
way into those related to the development and potential applications of
molecular devices and those concerning the novel physical phenomena that
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

The birth of molecular electronics 15

take place in molecular-scale junctions. In this monograph we are interested

in the latter type of topics and, in particular, we shall focus our attention
on the understanding of the basic mechanisms that dominate the electronic
transport at the molecular scale. To be precise, we shall concentrate on
the analysis of the properties of single-molecule junctions, although some
examples of junctions based on molecular assemblies will also be presented
and discussed.
Our main goal in this monograph is two-fold. On the one hand, we want
to provide a true textbook on molecular electronic for advanced undergrad-
uate and graduate students both in physics and chemistry. The book has
been designed so that, by the end of it, a student with a background in
quantum mechanics and some elementary notions of solid state physics9
and organic chemistry10 should be able to start doing research in the field
of molecular electronics. On the other hand, we also want to provide a
thorough review of the activities on single-molecule conduction over the
last ten years, from which both newcomers and researches working in the
field can profit.
With this double goal in mind, we have divided this monograph into
four parts that can be read independently.11 The first two are meant as
textbook material that can be used for a regular course, while the last two
ones are closer to a topical review. Part 1 includes, apart from this intro-
ductory chapter, a detailed description of the experimental techniques that
are currently being used to fabricate both atomic-scale wires and molecular
junctions as well as the basic principles of transport measurements. Here,
we have tried to explain both the basis of the different techniques as well as
their advantages and disadvantages. Moreover, we have included in section
3.2 a brief discussion about the main molecules used in molecular electron-
ics and their basic properties, which can be viewed as an accelerated course
in organic chemistry.
Part 2 contains an extensive theoretical background that provides a ba-
sic introduction both to the transport mechanisms in nanoscale systems
and to the standard theoretical techniques that are used to describe the
transport in molecular systems. We want to stress that this theory part is
not just meant for theoreticians and theory-inclined students, but for every-
9 For the students in chemistry we recommend the brief introduction to solid state

physics provided in chapter 4 of Ref. [30] or in chapter 3 of Ref. [31].

10 For the students in physics we recommend the brief introduction to organic chemistry

provided in chapter 5 of Ref. [31].

11 There is indeed a fifth part that contains an appendix about the second quantization

formalism of quantum mechanics.

January 12, 2010 11:27 World Scientific Book - 9in x 6in book

16 Molecular Electronics: An Introduction to Theory and Experiment

body. All the topics are discussed in a didactic and self-contained manner
so that students without a previous knowledge on these topics should be
able, after reading this part, to follow the theory papers in this field. To
be precise, this part starts in Chapter 4 with an introduction to the scat-
tering (or Landauer) approach, which provides an appealing framework to
describe coherent transport in nanostructures. Then, we go on with several
chapters devoted to Green’s function techniques (Chapters 5-8), which pro-
vide powerful tools to compute equilibrium and nonequilibrium properties
of atomic-scale junctions beyond the capabilities of the scattering approach.
Finally, Chapters 9 and 10 deal with the two most widely used electronic
structure methods in molecular electronics, namely the tight-binding ap-
proach and density functional theory. These methods in combination with
the Green’s function techniques provide the starting point for the realistic
description of the transport properties of atomic and molecular junctions.
Let us emphasize that at the end of every chapter one can find several
exercises that have been chosen to illustrate the main concepts.
Part 3 presents a basic description of the physics of atomic-sized con-
tacts. Although this is not the main topic of the book, it is crucial to
have a basic knowledge about the transport properties of the metallic wires
that are then used as electrodes in molecular junctions. We have divided
this part into two chapters where we describe the physics of non-magnetic
atomic contacts (Chapter 11) and magnetic ones (Chapter 12).
Finally, Part 4 presents a detailed review on the transport through
molecular junctions. We have organized the material according to the phys-
ical mechanism which dominates the transport properties. Thus, we start
this part with two chapters devoted to the coherent transport in molecular
junctions (Chapters 13 and 14). Then, we discuss in Chapter 15 the physics
of the so-called molecular transistors, which are nothing but weakly coupled
molecular junctions where the transport is dominated by electronic corre-
lations that lead to phenomena like Coulomb blockade or the Kondo effect.
We then proceed to discuss in Chapters 16 and 17 the role of molecular
vibrations in the electrical current through molecular junctions. Chapter
19 is devoted to other transport properties beyond conductance and we
discuss there, in particular, shot noise and thermal transport in molecular
conductors. The optical properties of current-currying molecular junctions
are the subject of Chapter 20. Chapter 18 deals with the electronic trans-
port in long molecules where the hopping (or incoherent) transport regime
is realized. Finally, we conclude this part in Chapter 21 with a list of topics
that have not been addressed in this monograph and we indicate where to
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

The birth of molecular electronics 17

find information about them. It is worth remarking that these chapters

have been written so that they can be read almost independently. This
way a reader can concentrate on those topics or chapters that are of special
interest for him/her.
Parts 3 and 4 are meant for both students and researchers working in
the field. We do not only review what has recently been done in the field,
but we also introduce the different topics at a elementary level. In this
sense, whenever it was possible, we have provided simple arguments and
suggested additional exercises. These two parts are intended for both exper-
imentalists and theoreticians and, most of the time, we have intentionally
avoided the typical separation between experiment and theory, which we
find particularly harmful in this field.
Let us close this chapter with some recommendations about the existent
literature. For those who want a quick overview about molecular electron-
ics, we recommend the short reviews of Refs. [2, 32–37]. A nice general
overview of the field can be found in chapter 20 of Ref. [31]. For more ex-
tensive introductions, we recommend Ref. [38] for the theory in molecular
systems and Refs. [39–41] for a discussion of the experimental techniques
used in molecular electronics. There already exist several books that deal
with different aspects of molecular electronics, see e.g. Refs. [42–49]. Most
of them consist of a collection of articles written by different authors, but
they are very useful if one wants a more detailed discussion of certain topics.
Concerning the theory of quantum transport or transport in nanoscale sys-
tems, which is one of the central subjects of this manuscript, we recommend
the monographs of Refs. [50–53].
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

18 Molecular Electronics: An Introduction to Theory and Experiment

January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Chapter 2

Fabrication of metallic atomic-size

contacts

2.1 Introduction

In this chapter we shall present the most common methods which have
been developed during the last years for the fabrication of metallic atomic-
size contacts. Both the contacting methods and the physical properties
of atomic contacts found the basis for contacting single molecules. On
the other hand, these techniques have been further refined for contacting
molecules. These refinements are now also used for studying atomic con-
tacts. Therefore, the decision in which chapter one or the other method
is described is somewhat arbitrary. Manifold variations of the techniques
exist and are permanently improved further. The aim of this chapter is to
introduce into the most important principles and to compare the techniques
regarding their advantages and drawbacks.
As important as the sample preparation is the quality of the electronic
transport measurements. When dealing with tiny contacts, care has to be
taken to reduce the influence of the measurement onto the contact itself.
We will therefore end this chapter with a few brief remarks about the most
common measurement setups and possible artifacts.

2.2 Techniques involving the scanning electron microscope

(STM)

One of the most versatile tools for the fabrication of atomic-size contacts
and atomic chains is the scanning tunneling microscope (STM) (for a re-
view, see Ref. [54]). It has been used for that purpose from the very be-
ginning of its invention [55]. While in the standard application of an STM
a fine metallic tip is held at distance from a counter electrode (in general

19
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20 Molecular Electronics: An Introduction to Theory and Experiment

motion

Fig. 2.1 Working principle of the fabrication of atomic contacts with an scanning tun-
neling microscope (STM). The electron micrograph shows a STM tip. The width at half
length is in the order of 100 to 200 µm. The lower inset gives an artist’s view of the
atomic arrangement of an atomic contact. Courtesy of C. Bacca.

a metallic surface) by making use of the exponential distance dependence

of the tunneling current, the tip can also be indented into the surface and
carefully withdrawn until an atomic size contact or short atomic wire forms.
An artist’s view of the STM geometry and the atomic configuration of a
contact is shown in Fig. 2.1. For many metals it has been shown that the
tip will be covered by several atomic layers of the metal of the counter elec-
trode upon repeated indentation such that clean contacts may be formed
consisting of the same metal for both electrodes.
The main advantages of the STM in this application are its speed and
versatility. When the electrodes forming the contacts are prepared in ultra
high vacuum conditions, the STM furthermore allows to gather information
about the topography of the two electrodes on a somewhat larger than the
atomic scale before or after the formation of the contact. Since however,
the tip is usually pressed into the substrate and the atomic-size contact is
formed when withdrawing, the exact atomic configuration of the atomic-
contact cannot be measured directly.
This problem is partially solved when the contact is formed upon ap-
proaching [56]. For good metals the distance dependence of the conduc-
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Fabrication of metallic atomic-size contacts 21

tance follows an exponential increase until a sudden “jump to contact”

occurs which is marked by a step-like increase of the conductance. The
jump indicates the formation of a chemical bond between the tip and the
electrode and thus the formation of a single-atom contact. The geometry
of the substrate side of the contact can be well controlled by first preparing
and characterizing a clean terrace of a single crystalline substrate and sub-
sequently evaporating a sub-monolayer small amount of metal atoms onto
it. The surface can then be scanned and the tip can be approached right
on top of one of the extra atoms. This technique enables to form hetero-
junctions, i.e. contacts between two different metals. The determination of
the atomic configuration on the tip-side of the contact remains unsolved,
though.
Spectroscopic measurements on the scale of electron volts allow one to
deduce information about the cleanliness and the electronic structure of the
metal [57].
The main drawbacks are its limited stability with respect to the change
of external parameters such as the temperature or magnetic fields and the
short lifetime of the contacts in general because of the sensitivity of the STM
to vibrations. In the early years of STM-based atomic-contact studies they
were furthermore limited to rather high temperatures in the range of 10 K
or higher. This drawback has been overcome in the last years. Nowadays
ultra high vacuum (UHV) STMs, which work with sufficient stability at
temperatures below 1 K and in strong magnetic fields are even commercially
available.

2.3 Methods using atomic force microscopes (AFM)

Another scanning probe technique which complements STM in many as-

pects is the atomic force microscope (AFM). Instead of the tunnel current
an AFM uses the distance dependence of the force between a fine tip and a
surface. Depending on the chemical nature of both the tip and the surface
this force consists of several contributions and its distance dependence may
be complex and even nonmonotonic. The working principle of the AFM
is based on measuring the force by recording the deflection of a cantilever
that carries the tip. The deflection can be detected by optical means or
by the detuning of an oscillator circuit due to the deflection. The AFM
has become a very versatile tool in surface science which works in various
environments and temperature ranges. In surface science the main advan-
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22 Molecular Electronics: An Introduction to Theory and Experiment

laser detector
(a) conductive (b)
cantilever

AFM cantilever
x-y motion
cantilever beam
metallic tip
metal sample

metal tip
conductive
sample

Fig. 2.2 Fabrication and characterization of atomic contacts with an atomic force mi-
croscope (AFM). (a) The conductive AFM uses a conductive cantilever and metallic
tip for recording the electrical signal. The deflection of the cantilever beam is detected
optically and used for recording the topographic information of sample. After Ref. [59].
(b) In the combined AFM-STM the sample is clamped to a cantilever. The metallic
contact is formed between the sample and the metal tip. The metal tip is part of an
STM and records the electrical signal. The deflection of the cantilever is recorded with a
separate AFM. This signal is used for measuring the force acting on the cantilever when
the atomic contact rearranges. After Ref. [58].

tage of AFM as compared to STM is its possibility to work on insulating

substrates. For the fabrication and characterization of atomic contacts the
AFM is in use in two different variations. The first one is the combination
with an STM which records the current while the AFM measures the force
that is necessary to form or break the contacts [58]. The second one is
the so-called conductive AFM which uses a metal-covered tip on a metallic
surface and both quantities, the current and the force, are available simul-
taneously, Fig. 2.2 [54]. The force signal can be used to determine the
topography.

2.4 Contacts between macroscopic wires

Transient atomic chains and contacts with lifetimes in the millisecond range
can also be fabricated in a table-top experiment first demonstrated by N.
Garcia and coworkers [60], which we call here “dangling-wire contacts”.
Two metal wires in loose contact to each other are excited to mechanical
vibrations, such that the contact opens and closes repeatedly. One end
of each wire is connected to the poles of a voltage source and the current
is recorded with a fast oscilloscope. This method is in principle particu-
larly versatile because it enables the formation of heterojunctions between
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Fabrication of metallic atomic-size contacts 23

Fig. 2.3 Experimental setup used to visualize contacts between macroscopic metallic
electrodes inside a scanning electron microscope (SEM). Adapted with permission from
[61]. Copyright 1997 by the American Physical Society.

various metals. However, in order to provide clean metallic contacts a thor-

ough cleaning of the wires would be required, similar to the tip and surface
preparation in a STM. Another drawback is the lack of control of the dis-
tance of the electrodes. It is thus mostly used as demonstration experiment
in schools with Au-Au contacts. The method has later been improved by
attaching the wires to piezo tubes. This realization thus resembles contacts
fabricated in the STM and have also been used within the chamber of an
scanning electron microscope for simultaneous imaging and conductance
measurements, see Fig. 2.3.

2.5 Transmission electron microscope

Another interesting method for preparing and imaging atomic contacts are
transient structures forming in a transmission electron microscope (TEM)
when irradiating thin metal films onto dewetting substrates [62, 63]. The
high energy impact caused by the intensive electron beam locally melts the
metal film causing the formation of constrictions which eventually shrink
down to the atomic size and finally pinch-off building a vacuum tunnel gap.
A typical system for these studies is Au on glassy carbon substrates. Several
variations of this principle have been developed that allows one to contact
both electrodes forming the contact, see Fig. 2.4. The high electron current
density necessary for imaging causes also high local temperatures resulting
in short lifetimes of these contacts. However, they offer the unique possi-
bility to simultaneously perform conductance measurements and imaging
with atomic precision. Similar results have been obtained with variations
of the STM inside a TEM [64]. This method enabled to directly prove
the existence of single-atom contacts, single-atom wide and several atom
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

24 Molecular Electronics: An Introduction to Theory and Experiment

Fig. 2.4 High resolution TEM images of short atomic wires fabricated with an STM
inside the vacuum chamber of the TEM. The arrows indicate the number of atomic rows.
In panel f the contact is broken and forms a tunnel contact. Reprinted by permission
from Macmillan Publishers Ltd: Nature [63], copyright 1998.

long chains as well as to establish a correlation between contact size and

conductance [63, 62, 65]. For Au and Ag contacts it has been shown that
preferably well ordered contacts with growing directions corresponding to
the symmetry axes of the crystal structure are formed.

2.6 Mechanically controllable break junctions (MCBJ)

Already before the development of the first STM another technique en-
abling the fabrication of atomic-size contacts and tunable tunnel contacts
has been put forward. The first realizations include the needle-anvil or
wedge-wedge point contact technique pioneered by Yanson and co-workers
(for a review see [66]) and the squeezable tunnel junction method described
by Moreland and Hansma [67] and Moreland and Ekin [68] who used metal
electrodes on two separate substrates which are then carefully adjusted with
respect to each other. The needle-anvil technique was mainly used to form
contacts with diameters of typically several nanometers and thus having
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Fabrication of metallic atomic-size contacts 25

L
counter supports

metal
wire
u

pushing rod dx elastic substrate

sacrificial layer

Fig. 2.5 Working principle of the MCBJ (not to scale) with the metal wire, the elastic
substrate, the insulating sacrificial layer, the pushing rod, the counter supports and the
dimensions used for calculating the reduction ratio (see text).

hundreds or thousands of atoms in the narrowest cross section. These two

techniques formed the starting point for the development of the mechani-
cally controllable break junctions (MCBJ) by C. Muller and coworkers [69],
which nowadays is applied for the fabrication of atomic contacts in vari-
ous subforms, the most common of which are the so-called notched-wire
[70] and thin-film MCBJs [71]. The working principle which is depicted
in Fig. 2.5 is the same for both variations: A suspended metallic bridge is
fixed on a flexible substrate, which itself is mounted in a three-point bend-
ing mechanism consisting of a pushing rod and two counter-supports. The
position of the pushing rod relative to the counter supports is controlled
by a motor or piezo drive or combinations of both. The electrodes on top
of the substrate are elongated by increasing the bending of the substrate.
The elongation can be reduced again by pulling back the pushing rod and
thus reducing the curvature of the substrate. In order to break a junction
to the tunneling regime, considerable displacements of the pushing rod and
thus important bending of the substrate is required. Therefore the most
common substrates are metals with a relatively high elastic limit like spring
steel or bronze. The substrates are covered by an electrically isolating ma-
terial such as polyimide before the junction can be fixed on it.
The notched-wire MCBJ, an example of which is shown in Fig. 2.6,
uses a thin metallic wire (diameter 50 µm to 200 µm) with a short, knife-
cut constriction to a diameter of 20 µm to 50 µm. The wire is glued at
both sides of the notch to the substrate and connected electrically to the
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26 Molecular Electronics: An Introduction to Theory and Experiment

Fig. 2.6 The 100 nm wide gold wire is glued with epoxy resin (black) onto the substrate.
The electrical contact is made by thin copper wires glued with silver paint. The inset
shows a zoom into the notch region between the two black drops of epoxy resin. Reprinted
from [15]. Copyright 2003, with permission from Elsevier.

measurement circuit at both ends. The distance between the glue drops is
of the order of 50 µm to 200 µm.
Variations of this method have been put forward which enable contact-
ing of reactive or brittle materials out of which no wires can be formed [72].
For this purpose the sample preparation is performed in protective environ-
ment. A beam-shaped piece of the material is cut in a non-reactive liquid
such as dodecanol or other slowly evaporating alcohols, or glycerine. Four
holes are drilled into the metal and a wedge is cut in the middle between
the holes. An example is shown in Fig. 2.7. The beam is screwed with the
help of two electrically isolating bolts to the substrate, one on each side of
the wedge. The remaining two holes serve for screwing metallic wires to
the beam for the conductance measurements.
For a version which enables scanning the two electrodes with respect to
each other, at first two piezo tubes are glued to the substrate. The metal
wire is then glued on top of the piezos. After mechanically breaking the
wire, the piezos are polarized such that they are bent and the two parts
of the wire are sliding along each other [73]. This realization corresponds
to a high-stability STM, but with very restricted scan possibility. It is
therefore used only sparsely. Finally, simultaneous force and conductance
measurements are possible when adding a tuning fork like in AFMs. Details
of this very sophisticated method are given in Ref. [75].
Fig. 2.8 shows two examples of thin-film MCBJs, which were fabricated
using the usual techniques of nanofabrication, i.e. electron beam lithography
and metal deposition by evaporation. There are mainly two differences to
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Fabrication of metallic atomic-size contacts 27

Fig. 2.7 Principle of the MCBJ technique adapted for reactive metals. Reprinted from
[15]. Copyright 2003, with permission from Elsevier.

(a) (b)

2 µm
100 nm
Fig. 2.8 Lithographic MCBJ. (a) Electron micrograph of a thin-film MCBJ made of
cobalt on polyimide taken under an inclination angle of 60o with respect to the normal.
The distance between the rectangular shaped electrodes is 2 µm, the thickness of the thin
film is 100 nm and the width of the constriction at its narrowest part is approximately
100 nm. (b) Electron micrograph of a thin-film MCBJ made of cobalt (medium grey) with
leads made of gold (light grey) taken under an inclination angle of 50o with respect to the
normal. The distance between the rectangular shaped electrodes is 2 µm, the thickness
of the Co film is 80 nm, of the Au film is 100 nm and the width of the constriction at its
narrowest part is approximately 100 nm. The sample has been fabricated using shadow
evaporation through a suspended mask such that two images of the mask exist. The Au
shadow of the bridge is broken off.

standard nanostructuring. The first one is the substrate, which in case

of MCBJs has to provide sufficient elastic flexibility without breaking or
irreversible bending. The second difference is the final etching step which
is needed to suspend the nanobridge (with typical dimensions of 2 µm in
length and 100 nm × 100 nm at the narrowest part of the constriction)
above the substrate by partial removal of a sacrificial layer underneath the
metal film. Fig. 2.9 summarizes the fabrication procedure. A piece of metal
with a typical thickness of a few hundred micrometers serves as substrate.
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28 Molecular Electronics: An Introduction to Theory and Experiment

The metal should have a high elastic deformation limit. Typical metals
are bronze or spring steel. For particular purposes, in particular when
capacitive effects have to be minimized, the metal is replaced by a plastic
substrate. Both metal or plastic are thoroughly polished to reduce the
roughness to less than a micrometer. The remaining corrugations are then
filled with a thin layer of polyimide (thickness 1-2 µm), which is spin-coated
and hardbaked in vacuum. The polyimide also serves as electrical insulator
between the nanostructure and the substrate. Subsequently the electron
resist is spin-coated and thermally treated as required for electron beam
structuring. Fig. 2.9(c) shows an example in which a double-layer resist
is used. The double-layer is necessary for, e.g. evaporation of the metal
under arbitrary angle. The next step is electron-beam writing in a scanning
electron microscope equipped with a pattern generator or in a commercial
electron-beam writer. After development of the resist in a selective solvent
the resist mask remains on top of the polyimide layer. The mask itself may
be partially suspended when using a double-layer resist. Subsequently the
metal will be deposited either by evaporation, sputtering, chemical vapor
deposition or other means. Shadow evaporation, i.e. evaporation of several
materials under different angles can be used for forming contacts between
different metals or for supplying nanobridges of one metal with electrodes
made of another metal. The advantage of the shadow-evaporation technique
lies at first in its self-alignment property because the same mask is used
for all metal depositions. The second advantage is given by the fact that
all depositions can be made in a single vacuum step, which enables one to
fabricate clean interfaces between the metals. After the metal deposition
the mask is stripped in a more aggressive solvent. Finally the structure is
exposed to an isotropic oxygen plasma which attacks the polyimide layer.
Consequently its thickness is reduced and all narrow metal parts, like the
nanobridge become suspended like a bridge.
Both versions of the technique - the notched-wire MCBJs and the litho-
graphic MCBJs - share the idea of enhanced stability due to the formation
of the contact by breaking the very same piece of metal on a single sub-
strate and by transformation of the motion of the actuator into a much
reduced motion of the electrodes perpendicular to it. The small dimensions
of the freestanding bridge-arms give rise to high mechanical eigenfrequen-
cies, much higher than the ones of the setup. As a result the system is less
sensitive to mechanical perturbations by vibrations.
Assuming homogeneous beam-bending of the substrate we can calculate
the reduction ratio r between the length change of the bridge u and the
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Fabrication of metallic atomic-size contacts 29

(a) (b)

(c) (d)

(e) (f)

(g) (h)

Fig. 2.9 Fabrication scheme of thin-film (lithographic) MCBJ. (a) The substrate (metal,
plastic) is polished mechanically. (b) the sacrificial wafer (polyimide) is spin-coated and
baked. (c) The resin (typically a bi-layer electron sensitive organic material) is spin-
coated and baked. (d) The resin is exposed in an electron beam writer or a scanning
electron microscope equipped with a pattern generator in the desired pattern. (e) The
chip is developed in a solvent which selectively removes the exposed parts of the resin.
The result is a mask, which resides on the sacrificial layer, in the shape of the exposed
pattern. (f) The metal is deposited by evaporation or sputtering. (g) The mask with the
metal on top of it is lifted-off in a more aggressive solvent which attacks the unexposed
parts of the resin. The result is a metal layer in the shape of exposed pattern. (h) Finally
the thickness of the sacrificial layer is reduced in an isotropic plasma. The narrow parts
of the metal pattern are suspended and form the bridge which will be broken in the
MCBJ mechanism.

motion of the pushing rod x (see Fig. 2.5).

6tu
r=, (2.1)
L2
where t is the thickness of the substrate, u the length of the free-standing
bridge arms and L the distance of the counter supports. This quantity
denotes the factor with which any motion of the pushing rod is reduced
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30 Molecular Electronics: An Introduction to Theory and Experiment

when it is transferred to the point contact. In a real MCBJ setup, however,

the beam-bending is in general non-uniform. Furthermore, also the sacrifi-
cial layer has a finite elasticity and is deformed when bending the MCBJ.
These effects can be accounted for by a correction factor, which enhances
r by a factor of roughly 4 [76]. The effective reduction ratio has a typical
value of 10−3 to 10−2 for the notched-wire MCBJs and 10−6 to 10−4 for the
thin-film MCBJs. The relatively weak reduction ratio of the notched-wire
MCBJs usually requires the use of a piezo drive for controlling and stabiliz-
ing single-atom contacts, while the lithographic MCBJs can be controlled
with purely mechanical drives, i.e. a dc-motor with a combination of gear
boxes, and a differential screw.
A common realization of a bending mechanism suitable for thin-film
MCBJs and use at low temperatures T < 1 K is shown in Fig. 2.10. A
rotary axis is connected to a differential screw which consists of a thread,
the two sections of which have a slightly different pitch. The typical values
for the pitches A and B are 0.7 to 0.8 mm and pitch differences 50 µm to
150 µm. Each full turn of the axis changes the distance between the sample
holder and the ground plate by the difference of the pitches. The shape
of the end of the pushing rod can be semi-cylindrical or wedge shaped,
depending on the desired deformation of the substrate. Because of the off-
line axis arrangement of rotary axis and pushing rod several guiding rods
are needed to reduce torque and ensure linear motion of the sample holder
with respect to the ground plate. The pushing rod can be designed such
that it hosts a piezo tube. The MCBJ is electrically contacted via spring
contacts or by gluing the wiring to it via silver paint. The thermal contact
of the sample to the thermal bath can additionally be provided by thick
wires and copper braid. Care has to be taken when choosing the materials
combination of the thread and its counterpart to avoid friction because
lubrification at low temperature and in vacuum is difficult.
Typical motion speeds of the piezo drive lie between 10 nm/s and
10 µm/s corresponding to results in 10 pm/s to 100 nm/s for the electrodes
forming the atomic contacts. For purely mechanical drive these values are
10 nm/s to 1 µm/s for the pushing rod and 10 fm/s to 10 nm/s for the
contact. Due to the in-built reduction also the piezo-driven setups are in
general slower than STM systems. The high stability enables comprehen-
sive studies on the very same atomic contact at various values of control
parameters such as fields and temperature.
On the other hand the small r values require considerable absolute mo-
tion of the pushing rod and deformation of the substrate in order to achieve
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Fabrication of metallic atomic-size contacts 31

to
to rotary axis
thermal bath

thread section
with pitch A
guiding wedge-ended
rod pushing rod
sample thread
t section
holder with pitch B
sample

Fig. 2.10 Sample holder with differential screw for thin-film MCBJ. A motor drives a
rotary axis which ends in a thread with two different pitches. Rotating the axis results
in varying distance between ground plate and sample holder. The sample resides on two
counter supports connected to the sample holder. It is bent by the pushing rod which
is attached to the ground plate. Three guiding rods (only one of which is shown) ensure
smooth and linear motion.

sufficient displacements of the electrodes. This reduces the possible choices

of the substrate material considerably.
MCBJ mechanisms have been developed for various environments in-
cluding ambient conditions, vacuum, very low temperatures [77] or liquid
solutions [78]. The latter one is of particular interest for the study of single-
molecule junctions and will be explained in detail in the following Chapter
3. The disadvantages of MCBJs as compared to STM techniques are the
small speed and the fact that the surrounding area of the contact cannot
easily be scanned. As for STM setups clean contacts can only be guaran-
teed when working in good vacuum conditions. The sample preparation
itself, however, does not require clean conditions because the atomic con-
tacts are only formed during the measurement by breaking the bulk of the
electrodes.

2.7 Electromigration technique

A third method for the formation of atomic-size contacts is controlled burn-

ing of a wire by electromigration (see Fig. 2.11). This technique has been
optimized for the formation of nanometer sized gaps for trapping individual
molecules or other nanoobjects [79, 80]. Before the wire finally fails and
the current drops drastically, atomic size contacts are formed for a rather
short time span [81–83]. During the electromigration process the trans-
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32 Molecular Electronics: An Introduction to Theory and Experiment

500 nm 500 nm

200 nm 200 nm

Fig. 2.11 Electromigration technique. Top: Fabrication via shadow evaporation

through a suspended mask of an electrode structure to be used for producing atomic
contacts by electromigration. The arrows indicate the directions from which the metal is
deposited. The electromigration will nucleate at the thinnest part of the electrode struc-
ture. Bottom: Series of atomic force microscope images taken in the tapping mode of
an electromigrated contact made of Au on Si in different phases of the electromigration
process. From left to right: before electromigration (R = 40 Ω), R = 105 Ω, R = 630 Ω,
R = 30.000 Ω. Courtesy of D. Stöffler and R. Hoffmann.

port changes from ohmic behavior, i.e. limited by scattering events of the
electrons to wave-like electronic transport, which can be described by the
Landauer picture (see Chapter 4).

The term electromigration denotes a process in which ions are moved

due to high electrical current densities. We concentrate here on the electro-
migration behavior of metals. It has been understood that several effects
contribute to the total force acting on a metal atom which forms the con-
ductor, the two most important being the so-called direct force due to the
electric field. It causes the electrical current and thus points into the di-
rection of the field. The second one is caused by momentum transfer of
the conduction electrons onto the ions. It has opposite sign and is called
the wind force. When the total force overcomes the binding force of the
ions, they start to diffuse but can be pinned again at defects or positions
where the current density and driving force falls below this threshold value.
Depending on the material, the temperature, the crystallinity, the surface
roughness, and many other parameters either the direct force may exceed
the wind force or vice versa [84]. Therefore the exact direction of the mate-
rial transport depends on the microscopic structure of the wires. In many
cases the motion of the material is such that the cross section of the con-
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Fabrication of metallic atomic-size contacts 33

ductor is locally reduced and its electrical resistance increases. The higher
resistance causes higher losses, enhanced dissipation, increasing tempera-
ture in the wire which further enhances the dissipation of ions. An impor-
tant role plays the temperature of the lattice because the diffusion and the
threshold current strongly depend on temperature. Electromigration has
become one of the most important origins of failures in integrated circuits,
due to the miniaturization of the metallic interconnects without reducing
the current by the same factor. Consequently, electromigration has widely
been studied in electrical engineering with the aim to achieve the highest
possible threshold current density for it to set in and the smallest diffusion
speed [85].
For the formation of atomic contacts a high threshold current is not
important but the possibility for controlling speed, shape and size of the
final structure. One of the most important preconditions is to define the
position at which the electromigration starts, and the contact forms. For
this purpose a short and thin metallic wire is fabricated by lithographic
methods as described in the previous section. Typical dimensions are a
length and width of 50 to 100 nm and a thickness of 10 to 20 nm. The
thin wire is connected to wider and thicker electrodes which consequently
have smaller resistivity. A convenient method to fabricate these structures
is shadow evaporation through a suspended mask as shown in Fig. 2.11.
First, thin layers of the metal (typical thickness 10 nm) are evaporated
under the angles Θ and −Θ. The angle is chosen such that both layers
slightly overlap underneath the suspended part of the mask. Afterwards a
thick layer of the electrode metal is deposited perpendicular to the substrate
plane. The ideal structure would consist of a single-crystalline wire in the
thin part of the wire, the boundaries of which are covered by the thick
electrodes in order to avoid electromigration of possible contaminants from
the grain boundaries. It is advantageous to work on a substrate with high
thermal conductivity in order to control the temperature.
The electromigration process itself is performed such that an electrical
current is continuously ramped up while the resistivity is monitored. As
soon as the resistance starts to increase a computer-controlled feedback
loop controls the current such that the rate of the resistance increase is
kept constant or slowed down. The resistance increase is partially due to
the temperature increase caused by the Joule heating of the driving current.
Although it has been shown that in the ohmic regime the current density is
the quantity which determines the diffusion of the ions, it is advantageous
to control the voltage in order to produce atomic size contacts. When the
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34 Molecular Electronics: An Introduction to Theory and Experiment

resistance increases the current becomes smaller, which helps to limit the
migration speed. The low-resistive electrodes ensure that the voltage drops
locally making the driving force acting only locally as well. Consequently,
the dissipation and Joule heat generation are local as well. The procedure
should be stopped when the desired resistance is achieved. For the study
of atomic contacts the interesting regime is reached when the resistance
exceeds roughly one kiloohm. For usual metals this corresponds to contacts
with a narrowest cross section of roughly 10 atoms. An important finding
is that the behavior changes markedly when the size of the smallest cross
section corresponds to a few atoms. However, the exact position of the
position at which the wire finally breaks is difficult to predict. As will be
explained in Chapter 11 the electrical transport of contacts of this size is
determined by the wave properties of the electrons rather than by collisions
with defects. If this happens the resistance may start to decrease again
before the wire finally is burned through. This non-monotonous behavior
complicates the control scheme further. Several control schemes have been
put forward which are optimized for various sample geometries, metals and
working conditions such as vacuum or low temperature [21, 81–83, 86]. So
far only a few studies exist in which the electromigration process has been
imaged in detail, although these kind of studies are very insightful. One
example is shown in Fig. 2.11, where AFM images have been taken after
discrete electromigration steps. A particularly nice series of TEM images
showing that the most dramatic shape changes occur during the final phase
can be found in Ref. [82].
An important difference to STM techniques and MCBJs is the fact that
the wire forming the contact is in solid contact with a substrate. The ad-
vantages are at first ultimate stability which will become important when
studying atomic or molecular junctions as a function of external fields (see
Chapters 12 and 20). The second advantage lies in the fact that no par-
ticular requirements exist for the properties of the substrate, besides the
fact that it should be sufficiently insulating. Often silicon - the standard
substrate in microelectronics - is used. With suitable doping it can be used
as back-gate for inducing an electric potential and building a three-terminal
device. This technique is important for studying effects like Coulomb block-
ade, which will be explained in Chapters 11 and 15.
The main drawback of the electromigration technique is the fact that it
is a single-shot experiment: Once an atomic contact has been established
there is only limited possibility to fine tune its atomic configuration, in par-
ticular coming back to a larger contact is almost impossible. After burning
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Fabrication of metallic atomic-size contacts 35

(a) (b)

Fig. 2.12 Electromigrated MCBJ with gate on silicon substrate. (a) Working principle
and (b) electron micrograph of an electromigrated MCBJ. The substrate is doped silicon
and can be used as back-gate. Reprinted with permission from [86]. Copyright 2005
American Chemical Society.

through the wire it cannot be closed again. As described before, the con-
trol of the final part of the electromigration process is tricky because the
character of the transport changes from ohmic to wave-like. A combina-
tion of electromigration with the lithographic MCBJ technique overcomes
this problem: a thin-film MCBJ is thinned-out by electromigration to a
narrow constriction with a cross section of less than 10 nm (see Fig. 2.12).
The substrate is then bent carefully for completely breaking the wire or
arranging single-atom contacts. This last step is reversible and repeatable
for studying small contacts [87] or trapped nanoobjects [86]. Because only
the very last part of the breaking requires mechanical deformation of the
substrate it is rather fast and enables the use of more brittle substrates
such as silicon.

2.8 Electrochemical methods

A completely distinct method for the formation of atomic-size contacts uses

electrochemical deposition and removal of metal atoms. Electrochemical
deposition of metals is a standard technique for surface treatment and in
micromachining. For the purpose of forming atomic contacts basically the
same principles are used. The main difference to the macroscopic techniques
is the shape of the starting electrodes and the feedback which controls
the deposition speed. Nanocontact formation by electrochemical methods
starts from metal electrodes with a gap or with a continuous wire that is
first broken either mechanically or by electromigration. The working prin-
ciple is depicted in Fig. 2.13. The electrode structure is then immersed into
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36 Molecular Electronics: An Introduction to Theory and Experiment

an electrolyte containing metal ions. The electrochemical setup is adapted

from the three-electrode cyclic voltammetry principle [88]. The deposition
and dissolution of metal is controlled by applying an electrical potential
difference between a so-called counter electrode and the electrodes forming
the nanocontact, which serve as “working electrodes”. A fourth electrode
defines the reference potential. The conductance is monitored and used as
control signal for the potentiostat which controls the deposition rate. The
typical control voltages are in the range of 20 mV to 1 V and can be ad-
justed to optimize the electrochemical process. It should exceed the bias
voltage if one aims at symmetric deposition on both electrodes forming the
contact. Obviously the place at which the fastest deposition and dissolu-
tion takes place can further be controlled by the size and the polarity of
the bias voltage. A typical metal combination is gold as electrode material
because of its weak chemical reactivity and silver for the formation of the
atomic contacts [89, 90]. Silver is easily dissolved in acids, like e.g. in ni-
tric acid, and simultaneously silver atomic contacts have well understood
transport properties, as will be further detailed in Chapter 11. One main
advantage of this technique is its versatility, since electrochemical deposi-
tion methods on the macroscale have been developed for almost all metals.
A further advantage is the simplicity of the working principle, in particular
the simplicity with which the starting electrodes can be produced: macro-
scopic wires as well as deposited thin films [91, 92] or STM setups [93]
are possible. Furthermore the contacts are mechanically stable because no
suspended parts are required.
Electrochemical contacts are often regarded to be three-terminal de-
vices: The two electrodes forming the contact correspond to source and
drain, the control electrode to the gate electrode in the language of semi-
conductor transistors. Since the electrochemical control involves diffusion
of ions, it is slower than the usual electrostatic gating in semiconductor
technology. It is however much faster than the purely mechanical control
used in the lithographic MCBJ technique. One obvious drawback is the
fact that the control mechanism requires liquid environment. It is not ob-
vious how one can bring the contacts into dry environment, vacuum or low
temperatures. Anyhow, after removal of the electrochemical environment
the contacts cannot be varied anymore (or one of the other techniques, e.g.
MCBJ or electromigration, have to be applied for this purpose).
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Fabrication of metallic atomic-size contacts 37

Fig. 2.13 Setup for the electrochemical fabrication and control of atomic contacts. For
particular choices of the control potential the atomic contact can be switched between
defined conductance values and thus a ”switching current” is recorded. Reprinted with
permission from [89]. Copyright 2003 by the American Physical Society.

2.9 Recent developments

As mentioned in the introduction of this chapter, many variations of the

standard methods described above have been developed. In particular,
combinations of the archetypical methods have been described. As an ex-
ample we present here two new versions of the MCBJ technique. The first
one has been introduced by Waitz et al. [94]. It uses thin-film-wires on
silicon membranes with a thickness of a few hundred nanometers. The
membrane is deformed by a fine tip on the rear side. At variance to the
MCBJ techniques on bulk substrates the elasticity of the membrane rather
than the bending determines the stretching of the metal wire, see Fig. 2.14.
The deformation of the substrate is applied locally and it is thus possible
to address particular positions while the rest of the circuit on the substrate
remains mainly unaffected. This is important when the MCBJ is embed-
ded in a more complex electronic circuit close to the atomic contact, which
should not be affected when changing the atomic contact. Such complex
circuits are required e.g. for studying Coulomb blockade, which we will de-
scribe in Chapter 15. Another advantage of this method as compared to
bulk substrates is that the membranes are electrically insulating or only
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38 Molecular Electronics: An Introduction to Theory and Experiment

poorly conducting. This reduces the capacitance of the circuit to ground

and is advantageous when fast measurements are required. A further differ-
ence to standard MCBJ techniques is that smaller suspended length of the
metal wire can be used. This enhances stability and reduces often unde-
sired effects such as magnetostriction when investigating magnetotransport
as explained in Chapter 12. Finally, by combining this membrane MCBJs
with electromigration it is possible to control atomic-size contacts at room
temperature without suspension at all [95].
The second recent improvement, which we want to describe here, is the
successful incorporation of a gate electrode into the lithographic MCBJ
techniques without combination with electromigration [96]. It is based on
the lithographic MCBJ technique on metallic substrates using two lithog-
raphy steps. In the first step a thin and rather narrow metallic gate strip is
patterned. The gate is then covered by an approximately 50 nm thick insu-
lating sacrificial layer and the resist system for the second lithography step
in which the nanobridge is patterned. After evaporation of the nanobridge
metal the sacrificial layer removed by dry etching as in the conventional pro-
cess for lithographic MCBJs. The result is shown in Fig. 2.15: a nanobridge
that is suspended approximately 50 nm above the gate electrode. With this
technique three-terminal devices with controllable source-drain coupling are
now possible.

2.10 Electronic transport measurements

Usually the first electrical characterization of nanoscale contacts is the mea-

surement of the linear conductance as a function of an outer parameter
such as temperature, magnetic field or size of the junction. The next more
complex quantity is the nonlinear conductance, i.e. measurements of the
current-voltage (I-V) characteristics or the differential conductance. Since
these quantities belong to the most common properties of any material
characterization their correct measurement is supposed to be trivial, and
manifold sophisticated equipment is on the market. In fact, several suppli-
ers of electronic measurement units offer information material or seminars
about low-level, high-resolution electronic measurements, and we encourage
our readers to access this literature. Therefore textbooks about nanoscience
only rarely address this issue. However, when dealing with nanoobjects it is
not easy how to perform a good conductance measurement. In this section
we will not give a complete overview over the various techniques. But since
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Fabrication of metallic atomic-size contacts 39

Fig. 2.14 MCBJ on silicon membranes. Top: Working principle of the membrane MCBJ
(not to scale). One or several lithographic MCBJs are defined on the front side of the
membrane. A glass or graphite tip is scanned along the rear side of the membrane with
the help of micromechanically controlled scan tables. The vertical motion of the tip
controls the deformation of the membrane. The close ups at the right side illustrate the
deformation of the membrane with a graphite tip, the rupture of the nanobridge, and
give an artist’s view of the atomic arrangement of a single-atom contact. The thickness
of the membrane is in the order of 300 nm, the lateral dimension of the membrane is
typically 1 mm × 1 mm. The length of the suspended bridge is smaller than the one for
lithographic MCBJs on massive substrates. The thickness of the sacrificial layer is in the
order of 100 nm only. When reducing the lateral size of the constriction first by electro-
migration, non-suspended metal bridges can be used. Bottom: optical micrograph of a
membrane carrying two MCBJs made of gold. The tip is positioned underneath the lower
bridge where the membrane is deformed. The size of the membrane is 0.6 mm × 0.6 mm.

the scope of this book is to serve as textbook for beginners in the field of
molecular electronics, we want to sensitize the reader to this issue. The par-
ticular facts which have to be taken into account in molecular conductance
measurements are the following:
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40 Molecular Electronics: An Introduction to Theory and Experiment

a
G

S D

500 nm
b
gold
electrodes counter
polyimide supports

phosphor
bronze pushing rod

Isd
100 MΩ
Vg Vb

Fig. 2.15 MCBJ with gate electrode on bulk substrate. (a) Scanning electron micro-
graph of a lithographic MCBJ with gate electrode, (b) working principle of the MCBJ,
(c) and electronic circuit for the gated MCBJ. Reprinted with permission from [96].
Copyright 2009 American Chemical Society.

• Wide range of conductances from nanosiemens (corresponding to

10−5 G0 (G0 = 2e2 /h is the conductance quantum with e the ele-
mentary charge and h Planck’s constant) to siemens.1
• Correct choice of bias voltage to assure working in the linear regime
is difficult because the effects giving rise to nonlinearities happen
on varying voltage scales ranging from microvolt to volt.
• Self-heating of the contacts due to Joule dissipation is not always
easy to detect and to discriminate from the intrinsic properties of
the sample.
• Sudden voltage spikes and jumps may destroy the sample. There-
fore abrupt switching actions in the electrical measurement circuit
have to be minimized, often hampering optimum range adjustment.
• Extreme variation of the differential conductance within small
changes of the bias.
• Limited lifetime of the junctions to study.

The typical signal sizes which have to be resolved are of the order of
a few nanovolts for the voltage and picoamperes for the current. For par-
11 Siemens is the inverse of 1 Ω = 1 Volt/Ampere and thus the unit of the conductance
in the international system of units (SI).
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Fabrication of metallic atomic-size contacts 41

ticular experiments the requirements might even be stronger. The relative

measurement accuracy which is required for most investigations is 10−4 or
better corresponding to a resolution of typically 14 bits when expressed in
digital units. These requirements mean that one often works at the resolu-
tion limit of commercial electronic equipment. When enhancing the size of
the excitation signal to obtain response signals well above the noise floor
one risks to at least smear out the electronic characteristics of the sample
by warming it up. In the worst case the sample is destroyed by the heat
dissipation.
When designing a measurement circuit the first choice that one has to
take is whether one feeds the current and measures the voltage or vice versa.
For measurements of the linear conductance, or when the I-Vs are mainly
linear, the most important criterion is to optimize the signal-to-noise-ratio.
The general rule is that measuring voltage is the better solution for small
conductances whereas measuring current is good for high conductance val-
ues. When, however, a well-defined energy difference between source and
drain is required, e.g. for investigating Coulomb blockade2 , a voltage bias is
obviously the best choice. For other purposes the transport current is the
decisive quantity and has to be defined. When dealing with hysteretic I-Vs
or junctions revealing negative differential resistance (NDR) (see section
13.7) the measurement strategy is crucial for reaching all interesting parts
of the I-Vs. Similar choices have to be made concerning the position of the
electric ground level of the circuit and whether one pole of the sample will
be directly connected to it.
Small nonlinearities in the I-Vs may easily disappear in the noise floor
of the electronic circuit. They are much easier to detect with a low-noise
lock-in amplifier working at a small but finite frequency. When the electric
circuit under study is biased with a harmonic voltage signal, the lock-
in detector measures directly the first derivative of the I-V when locking
it on the bias frequency. The second derivative (which is an important
quantity for detecting vibrational excitations (see chapter 16) can then be
determined by numerical differentiation of the dI/dV . Alternatively it can
be directly measured when recording the response at twice the excitation
frequency.3
In any case the energy scale given by the excitation voltage has to be
kept smaller than the width of the vibrational resonances under study.
2 Coulomb blockade and related effects shall be explained in Chapter refchap-transistors.
3 Practically all companies producing lock-in amplifiers offer tutorial material available
online.
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42 Molecular Electronics: An Introduction to Theory and Experiment

Furthermore the excitation energy has to be smaller than the temperature,

otherwise the spectra will be smeared out.
Abrupt changes of the conductance as a function of the bias or another
parameter, e.g. the conformation of the junction, result in abrupt changes of
the dissipated power as well. On the one hand this is a difficult task for the
measurement electronics to cope with. On the other hand this forces one
to take precautions, i.e. introduce measures for current limitation, which
themselves hamper a perfect voltage bias.
The limited lifetime of the junctions forces one to perform fast measure-
ments, a fact resulting in limited signal to noise ratios and limited statistical
information. Atomic and molecular junctions at room temperature reveal
intrinsic noise caused by atomic motion. Therefore low-temperature exper-
iments are very appealing. In standard cryostats the wires are rather long
and thermalization requires higher cable resistances. Additional measures
for high-frequency filtering are required. All these facts reduce the band-
width of the measurement circuit. As a result it is not trivial to perform
fast measurements at low temperatures.
As will be explained in Chapters 13 and 19, many important properties
of quantum transport cannot be revealed from conductance measurements
alone, but more complex transport properties such as shot noise or ther-
moelectric voltage have to be studied.
Obviously, for a meaningful noise measurement one has to discriminate
the shot noise signal from the undesired but unavoidable noise of the mea-
surement circuit. A fruitful method to do so is a correlation measurement
using two identical sets of cables [97, 98]. All noise signals which originate
from the wiring are uncorrelated to each other. Signals from the sample are
fed into both wires. They are correlated and are recorded in a spectrum
analyzer. Only those parts are processed further. An example of such a
wiring is shown in Fig. 2.16. It is particularly demanding to measure shot
noise at high frequency. A successful solution based on coupled quantum
dots has been reported in Ref. [99] and a version using superconducting
tunnel contacts in Ref. [100].
For measuring the thermopower a small voltage signal has to be detected
which is created by a small temperature gradient across the sample. This
means that this temperature difference has to be applied and detected with
high precision. One example where this has been successfully achieved
is given in Fig. 2.17. It is designed for detecting the conductance and
the thermopower of molecular junctions at room temperature [101, 102].
Another setup used for measuring the thermopower in atomic contacts at
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Fabrication of metallic atomic-size contacts 43

Fig. 2.16 Schematic experimental setup for measuring the voltage dependence of the
shot noise of an atomic contact. An atomic contact (double triangle symbol), of dynamic
resistance RD , is current biased through a resistance RB . The voltage V across the
contact is measured by two low noise preamplifiers through two nominally identical
lossy lines with total resistance RL in each line and the total capacitance C introduced
by the setup across the contact. The spectrum analyzer measures the cross-correlation
spectrum of the two voltage lines. The Si (i = B, Amp1 ,Amp2 ) are the known current
noise sources associated with the bias resistor and the two amplifiers. SI represents the
signal of interest, i.e. the shot noise associated with the current through the contact. Sv1
and Sv2 represent the voltage noise sources of each line (amplifier 1 connecting leads).
Reprinted with permission from [98]. Copyright 2001 by the American Physical Society.

low temperature is presented shown in Fig. 19.7 and explained there.

With these examples we will finish our short and incomplete list of
electronic measurement setups. Our aim was to make clear that although
the fabrication of atomic and molecular junctions is not simple, the correct
measurement of their electronic transport properties might be even more
demanding.

2.11 Exercises

2.1 Vacuum: Estimate the number of gas atoms per area impinging on a surface
at normal pressure, in high vacuum (p = 10−6 mbar), and in ultra high vacuum
(p = 10−10 mbar) during one minute. Let us assume that all incoming gas atoms
stick to the surface. How thick is the gas layer after 10 minutes?
2.2 Nanowires and atomic contacts: Let us consider a cylindrical nanowire
made of Au. Au has a lattice constant of a = 0.41 nm.
(a) Estimate the number of atoms in the cross section for a wire with diameter
10 nm, 5 nm, and 1 nm.
(b) Estimate the number of surface atoms for these wires with a length of 5
nm.
(c) Calculate the ratio between surface atoms and bulk atoms in these
nanowires.
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44 Molecular Electronics: An Introduction to Theory and Experiment

Fig. 2.17 Schematic description of the experimental setup for measuring thermoelec-
tric voltage based on an STM break junction. Individual molecules (symbolized by a
hexagon) are trapped between the Au STM tip kept at ambient temperature and a heated
Au substrate kept at temperature ∆T above the ambient. When the tip approaches the
substrate, a voltage bias is applied and the current is monitored to estimate the conduc-
tance. When the conductance reaches a threshold of 0.1 G0 , the voltage bias and the
current amplifier are disconnected. A voltage amplifier is then used to measure the in-
duced thermoelectric voltage, while the tip is gradually pulled away from the substrate.
Reprinted with permission from [102]. Copyright 2008 American Chemical Society.

2.3 Mechanically controllable break junctions: Let us consider a MCBJ

setup with a separation of the counter supports of L = 10 mm, a substrate
thickness of t = 0.5 mm and a suspended length of u = 2 µm. For simplicity
let us neglect the insulating sacrificial layer between substrate and metal wire.
Calculate the required displacement of the pushing rod for elongating the junction
by 10 nm assuming homogeneous bending, when the MCBJ is installed into a
differential screw with a pitch difference of 100 µm.
2.4 Joule heating: (a) Calculate the power dissipated in an atomic contact
(initially at room temperature) with a resistance of 10 kΩ when a voltage of 10
mV is applied.
(b) Assume that the dissipated power heats up a spherical volume containing
1000 atoms of a material with a specific heat of 130 J/(kg·K). Assume that the
sample is only possible to dissipate energy into the environment by radiation.
What is the temperature increase?
(c) Perform the same estimation when the sample is surrounded by a material
with heat conductivity of 300 W/(K·m).
(d) Repeat the set of estimations for a molecular contact with a resistance of
10 MΩ.
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Chapter 3

Contacting single molecules:

Experimental techniques

3.1 Introduction

In this chapter we shall present the most common methods for contacting
molecules. Although we are mainly interested in single molecule devices, we
shall also introduce the most basic methods which are in use for contacting
molecular ensembles, since many interesting effects in molecular electronics
have first been observed in devices containing these assemblies. Of course,
this list can never be complete because new methods and variations of
existing ones are constantly being developed. Let us remark that we shall
focus here on methods to contact molecules with metal electrodes. Devices
including at least one semiconductor electrode have also been realized and
examples will be briefly described in section 13.7. Finally, as in the previous
chapter, we shall compare the performance of the various techniques and
indicate their most common applications.
In the fabrication of molecular junctions not only the kind of the elec-
trodes used is crucial, but also the deposition method of the molecules.
Thus, any report about electric current through molecular junctions has
to address the “protocol”, i.e. the precise contacting scheme including the
way how, the moment when, and the conditions under which the molecules
are brought into electric contact with the electrodes. For this reason, we
shall introduce in this chapter the most common deposition methods, then
we shall turn to single-molecule contacting schemes and we shall end by
addressing the ensemble techniques.
Particularly interesting are techniques which enable the fabrication of
three-terminal devices. In these systems, two of the terminals serve to inject
the current and measure the voltage, while the third one acts as a gate that
controls the electrostatic potential in the molecule. The incorporation of

45
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46 Molecular Electronics: An Introduction to Theory and Experiment

this third electrode is crucial for revealing the transport mechanism and it
allows us to tune the current through a molecular junction, very much like
in the transistors fabricated with the standard semiconductor technology.
For the sake of completeness, the first part of this chapter will be devoted
to introduce the standard molecules in use in molecular electronics as well
as to describe their basic properties.

3.2 Molecules for Molecular Electronics

Part of the fascination of molecular electronics lies in the fact that the
molecular toolbox is almost infinite, which makes us believe that it is pos-
sible to find an appropriate molecule for any imaginable application. So
far, however, only a few classes of molecules have been explored in molec-
ular electronics. In this section we shall introduce some of these molecules
and discuss their basic properties. But before doing that, it is convenient
to recall the most common functional elements in digital electronic circuits
that molecules are supposed to mimic. The main elements and their re-
quirements are the following:

• Conducting wires: low resistance, high ampacity.

• Insulators: high resistivity, high breakdown voltage.
• Switches: high on/off resistance ratio, reliable switching, small leak
current in off position.
• Storage elements: long storage time, low loss.

When extending the scope to cover also logic circuits one additionally
has to consider:

• Diodes: high forward/backward current ratio.

• Amplifiers: high gain.

Finally, since most of the existing devices containing molecules are com-
posite devices in which the molecules are connected to either metal or semi-
conductor electrodes yet another function has to be realized:

• Anchoring groups: reliable contact between functional molecular

unit and electrode.

In order to be able to compete with standard semiconductor technology,

the time constants of all devices have to be small, i.e. capacitances and/or
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Contacting single molecules: Experimental techniques 47

H H H H
H C C H C C H C C H
H H H H

Ethene Ethyne
Ethane
(Ethylene) (Acetylene)

Fig. 3.1 Examples of hydrocarbons. Left: Ethane with C-C single bond. Middle:
Ethene with one C-C double bond. Right: Ethyne with one C-C triple bond.

resistances have to be small. Since dissipation is already one of the most se-
vere problems in nowadays semiconductor devices, signal sizes, i.e. the level
of the current should be considerably smaller than in those devices. Since
our main interest lies in exploring the fundamental properties of molecular
electronic devices, we shall not pay attention to those requirements for the
rest of this book.
From the very beginning of molecular electronics, it has been become
clear that carbon-based molecules offer the required versatility to realize
most of these desired functionalities. Carbon is the basis of a great variety
of solid structures including graphite, diamond, graphene, and molecules
like the cage-shaped fullerenes and - last but not least - the quasi one-
dimensional nanotubes.

3.2.1 Hydrocarbons
Another very rich class of carbon-based molecules is the hydrocarbons with
the possibility to tune their degree of conjugation. The electronic richness
of both classes stems from the fact that the degree of hybridization of the
molecular orbitals depends on the conformation and the environment. The
carbon atom has four valence electrons which in the case of diamond are
sp3 hybridized corresponding to a tetrahedral arrangement of the bonds in
space. This conformation is realized in the saturated hydrocarbons with
the sum formula Cn H2n+2 which are called alkanes.1 Each carbon atom has
four direct neighbors, either C or H atoms and all bonds are σ-bonds, see
Fig. 3.1. Bigger alkanes with n ≥ 4 exist in several isomers, some of which
are ring-shaped (cycloalkanes). Since all electrons are used for forming
chemical bonds they are basically localized and the alkanes are insulating.
1 The transport through alkane-based molecular junctions will be discussed in section
14.1.2.
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48 Molecular Electronics: An Introduction to Theory and Experiment

In graphite the valence electrons are sp2 hybridized in the graphite plane
with an angle of 120o between the bonds. The fourth electronic orbital has
p character with its lobes pointing perpendicular to the graphite plane. The
wave functions of neighboring carbon atoms overlap and form the electronic
π-system, which in case of graphite is responsible for the in plane and
the finite plane-to-plane conductance. The same situation takes place in
the alkene hydrocarbons containing one carbon-carbon double bond, see
Fig. 3.1. Interesting for molecular electronics are polyenes with the sum
formula Cn Hn+2 , which contain more than one double bond. When these
double bonds are alternating with single bonds, the wave function of π-
system is extended over the whole molecule. These molecules are called
conjugated or aromatic molecules. The criterion of aromaticity is 4n + 2
π-electrons.
The carbons in hydrocarbons may furthermore be triply bond in sp-
hybrids forming alkynes. When alternated with single-bonds these linear
bonds are very stable and give also rise to delocalized wave functions as in
the conjugated species with double bonds.
The delocalization of the wave function is broken when the double or
triple bonds do not alternate with single bonds. Furthermore, the con-
jugation can be tuned by introducing an angle between the planes of the
individual cyclic parts. The consequences of breaking the conjugation for
the conductance of a molecular junction will be discussed in section 13.5.
In a very common representation only the bonds are shown: single bonds
as single lines, double bonds as double lines, triple bonds as triple lines. The
carbon atoms themselves are not displayed. The positions of the carbon
atoms are at the kinks between these lines. Neither the hydrogen atoms
nor the bonds to them are drawn. The number and positions of them can
be deduced by fulfilling the valence four at each carbon. As an example we
show in Fig. 3.2(a) the polyene hexatriene (consisting of six carbons and
with three double bonds) in various representations.
As for the alkanes larger species of alkenes and alkynes arrive in several
isomers. When two doubly-bond carbon atoms are surrounded by different
groups one has to distinguish between the cis conformation, in which the
neighboring groups are on the same side of the double bond, and the trans
conformation with the neighbor groups being located on opposite sides of
the double bond. A cis-trans conformation change sets the basis for a class
of molecules with in-built switching functionality.2
2 The most popular species of molecular switches are those which can be addressed opti-
cally. Many realizations are based on two ground types of switching (cis/trans conforma-
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Contacting single molecules: Experimental techniques 49

(a) H H H H
(b) H H
H C H H C H
C C C C
C C C C C C
C C
, C C
H C H H C H
H H H H
H H

, :

H
H
H
H

H
H H
H

Hexatriene Benzene

Fig. 3.2 Various representations of the hexatriene and the benzene molecule. (a) The
polyene hexatriene is chosen as an example for a conjugated linear hydrocarbon molecule.
(b) The benzene molecule. Top and center panel: Because of the delocalization of the
π-electrons the positions of the double bonds are not defined. Therefore, they are often
symbolized by an inner ring.

The typical conformations of polyenes are zigzag-shaped lines reflecting

the preferred 120o orientation of the sp2 hybrid. When building the angle
to the same side cyclic molecules are formed. The ideal cyclic polyene
geometry is the benzene molecule consisting of six carbons forming planar
ring with perfect conjugation, see Fig. 3.2(b). Since the π-electrons are
delocalized over the whole ring, it is not obvious between which carbons
the double bonds and where the single bonds have to be drawn. Therefore,
one often uses a notation in which the π-electrons are symbolized by an
inner ring.
Molecules consisting of several benzene rings merged along one bond are
called polycyclic aromates. The most prominent examples are naphtalene,
consisting of two benzene rings, anthracene consisting of three rings in a
linear arrangement, tetracene with four and pentacene with five rings in
series. Also angular arrangements of the rings or combinations with rings
containing five carbon are used. Examples are shown in Fig. 3.3. Also
five-rings (cyclopentadiene) and less often seven-rings (cycloheptatriene)
are possible. They are aromatic if six π-electrons per ring exist. In the case
tion switching and ring opening/ring closure). These types of molecules are introduced
in section 20.7.
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50 Molecular Electronics: An Introduction to Theory and Experiment

Naphtalene Anthracene Tetracene Cyclopentadiene

anion

Biphenyl Phenanthrene Pyrene Cycloheptatriene

cation

Fig. 3.3 Examples of polycyclic molecules.

N N S O

Fig. 3.4 Examples of the most common heterocyclic aromates.

of cyclopentadiene this means that an extra electron has to be added to the

ring to provide a stable π-electron sextet (anion), while in cycloheptatriene
one electron charge has to be withdrawn (cation), see Fig. 3.3.
In heterocyclic molecules one or more carbon atoms are replaced by an
atom of another species. Some heterocycles in use in molecular electron-
ics are depicted in Fig. 3.4. The most common substituents are sulfur,
nitrogen and oxygen. Because of their chemical valence they posses more
electrons than the carbons. In hexagonal rings the additional electrons do
not contribute to the π-system, but may be used for forming bonds to other
atoms, e.g. to the metal electrodes. In five-rings they help stabilizing the
conjugation.

3.2.2 All carbon materials

As mentioned in the beginning, also pure carbon molecules are promising
for molecular electronics. Carbon nanotubes are sheets of graphite which
are rolled together. They have diameters ranging from 1 nm to several tens
of nanometers and length of up to millimeters. Depending on the orienta-
tion of the long axis with respect to the hexagons various nanotubes with
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Contacting single molecules: Experimental techniques 51

C60

Fig. 3.5 Line representation of the bonds of the fullerene molecule C60 .

varying electronic properties are possible.3 Since defect-free carbon nan-

otubes are ballistic conductors they may serve as interconnects for bridging
long distances.
Finally, the combination of pure carbon hexagons with pentagons, but
without hydrogen sets the basis for the fullerenes. Since the bond length
in pentagons is smaller than in hexagons, these molecules are not planar
but have a curvature. The most famous fullerene is C60 (see Fig. 3.5)
consisting of 20 hexagons and 12 pentagons in the same conformation as
in a soccer ball. It has a completely delocalized π system, making it also a
good candidate for molecular electronics applications.

3.2.3 DNA and DNA derivatives

A completely different class of molecules is based on our genetic information
carrying molecule DNA. It is very tempting to use DNA because of the rich
versatility, the possibility to tune the length from short to very long, and
its self-reproduction properties. After almost two decades of research on
DNA-based electronics it seems now to be clear that DNA by itself is too
poorly conducting for real electronic applications. However, it may serve
as template for assembling better conducting molecules or metal-molecule
combinations. Furthermore, DNA derivatives are under study which seem
to have more fortunate electronic properties. In section 18.3 we shall discuss
the transport properties of DNA-based molecular junctions.

3 An excellent review about the conformation and resulting electronic properties of

nanotubes is given in Ref. [103].
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52 Molecular Electronics: An Introduction to Theory and Experiment

3.2.4 Metal-molecule contacts: anchoring groups

A common problem in molecular electronics is the difficulty to form stable
and electronically transparent chemical bonds of the molecules to the metal
electrodes. Among the manifold possibilities one particular solution has
been chosen as standard system. This is the combination of a sulfur atom
to gold electrodes. The reason to choose gold lies in the fact that it is
inert to chemical reactions, which allows to prepare clean surfaces and tips.
The drawback of this inertia is the fact that it hardly undergoes chemical
reactions with other species. One of the rare exceptions is sulfur in its thiol
(sulfur-hydrogen) form. This bond is mechanically stable with a force in the
order of 1.5 nN [104]. The thiol-gold binding scheme has successfully been
tested in self-assembled monolayers (SAM) (see below) on flat surfaces as
well as in single-molecule contacts on tips. It provides sufficient electronic
transparency for most applications. This is the reason why alkanedithiols
(i.e. alkanes with thiol endgroups at both ends) and benzenedithiols (a
benzene ring with thiols usually at opposite ends) represent the testbeds
for molecular electronic circuits. The alkanedithiols are the archetypical
insulators, while benzenedithiol is the most simple aromatic molecule which
can be coupled to metal electrodes. However, alternatives to the thiol
bonding scheme are also under study, as it will be described in section 14.2.

3.2.5 Conclusions: molecular functionalities

We want to close this section by pointing out which molecules can be consid-
ered as possible candidates for various electronic components in molecular
circuits:

• Conducting wires: polyenes and alkynes.

• Insulators: alkanes.
• Switches: cis/trans conformation changes of manifold molecules,
the prototype being azobenzene, consisting of two benzene rings
connected via a C=C double bond. In many examples the con-
jugation is reduced in the trans isomer because the π-systems of
both parts are not coplanar. The second prototype of switches are
ring-opening-ring-closure transformations which can be triggered
optically, see section 20.7. In these switches one of the hydrocarbon
rings or heterocycles is opened thereby affecting the conjugation of
the π-system.
• Storage elements: all kinds of molecules with at least two states
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Contacting single molecules: Experimental techniques 53

may serve as storage elements, including among others conforma-

tions, redox states, spin states, and vibrational states. Examples
will be discussed in Chapters 13, 15, and 16.
• Diodes: molecules which consist of two different, and electronically
decoupled parts. An example is the famous suggestion by Aviram
and Ratner [8] mentioned in the first chapter.
• Amplifiers: in principle all molecules the electronic levels of which
can be tuned by a gate electrode might act as amplifiers. Although
electronic three-terminal devices following this principle of bipolar
transistors have been demonstrated, they do not provide current
amplification yet.
• Anchoring groups: thiols, amines, nitros, cyanos or heterocycles
with the substituent atoms serving as linkers to the metal electrodes
(see 14.2).

3.3 Deposition of molecules

Molecular deposition methods are manifold because of the rich variety of

molecules in use. In most experiments the molecules are deposited from
solution onto the metal films forming the electrodes. Various solvents and
a wide range of concentrations are used. The molecules are allowed to
chemisorb to the metal electrodes. After an incubation time the molecular
solution is rinsed away with pure solvent. For low-temperature measure-
ments the devices are then dried in a gas (nitrogen) flow. In some cases
the electronic measurements are performed without drying, in solution -
either in presence of the pure solvent or with the molecular solution. A
variation of this deposition from solution is spin-coating. A drop of the
molecular solution is given on the substrate which is mounted on the chuck
of spin-coater. Upon rotation of the substrate the solution is wide-spread
over the wafer such that a very small concentration of molecules on the sub-
strate is achieved. As an example we mention individual carbon nanotubes,
which after spin-coating can be localized by atomic force microscopy or
other techniques. A particular nanotube can subsequently be contacted via
lithographically defined metal electrodes.
Many molecules, in particular rod-like molecules form self-assembled
monolayers (SAM) on metal surfaces. For that purpose the substrate
covered with the metal layer is dipped into the molecular solution. The
mostly amphiphilic molecules are equipped with one anchoring group that
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54 Molecular Electronics: An Introduction to Theory and Experiment

facilitates the chemical adsorption on the surface. The most common com-
bination for molecular electronics devices is thiol-terminated molecules for
adsorption on gold surfaces. The molecules organize such that they form
ordered monolayers (see Fig. 3.6). This procedure sounds simple, but in
practice many parameters have to be well controlled for obtaining repro-
ducible SAM quality. A recent review of this technique is given in Ref. [105].
Another highly developed technique is monolayer formation via the
Langmuir-Blodgett (LB) technique [106, 107]. A LB film consists of one
or more monolayers of an organic material, deposited from the surface of
a liquid onto a metal surface by immersing the solid substrate into the liq-
uid. The molecules form a monolayer on the surface of the solution. The
monolayer is transferred to the substrate when dipping it into the solu-
tion. Upon repetition of the immersion a multilayer consisting of several
monolayers and, thus films with very accurate thickness can be formed (see
Fig. 3.6). The film formation relies on the fact that amphiphilic molecules
with a hydrophilic head and a hydrophobic tail are used. These molecules
assemble vertically onto the substrate. For other molecules a horizontal
adsorption may be favored, yielding low-density films. The density and or-
dering can be enhanced by concentrating the molecular layer on the surface
of the solution with a spatula before the substrate is dipped into it.
In particular, for the preparation of samples for low-temperature mea-
surements, remainders of the solvent may hamper the formation of clean
metal-molecule-metal junctions. Therefore, alternative “dry” deposition
methods have been developed. Gaseous molecules (like e.g. hydrogen, oxy-
gen, nitrogen, carbonmonoxide, methane) can be deposited directly from
the gas phase by condensation on the cold metal electrodes. Very stable
molecules, like the fullerenes or DNA bases may be evaporated thermally
from various sources including Knudsen cells or tungsten boats, which are
Joule heated by driving a current through them. More sensitive molecules
can be deposited using electrospray ionization (ESI). The method starts
with a solution in which the molecules to be ionized are dissolved. An
electrospray of this solution is created by a strong electric field, which orig-
inates from a voltage applied between the spray needle and the end of a
capillary. Due to the strong field at the tip apex, charged droplets are
created, which are directed towards the capillary, which forms the connec-
tion to a vacuum chamber where the already prepared metal electrodes are
located [108]. With this method well-controlled submonolayer molecular
films may be deposited onto substrates in ultra-high vacuum (UHV).
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Contacting single molecules: Experimental techniques 55

Au-covered substrate

Molecular solution
(thiol terminated)

Adsorption

Organization

Fig. 3.6 Top: Formation of a self-assembled monolayer (SAM) shown for two species of
alkanethiols on a gold-covered substrate. The substrate is immersed into the molecular
solution. The molecules adsorb assemble with the thiol-terminated end on the substrate.
After an incubation time a self-assembled monolayer is formed. Bottom: Fabrication of a
Langmuir-Blodgett (LB) film. The left panel shows a droplet of an amphiphilic molecule
dissolved in a volatile solvent. It is spread on the water-air interface of the trough. The
solvent evaporates and leaves a diluted and disordered monolayer behind which is then
compressed with the help of a moving barrier. The right panel shows how the monolayer
is transferred onto the substrate. Reprinted with permission from Ref. [106].

3.4 Contacting single molecules

The fabrication of single molecule electronic devices is a difficult task. The

main problem lies in the size of the molecules, which is usually smaller than
the resolution of lithographic methods. Thus, sophisticated techniques have
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56 Molecular Electronics: An Introduction to Theory and Experiment

to be applied for forming nanometer-size metal gaps. Most of the longer

molecules are not conductive enough to be studied in single-molecule de-
vices, but are rather investigated in ensembles.4 Furthermore, the coupling
between the molecule and the electrodes plays an important role. Another
consequence of the small size is the difficulty to image the geometry of the
junction and to prove that one deals indeed with a single-molecule device.
So far, no method exists which allows one to perform systematic measure-
ments of the electronic transport and to characterize the geometry of a
given junction with atomic precision. Therefore, several methods are used
and are permanently improved. This enables to distinguish between the
properties of the metal-molecule combination and the influence of the con-
tacting scheme. The methods may be divided into two main classes. The
first one produces rather stable devices, however, the geometry of it cannot
be varied and contamination cannot be excluded. Besides the stability, the
possibility to add a third electrode is an important advantage. For that
purpose, metal electrodes with small volume are desirable for reducing the
shielding of the electric field. The second class enables clean contacts and
modification of the junction geometry, but offers only limited stability.
The majority of methods in use for contacting individual molecules are
based on one of the techniques described in the previous chapter, since
contacting single molecules requires at least one atomically fine metal elec-
trode.

3.4.1 Electromigration technique

The electromigration technique described in the previous chapter is suc-
cessfully used for the fabrication of pairs of metal electrodes for contact-
ing single molecules [21, 109, 110]. For this purpose, the electromigration
has to be stopped when the contact is broken and the electrodes form a
nanometer-size gap. In vacuum this would be signaled by a sudden increase
of the resistance above the typical resistance of a single-atom contact. How-
ever, because clean interfaces are needed for achieving well-shaped single-
molecule junctions, the molecules are usually deposited - by one of the
methods mentioned above - before the electromigration process. This com-
plicates the control sequence needed for stopping the electromigration at
the right moment, because molecules short-cut the gap resistance. There-
fore, many junctions are prepared in parallel and the statistical behavior
is determined. Since the metal wire is at ambient conditions before the
4 The transport properties of long molecules will be addressed in Chapter 18.
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Contacting single molecules: Experimental techniques 57

VSD

Au Au
Vgate
SiO2
Si

a b

c d

e f

Fig. 3.7 Three-terminal devices and possible artifacts in molecular contacts. Top panel:
Schematic diagram of electromigration gap and measurement configuration. Bottom
panel: Six models describing possible geometries formed within the electromigration gap
by molecule(s) and contaminant metal particles. (a) Single-molecule contact as desired.
The molecule is chemisorbed with both ends at the metal electrodes. (b) Single-molecule
in the vacuum gap between the electrodes. The molecule is not chemisorbed. (c) Metal-
nanoparticle bridging the gap between the electrodes. (d) Multi-molecule contact. (e)
The molecules are coupled indirectly via a metal nanoparticle to the electrodes. (f)
The molecules are not chemisorbed to the electrodes but to a metal nanoparticle. After
Ref. [109].

deposition of the molecules, all kinds of contaminants might be present and

have to be carefully removed before deposition of the molecules.
With this technique, all kinds of current-voltage characteristics have
been measured ranging from ohmic behavior to Coulomb-blockade be-
havior.5 The tunnel contacts may be formed by vacuum gaps (without
molecules), single-molecule or multi-molecule contacts. One particular
problem of the method is the risk to form small metal grains, the transport
5 The various possible transport mechanisms will be described in Chapters 13 and 15.
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58 Molecular Electronics: An Introduction to Theory and Experiment

properties of which resemble molecular contacts [109, 110]. Some examples

of possible contact geometries are given in Fig. 3.7. Finally, the metal grain
may be contacted to both electrodes via one or several molecules. Thus, the
yield of this method, i.e. the probability to have a single-molecule contact
is in the order of a few percent only. On the other hand, the junctions are
extremely stable and well suited for systematic studies of their transport
behavior at varying temperature or magnetic field. Because the electrodes
are in direct contact to a substrate, it can be used as a back-gate form-
ing a three-terminal device.6 By applying a gate voltage the transport
mechanism can be detected and at least partial information of the contact
geometry can be obtained.

3.4.2 Molecular contacts using the transmission electron

microscope
In order to obtain very strong coupling between the metal electrodes and
the molecule, a particular method has been put forward. It includes further-
more the possibility to image the contact geometry, because the molecules
form suspended junctions over slits in thin membranes and can thus be
inspected by transmission electron microscopy (TEM). Several variations
have been reported, which are optimized for the various molecules. The
common point is that the metal electrodes, which have been pre-patterned
on a thin membrane or a TEM inspection grid, are rapidly heated up by
an intensive electron or laser beam above their melting temperature. Con-
tamination atoms are distilled out of the electrodes and defects are driven
out as well. The molecules are brought into contact while the metal is
liquid. During recrystallization parts of the molecule are soldered into the
electrodes resulting in small contact resistances.
This method has been demonstrated to work for long molecules like
DNA and carbon nanotubes [112] as well as for chains of clusters [113] (see
Fig. 3.8). Possible risks are, of course, destruction of the molecule by the
high-energy impact of the laser or electron beam or the hot metal electrodes
as well as formation of metal whiskers shorting the molecular junction.

6 Thephysical results obtained with these devices are discussed in particular in Chapters
15 and 16.
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Contacting single molecules: Experimental techniques 59

Fig. 3.8 Contacting individual molecules in a transmission electron microscope. Top:

Schematics of the sample geometry. A long molecule is suspended over a slit in a thin
membrane and soldered at its ends to two metal electrodes. Bottom: Transmission
electronic micrograph (TEM) of nanotubes, suspended across a slit between two metallic
pads, and detailed view of the contact region showing the metal molten by the laser beam.
Reprinted with permission from [112]. Copyright 2003 by the American Physical Society.

3.4.3 Gold nanoparticle dumbbells

A very elegant method for overcoming the size mismatch between the res-
olution of lithographic methods in use for the definition of the electrodes
and the molecules has been described by T. Dadosh et al. [114]. The au-
thors use gold nanoparticles (GNPs) with a typical diameter of 10 nm.
The molecules to be contacted are functionalized at both ends with thiol
anchoring groups, which have a high affinity to gold. By these thiol bonds
the molecules are attached to the GNPs such that two of them are com-
bined to form a dumbbell. Those dumbbells now have a suitable size for
bridging lithographically defined nanogaps and can be deposited onto them
straightforwardly. A further advantage of the method is that the statisti-
cal behavior of the molecules in contact with the GNPs can be studied by
various non-contact methods such as optical spectroscopic measurements
before deposition onto the electrodes (see Fig. 3.9).
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60 Molecular Electronics: An Introduction to Theory and Experiment

(a) (c)

(b)

Fig. 3.9 (a) The structures of three molecules studied with the dumbbell tech-
nique: 1,4-benzenedimethanethiol (BDMT), 4,4′ -biphenyldithiol (BPD) and bis-(4-
mercaptophenyl)-ether (BPE). (b) The dimer contacting scheme. (c) TEM image of
a BDMT dimer made of 10-nm colloidal gold particles. The separation between the
two particles corresponds approximately to the BDMT length (0.9 nm). Adapted with
permission from MacMillan Publishers Ltd: Nature [114], copyright 2005.

3.4.4 Scanning probe techniques

Conceptually, the most straightforward method for contacting a singe
molecule with a fine tip is to deposit the molecule on a metallic substrate
and to approach the molecule with the tip until one or several atoms of the
molecule are chemisorbed to the tip. However, this is not as simple as it
sounds and this method is only suitable for certain molecules. Even if the
process is successful, the interpretation of the subsequent conductance mea-
surements is not simple because in STM the electronic signal is convoluted
with the topographic information. Furthermore, the presence of the tip
may disturb or even destroy the molecule.7 Therefore, various variations
of the STM technique have been developed. They all have in common the
difficulty to add a third electrode for gating. A certain but nonlocal gate
effect can be achieved via electrochemical gates (see below). STM-based
techniques are particularly suitable for gathering statistical information be-
cause many contacts can be studied in relatively short time. As already
explained in the previous chapter, the price for the high flexibility is the
low stability and the, in general, short lifetimes of the junctions.

3.4.4.1 Direct contact

The direct contacting scheme mentioned above requires first a careful prepa-
ration and characterization of the surface. Subsequently a sub-monolayer
of the molecules is deposited. For stable molecules such as the fullerene
7 This can be checked by comparing topographic and spectroscopic results, though.
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Contacting single molecules: Experimental techniques 61

C60 this can be performed via evaporation [115, 116]. The surface is then
scanned and a suitable molecule is selected. Depending on the physical
question to study, an isolated molecule or a member of a larger aggregate
can be chosen. As described in the previous chapter, for single-atom con-
tacts, the formation of a single-molecule chemical junction is signaled by a
sudden increase of the conductance. When this is achieved, the approach
can be stopped and spectroscopic investigations can be performed. From
the electronic point of view this contacting method usually results in asym-
metric contacts, meaning that the molecule is electronically better coupled
to the substrate than to the tip. This is important for the interpretation of
the transport properties, which will be discussed in Part 4. Often the cou-
pling to the substrate is in the “strong” regime while the electrons have to
tunnel from the molecule onto the tip and vice versa, i.e. it is in the “weak
coupling” regime. Therefore, this method is most suitable for molecules
which are only loosely bound to the substrate, e.g. by a single atom or a
few atoms, like for C60 , where the binding is given through one pentagon
or one hexagon of carbon atoms.

3.4.4.2 Contacting rod-like molecules

Rod-like or planar molecules have the tendency to lay flat on the surface.
In that case the current will not flow along the molecule, but most probably
transverse it perpendicularly finding the path of smallest resistance. For
those molecules several variations of the scanning probe technique have
been put forward. The first method is particularly suitable for imaging
and spectroscopy on the molecular orbitals [117]. After preparation of the
clean metallic surface, a monomolecular layer of an insulator, e.g. a salt
is deposited. The molecules are then evaporated on top of this thin layer
which acts as a tunnel barrier between substrate and molecule.
Another possibility is to directly deposit the molecules onto the metal
surface, but to design the molecules such that they have edge atoms with
high chemical affinity to the tip metal. The tip is then approached to one of
these atoms until a chemical bond is formed. Upon carefully withdrawing
the tip the molecule is peeled off the substrate, as illustrated in Fig. 3.10.
During the peel-off process spectroscopic measurements can be performed
which enables to identify the varying charge-transport mechanisms and to
quantify the coupling strength [118, 119]. This will be explained in more
detail in section 14.4.
The spatial resolution of the STM imaging can be enhanced by suit-
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62 Molecular Electronics: An Introduction to Theory and Experiment

1 2 3 4

Fig. 3.10 Schematics of the contact formation process of a molecular junction with the
STM. Four stages of the contact formation during approach (1,2) and retraction (3,4) are
shown. At (3) the chemical bond between the contact atom and the substrate is broken
and the molecular wire is formed. Reprinted with permission from [119]. Copyright 2008
IOP Publishing Ltd.

able functionalization of the STM tip, e.g. with hydrogen molecules [122].
Recently, it has been demonstrated that molecular orbitals can be even
better resolved by atomic force microscopy when the tips are terminated
with carbon monoxide (CO) molecules [120, 121].
Finally, an elegant way to contact rod-like molecules is to embed the
molecules into a matrix of less conducting molecules, such that the long axis
of the molecules is almost perpendicular to the substrate, see Fig. 3.11(a).
With the techniques described in section 3.3 a self-assembled layer of weakly
conducting molecules is prepared. A standard combination would be alka-
nes with one thiol anchor group on a gold substrate. The thiol binds chem-
ically to the gold releasing the non-thiolated ends to the top of the SAM.
The properties of the SAM are chosen such that free places or defects ex-
ist at which the study molecule can be incorporated. When scanning the
sample with an STM tip the positions of the better conducting molecules
can be located and spectroscopic measurements can be performed [123].
In a variation of this technique the study molecules are equipped with
two highly reacting anchoring groups, e.g. thiols. One end attaches to
the gold surface, the other one pointing to the top of the SAM. These
thiols can the be used as binding places for gold nanoparticles (GNPs) , see
Fig. 3.11(b). Depending on the density of the study molecule and the size
of the GNPs, one or several molecules are contacted with the same GNP. In
this way a very stable molecular junction consisting of substrate, molecule
and GNP is fabricated. The prepared sample is then investigated with
an STM [124] or a conductive AFM [125]. The tip is either brought into
strong contact with the GNP, such that the tip-GNP contact has negligible
resistance. Or the transport properties due to the presence of the GNP have
to be incorporated in modeling the transport for deducing the properties of
the molecular junction. The obvious advantage of this latter method is the
high stability of the device. Both variations share the in-built possibility
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Contacting single molecules: Experimental techniques 63

(a) (b)
Tip Tip

metal
nanoparticle

Substrate Substrate

Fig. 3.11 (a) Scanning tunneling microscopy (STM) study of electron transport through
a target molecule inserted into an ordered array of reference molecules. (b) STM or
conducting atomic force microscopy (AFM) measurement of conductance of a molecule
with one end attached to a substrate and the other end bound to a metal nanoparticle.
After Ref. [40].

to perform statistical investigations because hundreds of junctions can be

prepared on the same chip. The main drawbacks are the complex sample
preparation and the limited versatility because successful embedding into
the matrix is not obvious.8

3.4.4.3 STM in liquid environment

A very powerful tool is the use of an STM in liquid environment. The
surface and the tip are prepared as usual for forming atomic contacts, but
immersed into a solvent, in which the molecules under study can be dis-
solved. The tip can be sharpened and covered with substrate atoms by
repeated indentation into the substrate. Then molecules are added to the
solvent. After an incubation time needed for chemical binding to the sub-
strate, the tip is repeatedly approached to the surface and withdrawn while
the conductance is recorded. Upon closing the gap a metal-molecule-metal
junction consisting of several molecules is formed. When withdrawing the
tip, the molecules loose the contact to the electrodes not all at once but in
an irregular series. The result is a step-like decrease of the conductance as
a function of the distance which varies from repetition to repetition. After
breaking the contact to the last molecule, a new junction can be formed.
The molecules which get stuck to either the substrate or the tip are re-
placed by fresh molecules diffusing in from the solution. After a while a
8 Examples of transport measurements performed with this technique will be described
in Chapter 13.
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64 Molecular Electronics: An Introduction to Theory and Experiment

new position on the substrate can be chosen. The method is very suitable
for gathering statistical information about the preferred conductance values
of molecular junctions. However, the stability of the molecular junctions is
usually not sufficient for spectroscopic measurements [36, 126].9

3.4.5 Mechanically controllable break-junctions (MCBJs)

Mechanically controllable break-junctions (MCBJs) (see section 2.6) are
used for contacting molecules in various environments. For measurements
at room temperature under ambient conditions, the molecules are usually
deposited from solution. After a reaction period, the remainder of the
molecular solution is rinsed in pure solvent and blew dry with nitrogen.
At variance to the electromigrated junctions, the molecules are usually de-
posited after forming the electrode gap by breaking the MCBJ. For polar-
izable molecules it might be helpful to apply a voltage in order to pull one
or several molecules into the junction. The junction is then carefully closed
until a measurable current flows. Depending on the molecule, the closing
traces show plateaus which signal the formation of a molecular junction
containing one or several molecules. At room temperature the electrode
atoms are rather mobile and the molecular junctions have only limited life-
time of a few minutes. This is, however, a much longer time span than
usually achieved with STM setups and is sufficient for measuring I-V char-
acteristics. On the other hand, only limited statistical information can be
acquired because of aging effects of the junctions. After several opening or
closing cycles no molecular junctions form any more. For recording con-
ductance histograms it is advantageous to perform the measurements in
liquid environment, as it was first proposed by Grüter et al. [78]. Fig. 3.12
shows a slightly different setup. A pipette is pressed onto the inner part
of MCBJ electrodes and sealed with gasket made of a flexible and solvent
resistant material (polydimethylsiloxane (PDMS)). The molecular solution
is continuously pulled through the pipette, while the MCBJ is opened and
closed and the conductance is recorded. Molecules which leave the junc-
tions are replaced by fresh ones from the solution as discussed earlier for
STM setups.
Much longer lifetimes of molecular junctions can be achieved at low
temperature. Furthermore, the thermal smearing of the electronic proper-
ties is considerably reduced. For that purpose several protocols have been
9 This technique has been applied for transport measurements through DNA. Examples
will be discussed in section 18.3.
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Contacting single molecules: Experimental techniques 65

electrical wiring
glass pipette contaiinng
molecular solution

spring-borne
contact
plug hosting pipette
and contacts
bolt

MCBJ sample PDMS gasket

Fig. 3.12 A PDMS-sealed glass pipette, in which the molecular solution circulates, is
pressed onto the central part of MCBJ chip with the help of a plug screwed to the sample
holder. The electrical contacts are realized in this case via spring-borne contacts outside
the gasket.

developed. When starting with a deposition of the molecules from solution,

the solvent and any humidity has to be carefully removed in order to obtain
clean molecular junctions without tunnel barriers due to ice formation. For
this purpose it is helpful to make use of strong metal molecule binding: A
molecular junction is formed at room temperature. When breaking it again
it may happen that the breaking does not occur between the molecule and
the metal electrode, but that one or several gold atoms remain attached to
the molecule leaving a gap between two metal atoms. The junction is then
cooled down and the metal-metal gap is closed again. Of course, water
films or other kinds of contamination may form on the metal surfaces as
well, but they can be pushed out of the contact such that a good electrical
contact can be established.
The problem of ice formation can be solved when forming the electrode
gap at low temperatures under cryogenic vacuum conditions. Even though
the surface of the native break-junction might be covered with water or
other contaminants fresh and clean metal tips are formed. Small molecules,
which at ambient conditions are in the gaseous phase (like e.g. hydrogen,
oxygen, carbonmonoxide, methane), may be condensed directly onto the
cold MCBJ electrodes with a nanometer-size separation [127]. Other small
molecules with low evaporation temperature (e.g. water) are first vaporized
and then condensed. Similarly, stable molecules like the fullerenes can be
evaporated on an opened MCBJ at low temperature [128].
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66 Molecular Electronics: An Introduction to Theory and Experiment

3.5 Contacting molecular ensembles

One main problem in single-molecule studies lies in the fact that the
electronic transport depends crucially on the exact coupling between the
molecule and the metal electrodes, i.e. on the precise atomic arrangement
of the contacts.10 As a result pronounced sample-to-sample and junction-
to-junction variations are observed. Repeated measurements are needed to
deduce the typical behavior of a given metal-molecule system. The influence
of varying contact geometry averages out in devices containing ensembles
of molecules. Furthermore, these ensembles are contacted with rigid and
robust electrodes. These devices usually provide better mechanical stability
and longer life-times allowing long-time systematic measurements and the
variation of outer control parameters like temperature or magnetic field.
However, when interpreting data recorded on ensemble devices one has
to bear in mind possible interaction effects between the molecules them-
selves which might affect their electronic properties. Furthermore, also
without interaction effects it is not straightforward to infer the single-
molecule junction behavior from the ensemble because the number of
molecules which contribute to the transport may be smaller than the total
number of molecules in the ensemble, if not all are contacted equally. For
instance, some of the molecules forming the ensemble might be in strong
coupling to the electrodes while others are only weakly coupled. As a result
the transport characteristics may show superpositions of various transport
mechanisms. Furthermore, ensemble structures are necessarily larger in
space than single-molecule devices giving limits to their maximum integra-
tion density. From the point of view of fundamental research the most
promising strategy is to compare the results from single-molecule contact
schemes with ensemble measurements for revealing the robust properties
of the given molecule-metal system. We shall restrict ourselves to methods
suitable for small ensembles ranging from roughly a few hundred molecules
to several thousand molecules. Very efficient methods have been developed
for contacting large area molecular films, which are however, out of scope
of this monograph.

3.5.1 Nanopores
One technique which produces rather small ensembles of molecules uses
pores in thin freestanding membranes. The method has been used in
10 This issue will be addressed in Chapters 13 and 14.
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Contacting single molecules: Experimental techniques 67

the 1980’s and 1990’s for fabricating nanometer-sized metallic contacts for
point contact spectroscopy [129]. However, no single-atom contacts can be
achieved. A single crystalline silicon wafer is covered from both sides with
a thin layer of silicon nitride with a typical thickness of 50 nm to 100 nm.
The rear side of the wafer is patterned by optical lithography with squares
of typical lateral size of 100 µm. Using first plasma etching for attacking
through the nitride, then wet-etching in hydrofluoric acid the squares are
etched through the bulk of the silicon wafer. The wet etching process is
anisotropic. It attacks particular crystal orientations of the silicon much
faster than others. As a result inclined etch walls are formed thereby re-
ducing the size of the squares. The inclined walls become covered with a
native silicon oxide layer during the following process steps. Furthermore,
the acid attacks silicon much faster than silicon nitride. The process can
thus be stopped controllably when a suspended silicon nitride membrane is
obtained. Now the membranes are patterned from the front side via elec-
tron beam lithography with a small dot in each membrane. Using plasma
etching a small pore is drilled into the membrane with a typical diameter
of 10 to 50 nm.
The formation of molecular junctions requires three further steps [130].
First, a metal electrode - usually gold - is evaporated from the top side.
The device is then immersed into the molecular solution until a SAM has
formed. After a suitable reaction time which depends on the molecule-
metal combination the sample is rinsed and dried and the second metal
electrode is deposited by evaporation onto the rear side, see Fig. 3.13. Care
has to be taken that the SAM is not destroyed by thermal impact com-
ing from the metal atoms. With this technique thermally stable molecular
ensemble junctions are obtained which are particularly suitable for studies
of the temperature dependence of the transport properties. A difficulty of
the method lies in the fact that the quality of the first deposited electrode
cannot be characterized; it might be covered with water or other contami-
nants which could hamper the formation of a high-quality SAM. A similar
objection was made concerning the second molecule-metal interface: The
molecular layer is exposed to ambient conditions before the deposition of
the second electrode.11

11 We will discuss data recorded with this sample species in Chapter 13.
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68 Molecular Electronics: An Introduction to Theory and Experiment

Au
Si3N4
Si
Si SiO2

Au

Au
Si3N4
Au

Alkanethiol

Fig. 3.13 Molecular junctions in nanopores. A small molecular ensemble is contacted

with metal electrodes in a nanometer-sized pore in a silicon-nitride membrane. Top
schematic is the cross section of a silicon wafer with a nanometer-scale pore etched
through a suspended silicon nitride membrane. Middle and bottom schematics show a
Au-SAM-Au junction formed in the pore area. The structure of octanethiol is shown as
an example. Reprinted with permission from [130]. Copyright 2003 by the American
Physical Society.

3.5.2 Shadow masks

Another method to fabricate small ensemble devices uses the self-alignment
property of shadow masks. The sample fabrication scheme is shown in
Fig. 3.14. Via e-beam lithography a suspended mask is produced with
a geometry of a wire that is interrupted by a small gap. A first metal
layer is evaporated perpendicularly through this mask. The next step is
the deposition of a SAM of the molecules. Alternatively molecules can
be evaporated on top of the metal under the same angle. Subsequently a
second metal layer is evaporated under an inclined angle such that the edge
of the metal film covers the molecular layer. The resolution limits of the
lithography used for the preparation of the mask restrict the contact size
to roughly 50 nm in width. The overlay length is given by the evaporation
angle and is usually chosen in the range of 20 to 50 nm. It has been
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Contacting single molecules: Experimental techniques 69

(a) Au (b)

SiO2

SiN Gate junction

Al

Fig. 3.14 Production of shadow mask on silicon substrate. (a) The shadow mask is
defined via electron-beam lithography in a Si3 N4 /SiO2 double layer using two dry etching
steps. (b) The bridge in the center of the structure is used to separate two metal
contacts, which are evaporated vertically onto the substrate. A SAM is deposited on
both electrodes. In a second step metal is evaporated under an angle that allows a small
overlap between this top electrode and one of the bottom electrodes. If this overlap is
small enough, transport through single or a few molecules can be possibly measured.
Reprinted with permission from [111]. Copyright 2005, American Institute of Physics.

shown that the smoothness of the first metal layer is mandatory for avoiding
shortcuts between both electrodes. A second problem of this method is the
risk of destroying the SAM by the heat impact during the evaporation of
the top electrode or of creating metal grains [109].

3.5.3 Conductive polymer electrodes

These problems are partially overcome by a technique described by Akker-
man et al. [131]. The fabrication method is shown in Fig. 3.15. In a first
lithography step metal lines are fabricated and then a second resist is spread
over the sample. In the next step this resist is patterned with holes via
electron-beam lithography. The molecular ensemble is deposited into these
holes. Next, the whole substrate is overcast with a highly conductive poly-
mer which provides the second electrode. The polymer is finally capped
by a planar top metal electrode. The result is a very robust molecular
junction because the SAM remains embedded into the resist. Furthermore,
the deposition of the conductive polymer is less aggressive to the SAM
than standard metal deposition techniques. At variance to most of the
previously described methods the contact scheme intrinsically gives rise to
asymmetric contacts.12 The fact that at least one of the metal electrodes
is not in direct contact with the molecular ensemble can be helpful when
exciting the molecular system optically, as described in Chapter 20.
12 The importance of the metal-molecule contact shall be discussed in detail in section
14.2.
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70 Molecular Electronics: An Introduction to Theory and Experiment

Fig. 3.15 Processing steps of a large-area molecular junction. (a) Gold electrodes are
vapor-deposited on a silicon wafer and a photoresist is spin-coated. (b) Holes are pho-
tolithographically defined in the photoresist. (c) An alkane dithiolSAM is sandwiched
between a gold bottom electrode and the highly conductive polymer PEDOT:PSS as
a top electrode. (d) The junction is completed by vapor-deposition of gold through a
shadow mask, which acts as a self-aligned etching mask during reactive ion etching of the
PEDOT:PSS. The dimensions for these large-area molecular diodes range from 10 to 100
mm in diameter. Reprinted with permission from MacMillan Publishers Ltd: Nature
[131], copyright 2006.

3.5.4 Microtransfer printing

A method which combines gentle deposition of the top electrode with the
ability to fabricate arrays of molecular junctions with similar contact prop-
erties is given by the micro- or nanotransfer printing technique. It produces
stable contacts on a substrate and involves also the formation of a SAM
(Fig. 3.16). At first an array of bottom electrodes is fabricated using litho-
graphic methods or evaporation through a mechanical mask. Subsequently
a SAM of the molecules to study is formed on the substrate. The molecules
are functionalized at their top ends with an anchoring group suitable for
binding to the metal of the top electrode. In a separate fabrication line a
stamp made of a flexible material such as PDMS is fabricated. The stamp is
topographically patterned in the geometry of the top electrodes. The metal
of the top electrode is evaporated onto it. This stamp is then pressed onto
the substrate. During this step the metal is transferred from the stamp to
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Contacting single molecules: Experimental techniques 71

(a) (b) Metal-

evaporation
PDMS

Si

(c) (d)

SAM
Substrate

Fig. 3.16 Production of nanoscale features by nano transfer printing (nTP). (a) The
features are defined by electron beam lithography in a polymethylmethacrylate (PMMA)
double layer on a silicon substrate. The elastomeric polydimethylsiloxane (PDMS) is cast
into the structures and cured at 60o C. Fluorination of the substrate before this step
ensures easy separation of PDMS and substrate after the curing. (b) Layers of 10-30 nm
metal gold are evaporated onto the PDMS stamp. (c) Alkanedithiols form a monolayer
on a GaAs substrate. The gold on the PDMS stamp binds to this monolayer and is
transferred to the substrate. (d) The patterned gold film that forms is transferred on
top of the GaAs substrate. Good binding to the monolayer is proved by the scotch
tape test. Reprinted with permission from [111]. Copyright 2005, American Institute of
Physics.

the substrate, thus forming an array of molecular junctions. This technique

enables junctions with areas ranging from less than a micrometer squared -
and thus named nanotransfer printing (nTP) - up to several hundred square
micrometers - microtransfer printing (µTP) [111, 132]. Besides the in-built
statistical information of molecular ensembles the quality of the SAM and
the contacts can be investigated by comparing contacts with varying area.
Furthermore, the contacts may be gated by applying voltages to the sub-
strate.

3.5.5 Gold nanoparticle arrays

Finally, it is possible to form networks of single-molecule junctions combin-
ing the robustness and statistical richness of ensemble studies with the fact
that each junction is formed by a single molecule or a very small number of
molecules only [133, 134]. The fabrication scheme is shown in Fig. 3.17. At
first gold nanoparticles (GNP) with a diameter of roughly 10 nanometers
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72 Molecular Electronics: An Introduction to Theory and Experiment

(a)
a l C8 OPE (b)
da
w
20mm 100nm

(c)
OPE

C8

C8 º S º S S OPE
Fig. 3.17 Contacting molecular networks with gold nanoparticles. (a) Electron mi-
croscopy image of a device: two square-shaped gold contacts were evaporated on top
of a nanoparticle array line of width w. (b) Electron micrographs of the array struc-
ture before and after OPE (oligo-phenylene-ethynylene) exchange. (c) Schematic of the
molecular-exchange process. Left: self-assembled alkanethiol-capped nanoparticles be-
fore exchange. Right: During the exchange process. The OPE molecules displace part of
the alkane chains and interlink neighboring nanoparticles to form a network of molecular
junctions. Adapted with permission from [133]. Copyright Wiley-VCH Verlag GmbH &
Co. KGaA.

are covered with a spherical ligand shell. The thickness of the ligand shell
corresponds to half the length of the molecules which shall be assembled
between the GNPs later. A dense-packed, well-ordered, two-dimensional
array with an approximate size of 10 µm × 20 µm of these dressed GNPs
is deposited onto a substrate which is subsequently patterned with metallic
electrodes for performing the contacts to the measurement circuit. The ar-
ray contains approximately a million nanoparticles. The molecules forming
the ligand shell can be replaced with an exchange reaction by the molecules
to be studied electrically. By using network analysis methods the typical
properties of an individual molecular junction can be at least partially de-
duced from the behavior of the network. Besides the particular stability
and in-built ensemble averaging, this method is suitable for the investiga-
tion of very small signals, such as electrical response to optical activation
of photochromic molecules [135], see section 20.7.
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Contacting single molecules: Experimental techniques 73

3.6 Exercises

3.1 Molecular ensembles: Estimate the number of alkanedithiol molecules in

the cross section of a nanopore with a diameter of 50 nm (see section 3.5.1). For
estimating the diameter of a molecule assume a C-C bond length of 0.15 nm, a
C-H bond length of 0.11 nm and a bonding angle of 110o between adjacent C-C
bonds. Furthermore, assume a densely packed SAM in a triangular arrangement.
3.2 Molecular arrays: Let us consider the technique shown in Fig. 3.17. As-
sume that the exchange reaction was perfect. Furthermore, assume that each
pair of nanoparticles is connected via a single molecule. How many molecules
will contribute to the transport if the array has a size of 20 µm × 10 µm. What
is the effective circuit diagram of this network? What happens when the exchange
reaction has a yield of 50%? What is the minimum rate for the exchange reaction
in order to obtain at least one conducting path between the ends of the array
(percolation threshold)?
3.3 Optical activation of molecules: In Chapter 20 we will present experi-
ments in which molecular contacts were excited by light irradiation. Therefore, we
want to estimate here the probability that a molecule in contact with metal elec-
trodes will be hit by a photon of the light source. Assume a single decanedithiol
molecule which spans the gap between two gold electrodes. The electrodes have
been fabricated with the MCBJ technique and have a cross section of 100 nm
times 100 nm. The break forms a slit with perfectly flat walls perpendicular to
the direction of light irradiation. The width of the slit is given by the length of
the molecule. Typical light intensities of the experiments are P = 1 mW focused
on an area of s = 100 µm2 with a light wavelength of λ = 400 nm. Consider
different positions of the molecule in the slit: (a) Top of the slit. (b) Center of
the slit. (c) Bottom of the slit.
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74 Molecular Electronics: An Introduction to Theory and Experiment

January 12, 2010 11:27 World Scientific Book - 9in x 6in book

PART 2

Theoretical background

75
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76
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Chapter 4

The scattering approach to

phase-coherent transport in
nanocontacts

4.1 Introduction

The electrical conduction in macroscopic metallic wires is described by

Ohm’s law, which establishes that the current is proportional to the ap-
plied voltage. The constant of proportionality is simply the conductance,
G, which for a given sample grows linearly with the transverse area S and
it is inversely proportional to its length L, i.e.
S
G=σ , (4.1)
L
where σ is the conductivity of the sample, which is a material specific
property. The conductance will be a key quantity in our analysis of the
transport properties of atomic and molecular junctions. However, concepts
like Ohm’s law are not applicable at the atomic scale. Atomic-size conduc-
tors are a limiting case of mesoscopic systems in which quantum coherence
plays a central role in the transport properties.
In mesoscopic systems one can identify different transport regimes ac-
cording to the relative size of various length scales. These scales are, in turn,
determined by different scattering mechanisms. A fundamental length scale
is the phase-coherence length, Lϕ , which measures the distance over which
the information about the phase of the electron wave function is preserved.
Phase coherence can be destroyed by inelastic scattering mechanisms such
as electron-electron and electron-phonon interactions. Scattering of elec-
trons by magnetic impurities, with internal degrees of freedom, also de-
grades the phase but elastic scattering by (static) non-magnetic impurities
does not affect the coherence length. Information on the coherence length
can be obtained experimentally, for instance, by studying the so-called weak
localization [50]. A typical value for Au at T = 1 K is around 1 µm [136],

77
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78 Molecular Electronics: An Introduction to Theory and Experiment

diffusive ballistic

Fig. 4.1 Schematic illustration of a diffusive (left) and ballistic (right) conductor.

while at room temperature it becomes of the order of a few tens of nm.

The mesoscopic regime is determined by the condition L < Lϕ , where L is
a typical length scale of our sample.
Another important length scale is the elastic mean free path ℓ, which
roughly measures the distance between elastic collisions with static impu-
rities. The regime ℓ ≪ L is called diffusive. In a semi-classical picture
the electron motion in this regime can be viewed as a random walk of step
size ℓ among the impurities. On the other hand, when ℓ > L we reach
the ballistic regime in which the electron momentum can be assumed to be
constant and only limited by scattering with the boundaries of the sample.
These two regimes are illustrated in Fig. 4.1.
In the previous discussion we have implicitly assumed that the typical
dimensions of the sample are much larger than the Fermi wavelength λF .
However, when dealing with atomic-scale junctions the contact width W is
of the order of a few nanometers or even less and thus we have W ∼ λF .
We thus enter into the full quantum limit which cannot be described by
semi-classical arguments. A main challenge for the theory is to derive the
conductance of an atomic-scale conductor from microscopic principles.
In this chapter we shall introduce the scattering (or Landauer) approach
, which is presently the most popular theoretical formalism to describe the
coherent transport in nanodevices. The central idea of this approach, al-
ready put forward by Rolf Landauer in the late 1950’s [137], is that if one
can ignore inelastic interactions, a transport problem can always be viewed
as a scattering problem. This means in practice that transport proper-
ties like the electrical conductance are intimately related to the transmis-
sion probability for an electron to cross the system. Our introduction to
the scattering approach will be divided into two main parts. First, using
heuristic arguments we shall show the relation between conductance and
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

The scattering approach 79

transmission, which is summarized in the so-called Landauer formula. This

formula will then be used to discuss basic concepts such as the tunnel effect
or resonant tunneling. Second, we shall present a more rigorous formulation
of this approach that will be used to compute various transport properties
such as shot noise and thermoelectric coefficients. Finally, we shall conclude
this chapter with a discussion of the limitations of the scattering formalism.

4.2 From mesoscopic conductors to atomic-scale junctions

On the basis of Ohm’s law one would expect the conductance of a metallic
wire to scale as R2 , where R is its radius. Deviations from such a scaling law
were already discussed by Maxwell [138], who studied with classical argu-
ments the conductance of a diffusive constriction, where the contact radius
is large compared to the mean free path. He found that the conductance
scales linearly with the contact radius, i.e.
G = 2Rσ. (4.2)
where σ is the conductivity.
As we shrink a conductor to well below the mean free path, the con-
ductance departs from the value expected from the previous expression. In
1965 Sharvin [139] considered the propagation of electrical current through
a ballistic contact by approximating it with a classical problem of dilute
gas flow through an orifice. He reasoned that if the potential difference be-
tween the two half-spaces is eV , the conduction electrons passing through
the orifice should change their velocity by the amount ∆v = ±eV /pF , where
pF is the Fermi momentum.1 The net current will be I = ne∆vS, where
S = πR2 is the contact area and taking into account the Fermi-Dirac statis-
tics for electrons, n = 4πp3F /(3h3 ), one gets the conductance for a circular
ballistic point-contact
¶2 ¶2
2e2 πR 2e2 kF R
µ µ
G= = , (4.3)
h λF h 2
where e is the electron charge and h is the Planck’s constant. Notice that
for ballistic contacts the conductance is proportional to the contact area,
like in Ohm’s law, but the proportionality constant 2e2 /h has a quantum
nature. An important difference between the two lies in the fact that G is
1 This is just an approximation and the exact treatment includes an integration of

the projection of ∆v along the orifice axis over the solid angle of 2π. Anyway, the
phenomenological result is only a factor 8/3 different from the exact one [140].
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80 Molecular Electronics: An Introduction to Theory and Experiment

independent of the length of the conductor and is determined only by its

cross-section radius R. It is remarkable that the Sharvin formula, being
based on semiclassical arguments, holds well for all ballistic contacts with
diameters down to a few nanometers. In the context of atomic contacts, it
is customary to use a slightly modified version of this equation in which the
so-called Weyl correction is introduced [141, 142]. This correction comes
from the fact that the Heisenberg uncertainty principle for Fermi electrons
in a narrow contact, 2pF R ≥ ~, gives a small correction to the conductance
and the resulting semiclassical formula takes the form
¶2 µ
2e2
µ ¶
kF R 2
G= 1− + ··· , (4.4)
h 2 kF R

where kF is the wave vector. This equation is valid for a contact in the
form of a wire. For an orifice the numerator of the last fraction should
be 1 instead of 2. Eq. (4.4), valid for contacts down to a few nanometers
in diameter [143], is often used to establish the relationship between the
conductance and the radius of a contact.
Due to limitations of the semiclassical approach, Eq. (4.4) does not
account for purely quantum effects which dominate when the size of the
contact becomes so small that the wave nature of an electron can no longer
be ignored. Rolf Landauer [137] showed, already back in the 1950’s, that
in the latter case “conductance is transmission”, i.e. in order to determine
the total conductance one has to solve the Schrödinger equation, find the
current-carrying eigenmodes, calculate their transmission values and sum
up their contributions. Mathematically, this is summarized by in the Lan-
dauer formula
N
2e2 X
G= Tn , (4.5)
h n=1

where the summation is performed over all available conduction modes and
Tn are their individual transmissions. If the transmission of a mode is per-
fect, it contributes exactly one quantum unit of conductance, G0 = 2e2 /h ∼
(12.9 kΩ)−1 . This formula shows that by changing the size of the contact,
one can change the number of modes contributing to the conductance and
thus the conductance itself in a step-like manner (see discussion below).
This is clearly at variance with the situations described above. The deriva-
tion of the Landauer formula and the discussion of its physical implications
is the subject of the rest of the next sections.
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The scattering approach 81

ikx −ikx
1111111111111111111111111111111111111111111
0000000000000000000000000000000000000000000
e + re
0000000000000000000000000000000000000000000
1111111111111111111111111111111111111111111
0000000000000000000000000000000000000000000
1111111111111111111111111111111111111111111
0000000000000000000000000000000000000000000
1111111111111111111111111111111111111111111
0000000000000000000000000000000000000000000
1111111111111111111111111111111111111111111
0000000000000000000000000000000000000000000
1111111111111111111111111111111111111111111
teikx
0000000000000000000000000000000000000000000
1111111111111111111111111111111111111111111
0000000000000000000000000000000000000000000
1111111111111111111111111111111111111111111
0000000000000000000000000000000000000000000
1111111111111111111111111111111111111111111
0000000000000000000000000000000000000000000
1111111111111111111111111111111111111111111
0000000000000000000000000000000000000000000
1111111111111111111111111111111111111111111
0000000000000000000000000000000000000000000
1111111111111111111111111111111111111111111
0000000000000000000000000000000000000000000
1111111111111111111111111111111111111111111
2 x
T = |t|
Fig. 4.2 Wave function (plane wave) impinging on a potential barrier. The wave is
partially reflected with a probability amplitude r and partially transmitted with a prob-
ability T = |t|2 .

4.3 Conductance is transmission: Heuristic derivation of

the Landauer formula

In a typical transport experiment on a nanoscale device, the sample is

connected to macroscopic electrodes by a set of leads (or electrodes) which
allow us to inject currents and fix voltages. The electrodes act as ideal
electron reservoirs in thermal equilibrium with a well-defined temperature
and chemical potential. The basic idea of the scattering approach is to relate
the transport properties with the transmission and reflection probabilities
for carriers incident on the sample. In this one-electron approach phase-
coherence is assumed to be preserved on the entire sample and inelastic
scattering is restricted to the electron reservoirs only. Instead of dealing
with complex processes taking place inside the reservoirs, they enter into
the description as a set of boundary conditions. In spite of its simplicity,
this approach has been very successful in explaining many experiments on
nanodevices.
Before turning to the description of the general scattering formalism, it
is instructive to understand the relation between current and transmission
with a simple heuristic argument. Let us consider a one-dimensional situ-
ation, like the one depicted in Fig. 4.3. Here, the potential simulates the
central part of a junction, where electrons are elastically scattered before
reaching one of the electrodes. We assume that when the electrons are in-
side the reservoirs, they are in thermal equilibrium at the temperature
√ of
ikx
the corresponding electrode. Let us now consider a plane wave, (1/ L)e ,
that is impinging on the potential barrier from the left (L represents the
length of the system). This wave is partially reflected with a probability
amplitude r and partially transmitted with a probability T = |t|2 . We
can now compute the electrical current density, Jk , carried by an electron
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82 Molecular Electronics: An Introduction to Theory and Experiment

described by this wave function. It is given by the quantum-mechanical

expression
dψ ∗
· ¸
~ ∗ dψ e
Jk = ψ (x) − ψ(x) = v(k)T (k), (4.6)
2mi dx dx L
where v(k) = ~k/m is the group velocity and we have computed the current
on the right hand side of the scattering potential (remember that the current
is conserved and thus its value is independent of where it is evaluated).
In a solid state device there are many electrons contributing to the
current. Therefore, we have to introduce a sum over k (strictly speak-
ing over the positive values). Moreover, we have to take into account the
Pauli principle, which means in practice that we have to introduce a factor
fL (k)[1 − fR (k)], where fL,R is the Fermi function of the electron reservoir
on the left (L) or on the right (R) of the potential barrier. These Fermi
functions take also into account the fact that the corresponding chemical
potential can be shifted by an applied bias voltage, V . The blocking factor
above ensures that only those states that were initially occupied on the left
and empty on the right contribute to the current flowing from left to right,
JL→R , which adopts the form
eX
JL→R = v(k)T (k)fL (k)[1 − fR (k)]. (4.7)
L
k

Now, we can convert theRsum into an integral with the usual replacement:
P
(1/L) k g(k) → 1/(2π) g(k)dk. Thus,
e
Z
JL→R = dk v(k)T (k)fL (k)[1 − fR (k)]. (4.8)
2π
We now change from the variable k to energy, E, introducing the density
of states dk/dE = (dE/dk)−1 = m/(~2 k), since E = ~2 k 2 /(2m).2 Due to
the cancellation between the group velocity and the density of states, the
left-to-right current can be written as
e
Z
JL→R = dE T (E)fL (E)[1 − fR (E)]. (4.9)
h
Analogously, we can show that the current from right to left can be
written as
e
Z
JR→L = dE T (E)fR (E)[1 − fL (E)], (4.10)
h
2 Here, we are assuming that the conduction electrons can be described by a non-
interacting electron (or Fermi) gas.
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The scattering approach 83

where we have used the fact that the transmission probability is the same,
no matter in which direction the barrier is crossed.
Now, the total current3 I(V ) = JL→R − JR→L can be simply expressed
as
2e ∞
Z
I(V ) = dE T (E)[fL (E) − fR (E)]. (4.11)
h −∞

Here, we have introduced an extra factor 2 to account for the spin degener-
acy that usually exists in the systems that we shall analyze. This expression
is the simplest version of the so-called Landauer formula and it illustrates
the close relation between current and transmission. At zero temperature
fL (E) and fR (E) are step functions, equal to 1 below EF + eV /2 and
EF − eV /2, respectively, and 0 above this energy. If we moreover assume
low voltages (linear regime), this expression reduces to I = GV , where the
conductance is G = (2e2 /h)T , where the transmission is evaluated at the
Fermi energy.
This simple calculation demonstrates that a perfect single mode conduc-
tor between two electrodes has a finite resistance, given by the universal
quantity h/2e2 ≈ 12.9 kΩ. This is an important difference with respect to
macroscopic leads, where one expects to have zero resistance for the per-
fectly conducting case. The proper interpretation of this result was first
pointed out by Imry [144], who associated the finite resistance with the
resistance arising at the interfaces between the leads and the sample.

4.4 Penetration of a potential barrier: Tunnel effect

As it is clear from Eq. (4.11), the transmission probability plays a central

role in Landauer approach. For this reason, it is worth reminding how this
quantum mechanical quantity can be computed in some simple situations of
special interest. For the sake of concreteness, we shall focus our discussion
in this section on the analysis of the transmission through a single potential
barrier. This simple problem not only illustrates some fundamental issues,
but it also provides a basic model widely used for the understanding of
tunneling currents in a great variety of situations such as tunnel junctions
based on insulating barriers, STM and even single-molecule junctions, as
we shall show later in this book.
3 Since we are in a 1D situation, there is no difference between total current and current
density.
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84 Molecular Electronics: An Introduction to Theory and Experiment

V(x)
V0
I II III

0 L x

Fig. 4.3 Rectangular potential barrier of height V0 and width L.

Let us consider the potential barrier of height V0 depicted in Fig. 4.3.

Our goal is to compute the probability to cross such a barrier as a function
of the energy, E, of an incoming electron. Classical mechanics tell us that
an incident particle will always be reflected when E < V0 , and it will always
be transmitted when E > V0 . We all know that in quantum mechanics a
particle can pass through a barrier, even when its energy is lower than the
barrier height. This phenomenon is known as quantum tunneling or simply
tunnel effect and it lies at the heart of the whole physics discussed in this
book.
In order to compute the transmission we proceed in the standard way.
We first determine the wave functions in the three different regions defined
in Fig. 4.3, and then we match these functions and their first spatial deriva-
tives at the boundaries (x = 0 and x = L). Let us first consider the case of
E < V0 . In this case, the solutions of the Schrödinger equation in the three
regions are of the form

ψI = a1 eik1 x + b1 e−ik1 x , ψII = a2 ek2 x + b2 e−k2 x , ψIII = a3 eik3 x , (4.12)

where
√ p
2mE 2m(V0 − E)
k1 = k3 = and k2 = . (4.13)
~ ~
Note that we have assumed that the effective mass is the same everywhere
and we have discarded the incoming term (b3 e−ik3 x ) in ψIII because we are
considering here the problem of an a wave function impinging on the barrier
from the left.
Using now the continuity of the wave function and its first derivative at
x = 0 and x = L, we arrive at the following relationships

a1 + b1 = a2 + b2 ; ik1 a1 − ik1 b1 = k2 a2 − k2 b2 (4.14)

k2 L −k2 L ik1 L k2 L −k2 L ik1 L
a2 e + b2 e = a3 e ; k2 a2 e − k2 b2 e = ik1 a3 e .
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The scattering approach 85

1 1
0.8
(a) (b)
0.01
T(E)

0.6 1
0.001 0.0001
0.4 1e-06 E = 1 eV
1e-06 E = 2 eV
0.2 1e-09 E = 3 eV
0 0.2 0.4 0.6 0.8 1
0 1e-08
0 1 2 3 4 0.2 0.4 0.6 0.8 1
E/V0 L (nm)
Fig. 4.4 (a) Transmission probability vs. energy for a symmetric potential barrier of
height V0 = 4 eV and width L = 1 nm. The inset shows a blow-up of the region E < V0 .
(b) Transmission as a function of the width of the potential barrier (V0 = 4 eV) for
different values of the energy. In both cases the mass is assumed to be the electron mass.

Solving these equations, we obtain the following expression for the energy
dependence of the transmission coefficient
¯ ¯2
¯ a3 ¯ 1 4E(V0 − E)
T = ¯¯ ¯¯ = ³ 2 2 ´2 = .
a1 k1 +k2
1 + 2k sinh2
(k L) 4E(V0 − E) + V02 sinh2 (k2 L)
1 k2
2
(4.15)
Proceeding in a similar way, one can compute the transmission for E >
V0 and the result is (see Exercise 4.2)
1 4E(E − V0 )
T = ´2 = . (4.16)
4E(E − V0 ) + V02 sin2 (k2 L)
³
k12 −k22
1+ 2k1 k2 sin2 (k2 L)
The energy and length dependence of the transmission of this potential
barrier are illustrated in Fig. 4.4. The most prominent feature is maybe
the exponential dependence of the transmission on the barrier width for
energies E < V0 , see Fig. 4.4(b). According
p to Eq. (4.15), this decay is
given by T ∝ exp(−2k2 L) = exp(−2L 2m(V0 − E)/~), i.e. the slopes
in Fig. 4.4(b) are mainly determined by the square root of the difference
between the electron energy and the barrier height. Since the transmission
determines the conductance, this model provides a natural explanation for
the exponential decay of the low-bias conductance as a function of the
distance between the electrodes in all kind of tunnel barriers. It also tells
us that such decay is simply governed by the work function of the metals
involved.
Landauer formula shows that the linear conductance at low tempera-
tures is determined by the transmission at the Fermi energy. However, the
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86 Molecular Electronics: An Introduction to Theory and Experiment

analysis of the current-voltage (I-V) characteristics requires the knowledge

of the energy dependence and, strictly speaking, also of the voltage de-
pendence of the transmission probability, see Eq. (4.11). In the case of a
rectangular barrier, the voltage can be introduced in an approximate way
as shown in Fig. 4.5(a). The computation of the transmission and in turn of
the I-V curves is then a simple problem, see Exercise 4.3. A more appropri-
ate way of describing the effect of the voltage is shown in Fig. 4.5(b), where
a linear drop in the potential with the barrier region has been assumed.

(a) (b) eV

eV eV
Fig. 4.5 Rectangular potential barrier under the application of a voltage: (a) approxi-
mation and (b) actual potential profile.

The analysis of the transmission through a potential like the one of

Fig. 4.5(b), or any other smooth barrier, can be tackled with the help of
the WKB approximation [145, 146] (see Exercise 4.4). This is precisely
what Simmons did in 1963 [147] in his celebrated model. He considered the
problem of the tunnel effect between metallic electrodes separated by a thin
insulating film. He derived a general formula for the I-V curves for a barrier
of arbitrary shape, and we reproduce here his result for the particular case of
a rectangular barrier. Simmons showed that zero-temperature net current
density in this case can be written as [147]
n √ p o
J = J0 ϕB exp(−A ϕB ) − (ϕB + eV ) exp(−A ϕB + eV ) , (4.17)
where ϕB is the average barrier height relative to the negative electrode
and sB is the barrier width sB , see Fig. 4.6. Moreover,
2αsB √ e
A= 2m and J0 = , (4.18)
~ 2πhα2 s2B
where α is a dimensionless correction factor of order unity. Eq. (4.17) can
be simplified in three distinct cases depending on the applied voltage:
Low-voltage range. For very small voltages (eV ∼ 0), see Fig. 4.6(a),
the average barrier height ϕB is independent of the applied voltage and
equals the zero voltage barrier height ϕ0 = (ϕ1 + ϕ2 )/2. Then, Eq. (4.17)
can be simplified into
√
e2 2mϕB √
J = JL V with JL = exp(−A ϕB ). (4.19)
4π 2 α~2 sB
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The scattering approach 87

(a) (b) (c)

ϕ
1 sΒ ϕΒ ϕ ϕ ϕ
Β
ϕ
2 1
sΒ ϕ 1 ϕ
2 Β

Metal 1 Metal 2 eV sΒ
ϕ eV
Insulator 2

Fig. 4.6 Tunneling through a junction in which two metallic electrodes are separated
by a thin insulating film, which is modeled as a rectangular potential barrier. The three
panels show the three distinct voltage ranges discussed in the text.

Here, α = 1. As it can be seen in Eq. (4.19), the current density is a linear

function of the applied voltage V (Ohmic regime).
Intermediate-voltage range. For a medium applied voltage eV < ϕ0 ,
see Fig. 4.6(b), the average barrier height ϕB is given by (ϕ1 + ϕ2 − eV )/2.
The current density can then be simplified to (assuming that α = 1)
(Ae)2 Ae2
J = JL (V + γV 3 ) with γ = − 3/2
. (4.20)
96ϕ0 32ϕ0
This expression can be used to determine both the height and the barrier
width in terms of the coefficients γ and JL .
High-voltage range. For voltages eV > ϕ0 , see Fig. 4.6(c), the aver-
age barrier height is reduced to ϕ1 /2 and even the barrier width is reduced.
Eventually, the voltage is high enough so that the Fermi level of electrode
2 is lower than the conduction band of electrode 1. In this case, tunneling
from electrode 2 in electrode 1 is not possible since there are no empty
states in electrode 1 to tunnel to. As for electrons tunneling from electrode
1 into electrode 2, all states in electrode 2 are empty. This is analog to
field emission from a metal into vacuum. Then, the current density can be
simplified to
à √ 3/2
!
2.2e3 F 2 8π 2mϕ1
J= exp − , (4.21)
8πh ϕ1 2.96ehF

with the electric field strength in the insulator F = V /s, where s is the
thickness of the insulating field.
In the case of vacuum tunneling (or tunneling through an insulator), we
should be aware of the fact that whilst the electron is in the tunnel gap,
it will induce image charges in the two electrodes. This serves to modify
the barrier potential. The net effect of this is to reduce the average barrier
height and hence increase the transmission probability. For an analysis of
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88 Molecular Electronics: An Introduction to Theory and Experiment

V(x)

region 1 region 2

−L/2 0 +L/2 x

Fig. 4.7 The potential V (x) under consideration varies in an arbitrary way within the
interval −L/2 ≤ x ≤ +L/2 and goes to zero outside this interval.

these “image forces” for the case of the rectangular barrier discussed here,
see Ref. [147].
It is worth mentioning that the problem of the rectangular barrier under
an applied voltage, see Fig. 4.5(b), can be solved exactly using the full Airy
functions. This was done by Grundlach [148], who showed that the current
exhibits oscillations as a function of voltage that are superimposed in the
WKB result discussed above.

4.5 The scattering matrix

In the next section we shall present a more rigorous discussion of the scat-
tering formalism, where the concept of scattering matrix plays a central
role. The definition and properties of this matrix are described in many
quantum mechanics textbooks, but for the sake of completeness, we have
included here a brief discussion of this subject.

4.5.1 Definition and properties of the scattering matrix

In order to keep our discussion at a simple level, we study here a one-
dimensional situation. Let us consider a potential V (x) which is zero out-
side the region defined by |x| > L/2, but which varies in an arbitrary
way inside this interval, see Fig. 4.7. The equation satisfied by every wave
function ψ(x) associated with a stationary state of energy E is
½ 2 ¾
d 2m
+ [E − V (x)] ψ(x) = 0. (4.22)
dx2 ~2
The most general solution ψ(x) of Eq. (4.22) in the region x < −L/2
(region 1) for a given value of E can be written as
ψk (x) = a1 eikx + b1 e−ikx , (4.23)
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The scattering approach 89

p
where k = 2mE/~2 , while in the region x > +L/2 (region 2) it has the
form
ψk (x) = a2 e−ikx + b2 eikx , (4.24)
Here, the different coefficients depend on k, as well as on the shape of the
potential under study. Notice that with our notation, the amplitudes ai
(i = 1, 2) correspond to the incoming waves impinging on the potential
region, whereas the amplitudes bi correspond to the outgoing waves.
The scattering matrix is defined as the 2 × 2 matrix that relates the
incoming and outgoing amplitudes as follows
µ ¶ µ ¶
b1 a1
= Ŝ , (4.25)
b2 a2
where Ŝ is usually written as
r t′
µ ¶
Ŝ = . (4.26)
t r′
Here, r and r′ are reflection amplitudes and t and t′ are the transmission
amplitudes associated to this potential.
Are all these four elements independent? What are the properties of the
scattering matrix? A first property of the S-matrix can be deduced from
the conservation of the current. Let us remind that in quantum mechanics,
the current associated with a wave function ψ(x) is given by
dψ ∗
· ¸
~ dψ
J(x) = ψ ∗ (x) − ψ(x) . (4.27)
2mi dx dx
Differentiating, we find
d2 ψ d2 ψ ∗
· ¸
d ~ ∗
J(x) = ψ (x) 2 − ψ(x) 2 . (4.28)
dx 2mi dx dx
Taking into account Eq. (4.22), we obtain
d
J(x) = 0. (4.29)
dx
Therefore, the current J(x) associated with a stationary state is the same
at all points of the x-axis. Note, moreover, that Eq. (4.29) is simply the
one-dimensional analog of the relation (continuity equation)
∇ · J(r) = 0, (4.30)
which is valid for any stationary state of a particle moving in three-
dimensional space. According to Eq. (4.29), the current J(x) has the same
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90 Molecular Electronics: An Introduction to Theory and Experiment

value, no matter in which region it is evaluated. Then, computing the

current in regions 1 and 2 we have
~k £ ¤ ~k £ 2
|a1 |2 − |b1 |2 = |b2 | − |a2 |2 ,
¤
J(x) = (4.31)
m m
which implies that
|a1 |2 + |a2 |2 = |b1 |2 + |b2 |2 . (4.32)
This relation can be used to establish the first property of the scattering
matrix in the following way
µ ¶ µ ¶
b a1
|b1 |2 + |b2 |2 = (b∗1 , b∗2 ) 1 = (a∗1 , a∗2 ) Ŝ † Ŝ =
b2 a2
µ ¶
∗ ∗ a1
(a1 , a2 ) = |a1 |2 + |a2 |2 , (4.33)
a2

which simply implies that Ŝ is a unitary matrix, i.e.

Ŝ † = Ŝ −1 . (4.34)
In terms of the matrix elements, this relation reads
|r|2 + |t|2 = 1 ; r∗ t′ + t∗ r′ = 0
(t′ )∗ r + (r′ )∗ t = 0 ; |r′ |2 + |t′ |2 = 1. (4.35)
Notice that the second and third relations are indeed the same.
If the potential V (x) is real, which means in particular that there is no
magnetic field applied, an additional property can be derived as follows. If
ψ(x) is a solution of Eq. (4.22), then ψ ∗ (x) is also a solution. This new
solution can be written as
ψ ∗ (x) = a∗1 e−ikx + b∗1 eikx if x < −L/2
∗
ψ (x) = a∗2 eikx + b∗2 e−ikx if x > +L/2.
Notice that in this solution the coefficients a∗i correspond to the outgoing
amplitudes, while b∗i represent the incoming amplitudes. Therefore, by
definition they are related via the scattering matrix as follows
µ ∗¶ µ ∗¶
a1 b1
= Ŝ , (4.36)
a∗2 b∗2
which can be rewritten as
µ ¶ µ ¶
b1 ∗ −1 a1
= (Ŝ ) , (4.37)
b2 a2
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The scattering approach 91

If we now compare this relation with Eq. (4.25), we arrive at

(Ŝ)−1 = Ŝ ∗ . (4.38)
If we now combine this with the fact that the scattering matrix is unitary,
we have that Ŝ is symmetric
(Ŝ)T = Ŝ ⇒ t′ = t. (4.39)
In the presence of a magnetic field, this latter relation changes and one
can show that reversing the magnetic field B transposes the S-matrix
Ŝ(B) = Ŝ T (−B) ⇒ t′ (B) = t(−B). (4.40)
The demonstration is left to the reader as an exercise (see Exercise 4.7).

4.5.2 Combining scattering matrices

It is interesting to discuss how one can combine different scattering matri-
ces in a problem in which there are several scattering potentials. Let us
for instance consider the case of two potential barriers of arbitrary shape.
This situation is schematically represented in Fig. 4.8. We shall include in
the scattering matrix a superindex indicating to which potential barrier it
corresponds, Ŝ (i) (i = 1, 2). These matrices Ŝ (i) relate the incoming and
outgoing amplitudes across the corresponding potential barrier as follows
(see Fig. 4.8)
µ ¶ µ ¶ µ ¶ µ ¶
b1 (1) a1 a2 (2) b2
= Ŝ ; = Ŝ . (4.41)
b2 a2 b3 a3
Notice that we have already used the fact that a2 is at the same time
the incoming amplitude for the potential 1 and the outgoing amplitude for
potential 2. Something similar happens with b2 .
Our problem is to find in terms of the matrix elements of Ŝ (i) the total
scattering matrix ŜTot that relates the incoming and outgoing amplitudes
of the two scatterers, i.e.
µ ¶ µ ¶ µ ′¶
b1 a1 rt
= ŜTot ; ŜTot = . (4.42)
b3 a3 t r′
This can be easily done eliminating a2 and b2 from Eq. (4.41) and the
final result can be written as
h i−1 h i−1
r = r(1) + t′(1) r(2) 1 − r′(1) r(2) t(1) ; t = t(2) 1 − r′(1) r(2) t(1)
h i−1 h i−1
r′ = r′(2) + t(2) 1 − r′(1) r(2) r′(1) t′(2) ; t′ = t′(1) 1 − r(2) r′(1) t′(2) .
(4.43)
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92 Molecular Electronics: An Introduction to Theory and Experiment

V(x)
a1 b2 b3

b1 a2 a3
potential 1 potential 2

Fig. 4.8 Combination of two potential barriers of arbitrary shape. The coefficients ai
and bi represent the different incoming and outgoing amplitudes with respect to the
potential barrier i.

This result allows us to compute now very easily, for instance, the total
transmission through the combined structure. According to the previous
equations
T1 T2
T = |t|2 = √ , (4.44)
1 − 2 R1 R2 cos θ + R1 R2
where Ti = |t(i) |2 = |t′(i) |2 , Ri = |r(i) |2 = |r′(i) |2 and θ = phase(r′(1) ) +
phase(r(2) ) is the phase shift acquired in one round-trip between the scat-
terers.
This result can be used to study a very important phenomenon for us,
namely the resonant tunneling. In a double barrier system (or in a potential
well) one can have bound states in the region between the two scattering
centers. Then, the transmission probability in this system exhibits reso-
nances at energies close to the position of those bound states. The width
of the transmission peaks depends upon the transmissivity of the barriers,
while the distance between peaks is mainly determined by the distance be-
tween the barriers. These facts can be shown with the help of Eq. (4.44),
as it is illustrated in Exercise 4.8.

4.6 Multichannel Landauer formula

We present in this section a more rigorous derivation of Landauer formula,

where the important concept of conduction channel will arise. This for-
mulation will also be the starting point for the extension of the scattering
formalism to the description of other transport properties such as shot noise
or thermoelectric coefficients. This section is based on Refs. [149, 150] and
we refer the reader to them for more details.
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The scattering approach 93

We consider a mesoscopic sample connected to two reservoirs (terminals,

probes), to be referred to as “left” (L) and “right” (R). It is assumed that
the reservoirs are so large that they can be characterized by a temperature
TL,R and a chemical potential µL,R ; the distribution functions of electrons
in the reservoirs, defined via these parameters, are then Fermi distribution
functions
fα (E) = [exp[(E − µα )/kB Tα ] + 1]−1 , α = L, R (4.45)
(see Fig. 4.9). Far from the sample, we can assume that transverse (across
the leads) and longitudinal (along the leads) motion of electrons are sepa-
rable. In the longitudinal (from left to right) direction the system is open,
and is characterized by the continuous wave vector kl . It is advantageous
to separate incoming (to the sample) and outgoing states, and to introduce
the longitudinal energy El = ~2 kl2 /2m as a quantum number. Transverse
motion is quantized and described by the discrete index n (corresponding
to transverse energies EL,R;n , which can be different for the left and right
leads). These states are in the following referred to as transverse (quan-
tum) channels. We write thus E = En + El . Since El needs to be positive,
for a given total energy E only a finite number of channels exists. The
number of incoming channels is denoted NL,R (E) in the left and right lead,
respectively.

L ^
aL ^
aR
R

TL sample
TR
µL µR
^ ^
bL bR

Fig. 4.9 Two-terminal scattering problem for the case of one transverse channel.

We now introduce creation and annihilation operators of electrons in

the scattering states.4 In principle, we could have used the operators which
refer to particles in the states described by the quantum numbers n, kl .
However, the scattering matrix relates current amplitudes and not wave
function amplitudes. Thus, we introduce operators â†Ln (E) and âLn (E)
which create and annihilate electrons with total energy E in the transverse
4 The second quantization language will be used here at a very simple level. A discussion

of this formalism is included in Appendix A and it will be widely used in the following
chapters.
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94 Molecular Electronics: An Introduction to Theory and Experiment

channel n in the left lead, which are incident upon the sample.5 In the
same way, the creation b̂†Ln (E) and annihilation b̂Ln (E) operators describe
electrons in the outgoing states. They obey anti-commutation relations

â†Ln (E)âLn′ (E ′ ) + âLn′ (E ′ )â†Ln (E) = δnn′ δ(E − E ′ )

âLn (E)âLn′ (E ′ ) + âLn′ (E ′ )âLn (E) = 0
â†Ln (E)â†Ln′ (E ′ ) + â†Ln′ (E ′ )â†Ln (E) = 0. (4.46)

Similarly, we introduce creation and annihilation operators â†Rn (E) and

âRn (E) for incoming states and b̂†Rn (E) and b̂Rn (E) for outgoing states in
the right lead (Fig. 4.9).
The operators â and b̂ are related via the scattering matrix Ŝ,
   
b̂L1 âL1
 .   . 
 ..   .. 
   
 b̂LNL   âLNL 
   
 = Ŝ  . (4.47)
 b̂R1   âR1 

 ..   .. 
   
 .   . 
b̂RNR âRNR

The creation operators ↠and b̂† obey a similar relation with the Hermitian
conjugated matrix Ŝ † .
The matrix Ŝ has dimensions (NL + NR ) × (NL + NR ). Its size, as well
as the matrix elements, depends on the total energy E. It has the block
structure
µ ′¶
r̂ t̂
Ŝ = . (4.48)
t̂ r̂′

Here the square diagonal blocks r̂ (size NL × NL ) and r̂′ (size NR × NR )

describe electron reflection back to the left and right reservoirs, respectively.
The off-diagonal, rectangular blocks t̂ (size NR × NL ) and t̂′ (size NL ×
NR ) are responsible for the electron transmission through the sample. The
properties of the matrix Ŝ are a straightforward generalization to a multi-
mode case of those discussed in the previous section. Thus for instance, the
flux conservation in the scattering process implies that Ŝ is quite generally
unitary. In the presence of time-reversal symmetry the scattering matrix is
also symmetric.
5 We shall denote here the operators with a “hat” to distinguish them from the ampli-
tudes of the previous section.
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The scattering approach 95

The current operator in the left lead (far from the sample) is expressed
in a standard way,
· µ ¶ ¸
∂ ∂ †
Z
~e
IˆL (z, t) = dr⊥ Ψ̂†L (r, t) Ψ̂L (r, t) − Ψ̂L (r, t) Ψ̂L (r, t) ,
2im ∂z ∂z
(4.49)
where the field operators Ψ̂ and Ψ̂† are defined as
NL (E)
χLn (r⊥ )
Z X h i
−iEt/~ ikLn z −ikLn z
Ψ̂L (r, t) = dEe âLn e + b̂ Ln e
n=1
(2π~vLn (E))1/2
(4.50)
and
NL (E)
χ∗Ln (r⊥ )
Z h i
Ψ̂†L (r, t) = dEeiEt/~ â†Ln e−ikLn z + b̂†Ln eikLn z .
X
(2π~vLn (E)) 1/2
n=1
(4.51)
Here r⊥ is the transverse coordinate(s) and z is the coordinate along the
leads (measured from left to right), χL n are the transverse wave functions,
and we have introduced the wave vector, kLn = ~−1 [2m(E − ELn )]1/2
(the summation only includes channels with real kLn ), and the velocity of
carriers vn (E) = ~kLn /m in the n-th transverse channel.
After some algebra, the expression for the current can be cast into the
form6
eX
Z h i
IˆL (t) = dEdE ′ ei(E−E )t/~ â†Ln (E)âLn (E ′ ) − b̂†Ln (E)b̂Ln (E ′ ) .
′

h n
(4.52)
Using Eq. (4.47) we can now express the current in terms of the â and â†
operators alone,
e XX
Z
IˆL (t) =
′
dEdE ′ ei(E−E )t/~ â†αm (E)Amn ′
αβ (L; E, E )âβn (E ).
′
h mn
αβ
(4.53)
Here the indices α and β label the reservoirs and may assume values L or
R. The matrix A is defined as
X †
Amn ′
αβ (L; E, E ) = δmn δαL δβL − SLα;mk (E)SLβ;kn (E ′ ), (4.54)
k

and SLα;mk (E) is the element of the scattering matrix relating b̂Lm (E) to
âαk (E). Note that Eq. (4.53) is independent of the coordinate z along the
lead.
6 Here, we have used the fact that the velocities vn (E) vary with energy quite slowly,
typically on the scale of the Fermi energy, and neglected their energy dependence.
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96 Molecular Electronics: An Introduction to Theory and Experiment

Let us now derive the average current from Eq. (4.53). For a system at
thermal equilibrium the quantum statistical average of the product of an
electron creation operator and annihilation operator of a Fermi gas is
­ †
âαm (E)âβn (E ′ ) = δαβ δmn δ(E − E ′ )fα (E).
®
(4.55)
Using Eq. (4.53) and Eq. (4.55) and taking into account the unitarity of
the scattering matrix Ŝ, we obtain
e ∞
Z
dE Tr t̂† (E)t̂(E) [fL (E) − fR (E)] .
£ ¤
I ≡ hIL i = (4.56)
h −∞
Here the matrix t is the off-diagonal block of the scattering matrix, tmn =
SRL;mn . In the zero-temperature limit and for a small applied voltage
Eq. (4.56) gives a conductance
e2 £ † ¤
G= Tr t̂ (EF )t̂(EF ) , (4.57)
h
where EF is the Fermi energy. Eq. (4.57) establishes the relation between
the scattering matrix evaluated at the Fermi energy and the conductance.
It is a basis invariant expression. The matrix t̂† t̂ can be diagonalized;
it has a real set of eigenvalues (transmission coefficients) Tn (E) (not to be
confused with temperature), each of them assumes a value between zero and
one. The corresponding eigenfunctions will be referred to as eigenchannels
or conduction channels. In this natural basis we have instead of Eq. (4.56)
eX ∞
Z
I= dE Tn (E) [fL (E) − fR (E)] . (4.58)
h n −∞

and thus for the conductance

e2 X
G= Tn . (4.59)
h n

Eq. (4.59) is known as a multi-channel generalization of Landauer formula.

Notice also that in the last formulas there is a difference of a factor 2 with
respect to Eq. (4.11). The reason is that in the discussion above we have
not assumed spin degeneracy.
For a constriction of only one atom in cross section one can estimate
the number of conductance channels as N ≃ (kF R/2)2 , which is between
1 and 3 for most metals. We shall see that the actual number of channels
is determined by the valence orbital structure of the atoms. In the case of
molecular junctions, it turns out that, apart from a few notable exceptions,
the conductance is dominated by a single conduction channel.
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The scattering approach 97

(a) (b)
(b) ky
y
Gate voltage
kF
x
π/W

2DEG W 2DEG kx

Fig. 4.10 (a) Schematic representation of a point contact defined in a two-dimensional

electron gas (2DEG) by means of a split gate on top of the heterostructure. (b) Allowed
states in the point contact constriction, which correspond to quantized values for ky =
±nπ/W , and continuous values for kx . The formation of these 1D subbands gives rise
of a quantized conductance.

Let us emphasize that we have focused our discussion on a two-

terminal configuration. The scattering approach was extended by Büttiker
to describe the electronic transport in multi-terminal situations and this
formalism (generally referred to as Landauer-Büttiker’s formalism) has
been widely used in the interpretation of mesoscopic experiments. We
shall not discuss this generalization here and we refer the reader to
Refs. [50, 149, 150, 171] for more details about this formalism.

4.6.1 Conductance quantization in 2DEG: Landauer for-

mula at work
As a simple illustration of the use of Landauer formula, we shall now briefly
discuss the conductance quantization in quantum point contacts defined in
semiconductor hetero-structures (for a detailed discussion of this topic, see
Refs. [151, 152]). It is well-known that in a semiconductor heterostructure
like GaAs-AlGaAs one can confine the electrons in the two-dimensional
interface between the two materials. Additionally, one can define electro-
statically a point contact by means of a split gate on top of the heterostruc-
ture. This is schematically represented in Fig. 4.10(a). In this way one can
define short and narrows constrictions in the two-dimensional electron gas
(2DEG), of variable width 0 < W < 250 nm comparable to the Fermi
wavelength λF ≈ 40 nm and much shorter than the mean free path l ≈ 10
µm.
Van Wees et al. [153] and Wharam et al. [154] independently discov-
ered a sequence of steps in the conductance of such a point contact as its
width was varied by means of the voltage on the split gate (see Fig. 4.11).
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98 Molecular Electronics: An Introduction to Theory and Experiment

Fig. 4.11 Point contact conductance as a function of gate voltage at 0.6 K, demonstrat-
ing the conductance quantization in units of 2e2 /h. The constriction width increases
with increasing voltage on the gate (see inset). Reprinted with permission from [153].
Copyright 1988 by the American Physical Society.

The steps are near integer multiples of 2e2 /h, after correction for a gate-
voltage-independent series resistance from the wide 2DEG regions. This
phenomenon is referred to as conductance quantization.
An elementary explanation of this effect relies on two facts: (i) the
2DEGs are ballistic systems (at least along the constriction) and the only
scattering takes place against the potential walls defined by the split gates
and (ii) the momentum of the electron is quantized in the transverse direc-
tion giving rise to 1D subbands. Since every subband that contributes to
the transport (or conduction channel) has a perfect transparency and the
number of them is obviously an integer, it follows from the two-terminal
Landauer formula that the low temperature conductance G is quantized,
G = (2e2 /h)N, (4.60)
as observed experimentally. Here, N is the total number of open conduc-
tion channels and the prefactor 2 accounts for the spin degeneracy. This
number can be simply calculated assuming a square-well lateral confining
potential of width W . In the constriction, the electron momentum along
the transport direction (x-direction) can take any value, while the trans-
verse momentum ky is quantized and can only take the following values:
ky = ±nπ/W with n = 1, 2, ..., N , see Fig. 4.10(b). Since the current is only
carried by those electrons at the Fermi energy (or with momentum equal
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

The scattering approach 99

to the Fermi momentum kF ), the number of subbands is simply given by

N = Int[kF W/π]. Therefore, a new subband is made available for transport
every time the width of the gate is increased by approximately half of the
Fermi wavelength. This explains the stair-like behavior seen in Fig. 4.11.
A detailed explanation of the necessary conditions to observe the con-
ductance quantization requires a more rigorous treatment of the confine-
ment potential and the corresponding analysis of the mode coupling at the
entrance and exit of the constriction. A more realistic model is discussed
in Exercise 4.9.

4.7 Shot Noise

Shot noise is another important quantity for characterizing the transport

properties of nanoscale systems [150, 155]. It refers to the time-dependent
current fluctuations due to the discreteness of the electron charge. In a
mesoscopic conductor these fluctuations have a quantum origin, arising
from the quantum mechanical probability of electrons being transmitted or
reflected from the sample. In contrast to thermal noise, shot noise only
appears in the presence of transport, i.e. in a non-equilibrium situation.
Shot noise measurements provide information on temporal correlations
between the electrons. In a tunnel junction, where the electrons are trans-
mitted randomly and correlation effects can be neglected, the transfer of
carriers of charge q is described by Poisson statistics and the amplitude of
the current fluctuations is 2qI. In nanoscale conductors correlations may
suppress the shot noise below this value. Even when electron-electron inter-
actions can be neglected the Pauli principle provides a source for electron
correlations.
The relation between shot noise and the transmitted charge unit q has
allowed the detection of the carrier charge in exotic situations such as the
fractional quantum Hall regime [156, 157], where the charge can be frac-
tional and depends on the filling factor. It has also allowed to show that
the sub-gap transport in superconducting atomic contacts takes place in
big shots of multiple ne charges associated with multiple Andreev reflec-
tion processes [158, 159, 98].
The interest in shot noise in molecular electronics lies in the fact that
this quantity depends on the transmission coefficients in a nonlinear man-
ner. Thus, the shot noise can provide valuable information, not contained
in the conductance, about the number of conduction channels and their
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100 Molecular Electronics: An Introduction to Theory and Experiment

n(E) n(E)
1 Tn 1
Tn
E E
0 0
EF EF+eV EF EF+eV

Fig. 4.12 In a quantum point contact with bias voltage, V , the transmission probability,
Tn , determines the distribution function, n(E), of a transmitted state as a function of its
energy, E. In the right reservoir, states with energy lower than the Fermi energy are all
occupied, while right-moving states with higher energy can only be coming from the left
reservoir, and therefore their average occupation is equal to the transmission probability,
Tn . This argument applies to every individual conduction channel.

transmission coefficients. This will be discussed in detail in Chapter 19.

Qualitatively, the shot noise in nanocontacts can be understood from
the diagram in Fig. 4.12. Let us consider the right moving states in this
contact, which have been transmitted through the junction with an excess
energy between 0 and eV . Their average occupation number, n, is given
by the transmission probability Tn . For the fluctuations in this number we
find

∆n2 = n2 − n2 = Tn (1 − Tn ), (4.61)

where in the last step we used the fact that n2 = n, since for fermions n
is either zero or one. Hence, the fluctuations in the current are suppressed
both for Tn = 1 and for Tn = 0. According to Eq. (4.61) the fluctuations
will be maximal when the electrons have a probability of one half to be
transmitted. The shot noise is thus a non-linear function of the transmission
coefficients, as we anticipated above.
We shall now derive in a rigorous manner the main results concerning
shot noise in a two-terminal device within the scattering formalism. For
this purpose, we follow again Ref. [150]. Since are concerned with the
fluctuations of the current away from its average value, we then introduce
ˆ ≡ I(t)
the operators ∆I(t) ˆ − hIi, where Iˆ is the current operator evaluated
in a given reservoir, let us say, the left one. We define the correlation
function P (t − t′ ) of the current in a given contact as
1D ˆ ˆ ′ ) + ∆I(t
ˆ ′ )∆I(t)
ˆ
E
P (t − t′ ) ≡ ∆I(t)∆I(t . (4.62)
2
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The scattering approach 101

Note that in the absence of time-dependent external fields, as we assume

here, the correlation function must be function of only t − t′ . Its Fourier
transform,
D E
ˆ
2πδ(ω + ω ′ )P (ω) ≡ ∆I(ω)∆ ˆ ′ ) + ∆I(ω
I(ω ˆ ′ )∆I(ω)
ˆ , (4.63)
is sometimes referred to as noise power.
To find the noise power we need the quantum statistical expectation
value of products of four operators â. For a Fermi gas at equilibrium this
expectation value is
D E
â†αk (E1 )âβl (E2 )â†γm (E3 )âδn (E4 ) −
D E­
â†αk (E1 )âβl (E2 ) â†γm (E3 )âδn (E4 )
®

= δαδ δβγ δkn δml δ(E1 − E4 )δ(E2 − E3 )fα (E1 ) [1 − fβ (E2 )] . (4.64)
Here fα (E) is the corresponding Fermi distribution. Now, making use of
the current operator of Eq. (4.53) and of the expectation value of Eq. (4.64),
we arrive at the following expression for the noise power
e2 X X
Z
P (ω) = dEAmn nm
γδ (L; E, E + ~ω)Aδγ (L; E + ~ω, E)
h mnγδ
× {fγ (E) [1 − fδ (E + ~ω)] + [1 − fγ (E)] fδ (E + ~ω)} . (4.65)
Note that with respect to frequency, it has the symmetry properties P (ω) =
P (−ω). In the rest of this discussion, we shall only be interested in the
zero-frequency noise.7 For the noise power at ω = 0 we obtain
e2 X X
Z
P ≡ P (0) = dEAmn nm
γδ (L; E, E)Aδγ (L; E, E) (4.66)
h mnγδ
× {fγ (E) [1 − fδ (E)] + [1 − fγ (E)] fδ (E)} .
Eq. (4.66) can now be used to predict the low frequency noise properties of
arbitrary multi-channel phase-coherent conductors. But before presenting
the general result, let us first discuss two limiting cases of special interest:
Equilibrium noise. If the system is in thermal equilibrium at tem-
perature T , the distribution functions in both reservoirs coincide and are
equal to f (E). Using the property f (1 − f ) = −kB T ∂f /∂E and employing
the unitarity of the scattering matrix, one can arrive at the following result
P = 4kB T G, (4.67)
7 Zero-frequency noise actually means that the frequency is small in comparison with
the relevant frequency scales of the problem, but large enough to neglect the 1/f noise
that is present in almost any system.
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102 Molecular Electronics: An Introduction to Theory and Experiment

where G is the linear conductance given by

e2 ∞
µ ¶
∂f
Z
Tr t̂† (E)t̂(E) .
£ ¤
G= dE − (4.68)
h −∞ ∂E
This is the thermal, or Nyquist-Johnson noise. In the approach discussed
here it is a consequence of the thermal fluctuations of occupation numbers
in the reservoirs. This is the manifestation of the fluctuation-dissipation
theorem: equilibrium fluctuations are proportional to the corresponding
generalized susceptibility, in this case to the conductance.
Zero-temperature shot noise. In the zero-temperature limit the
Fermi distribution in each reservoir is a step function fα (E) = θ(µα − E).
Utilizing the representation of the scattering matrix (4.48), and taking into
account that the unitarity of the matrix Ŝ implies r̂† r̂ + t̂† t̂ = 1, after some
algebra we can rewrite Eq. (4.66) as
2e2
P = Tr (r̂† r̂t̂† t̂) e|V |, (4.69)
h
where the scattering matrix elements are evaluated at the Fermi level. Like
the expression of the conductance, Eq. (4.57), we can express this result in
the basis of eigenchannels with the help of the transmission probabilities
Tn and reflection probabilities Rn = 1 − Tn ,
2e3 |V | X
P = Tn (1 − Tn ) . (4.70)
h n

We see that the non-equilibrium (shot) noise is not simply determined by

the conductance of the sample. Instead, it is determined by a sum of prod-
ucts of transmission and reflection probabilities of the conduction channels.
Only in the limit of low-transparency Tn ≪ 1 in all conduction channels is
the shot noise given by the Poisson value, discussed by Schottky,
2e3 |V | X
P = Tn = 2ehIi. (4.71)
h n

It is clear that zero-temperature shot noise is always suppressed in com-

parison with the Poisson value. In particular, neither closed (Tn = 0) nor
open (Tn = 1) channels contribute to shot noise; the maximal contribution
comes from channels with Tn = 1/2. The suppression below the Poissonian
limit given by Eq. (4.71) was one of the aspects of noise in mesoscopic sys-
tems which triggered many of the subsequent theoretical and experimental
works. A convenient measure of sub-Poissonian shot noise is the Fano fac-
tor F , which is the ratio of the actual shot noise and the Poisson noise that
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The scattering approach 103

would be measured if the system produced noise due to single independent

electrons,
P
F = . (4.72)
2ehIi
For energy-independent transmission and/or in the linear regime the Fano
factor is
P
Tn (1 − Tn )
F = nP . (4.73)
n Tn

The Fano factor assumes values between 0 (all channels are transparent)
and 1 (Poissonian noise). In particular, for one channel it becomes (1 − T ).
The general result for arbitrary temperature and voltage for the noise
power of the current fluctuations in a two-terminal conductor is
2e2 X ∞
Z
P = dE { Tn (E) [fL (1 − fL ) + fR (1 − fR )] +
h n −∞
o
2
Tn (E) [1 − Tn (E)] (fL − fR ) . (4.74)

Here the first two terms are the equilibrium noise contributions, and the
third term is the non-equilibrium or shot noise contribution to the power
spectrum. Note that this term is second order in the distribution function.
At high energies, in the range where the Fermi distribution function is well
approximated by a Maxwell-Boltzmann distribution, it is negligible com-
pared to the equilibrium noise described by the first two terms. According
to Eq. (4.74) the shot noise term enhances the noise power compared to
the equilibrium noise.
In the practically important case, when the scale of the energy depen-
dence of transmission coefficients Tn (E) is much larger than both the tem-
perature and applied voltage, these quantities in Eq. (4.74) may be replaced
by their values taken at the Fermi energy. We obtain then
" #
2e2
µ ¶X
X eV
P = 2kB T Tn2 + eV coth Tn (1 − Tn ) , (4.75)
h n
2kB T n

where V is again the voltage applied between the left and right reservoirs.
The full noise is a complicated function of temperature and applied voltage
rather than a simple superposition of equilibrium and shot noise. For low
voltages eV ≪ kB T one recovers the result of pure thermal noise, i.e. P =
4kB T G. Eq. (4.75) is the starting point for the analysis of experimental
results on noise in atomic and molecular junctions, see section 19.1.
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104 Molecular Electronics: An Introduction to Theory and Experiment

4.8 Thermal transport and thermoelectric phenomena

The scattering formalism is by no means restricted to the description of

the electronic transport. It has also been extended to describe thermal
transport and thermoelectric cross-effects [160–163] and in what follows
we present a discussion of these transport properties within the scattering
approach.8
Let us consider a generic two-terminal device like in the previous sec-
tions. In equilibrium, the electron reservoirs are at chemical potential µ
and temperature T . In the regime of linear response, the current I and
heat flow Q are related to the chemical potential difference ∆µ and the
temperature difference ∆T by the constitutive equations
µ ¶ µ ¶µ ¶
I G L ∆µ/e
= . (4.76)
Q M K ∆T
The thermoelectric coefficients L and M are related by an Onsager relation,
which in the absence of a magnetic field is
M = −LT. (4.77)
Equation (4.76) is often re-expressed with the current I rather than the
electrochemical potential ∆µ as an independent variable,
µ ¶ µ ¶µ ¶
∆µ/e R S I
= . (4.78)
Q Π −κ ∆T
The resistance R is the reciprocal of the isothermal conductance G. The
thermopower S is defined as
µ ¶
∆µ/e
S≡ = −L/G. (4.79)
∆T I=0
The Peltier coefficient Π, defined as
µ ¶
Q
Π≡ = M/G = ST, (4.80)
I ∆T =0
is proportional to the thermopower S in view of the Onsager relation (4.77).
Finally, the thermal conductance κ is defined as
S 2 GT
µ ¶ µ ¶
Q
κ≡− = −K 1 + . (4.81)
∆T I=0 K
In order to compute all the thermoelectric coefficients, we still need to
determine the heat current, which in the spirit of the scattering formalism
8 Itis worth stressing that we shall only consider the contribution of the electrons to
the thermal transport properties. In general, phonons can also play an important role.
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

The scattering approach 105

will be expressed in terms of the transmission and reflections coefficients

of the system. Let us assume that the left electrode has a temperature T1 ,
while the right one has a temperature T2 . Following Ref. [160], the total
entropy current moving to the right on the left lead will be given by9
kB
Z
→
J1S = − [f1 ln f1 + (1 − f1 ) ln(1 − f1 )] dE, (4.82)
h
where f1 = f (E, µ1 , T1 ) denotes the Fermi function on the left electrode.
On the other hand, the entropy current going to the left on the same lead
is given by
kB
Z
←
J1S = − [(R11 f1 + T12 f2 ) ln(R11 f1 + T12 f2 )+
h
(1 − R11 f1 − T12 f2 ) ln(1 − R11 f1 − T12 f2 )] dE, (4.83)
where T12 ≡ Tr{t̂† t̂} is the total transmission of the contact and R11 ≡
Tr{r̂† r̂} is the corresponding reflection coefficient.
By subtracting (4.82) and (4.83) the following expression for the heat
current is obtained [160]
1
Z
Q1 = T J1S = T12 (E)(E − µ) [f1 − f2 ] dE, (4.84)
h
where T and µ are the average temperature and chemical potential.
Therefore, the thermoelectric coefficients are given in the scattering for-
malism by [160, 162]
2e2 ∞ ∂f
Z
G=− dE T12 (E), (4.85)
h −∞ ∂E
2e2 kB ∞ ∂f E−µ
Z
L=− dE T12 (E) , (4.86)
h e −∞ ∂E kB T
µ ¶2 Z ∞ ¸2
2e2 kB
·
K ∂f E−µ
= dE T12 (E) .
T h e −∞ ∂E kB T
(4.87)
These integrals are convolutions of T12 (E), which characterizes the conduc-
tor, and a kernel of the form ǫm df /dǫ, m = 0, 1, 2, with ǫ ≡ (E − µ)/kB T ,
and f the Fermi function f (ǫ) = [exp(ǫ) + 1]−1 . Both df /dǫ and ǫ2 df /dǫ are
symmetric functions of ǫ, which is why the conductance, G, and the thermal
conductances K and κ are determined to first order by T12 (µ). (The term
9 Notice that the expression in the square bracket is the entropy density of noninteracting

electrons, distributed according to an arbitrary non-equilibrium distribution function f1 ,

see pag. 54 of Ref. [164].
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106 Molecular Electronics: An Introduction to Theory and Experiment

within brackets in equation (4.81) is usually small.) In contrast, ǫdf /dǫ is

an antisymmetric function of ǫ, so that the thermoelectric cross-coefficients
L, S, M , and Π are determined mainly by the derivative dT12 (E)/dE at
E = µ. This is substantiated by a Sommerfeld expansion of the integrals
in Eqs. (4.85)-(4.87), valid for a smooth function T12 (E) to lowest order in
kB T /µ [162]
2e2
G≈ T12 (µ) (4.88)
h
2e2
µ ¶
dT12 (E)
L≈ L0 eT (4.89)
h dE E=µ
2e2
K≈− L0 T T12 (µ), (4.90)
h
2 2
with L0 ≡ (kB /e) π /3 the Lorentz number. In this approximation K =
−L0 T G, so that for S 2 ≪ L0 one finds from Eq. (4.81) the Wiedemann-
Franz relation: κ ≈ L0 T G.
Thermoelectrical effects have been experimentally studied in detail in
2DEG quantum point contacts by van Houten et al. [163]. In the context
of atomic and molecular junctions, special attention has been paid to the
thermopower. As we shall discuss in section 19.3, the thermopower con-
tains valuable information about these systems that is not contained in the
electrical conductance.

4.9 Limitations of the scattering approach

The scattering formalism has been very successful explaining many basic
transport phenomena in a great variety of nanostructures. It has also been
extended to other situations of interest for the purpose of this book, such
as e.g. photon-assisted transport [165]. For space reasons we have to end
here our discussion of this formalism, and for more details we recommend
the the reviews of Refs. [150, 151, 166–168], the didactic book of S. Datta
[50] and the book on mesoscopic physics of Y. Imry [169].
In spite of its great success, the scattering approach is far from being a
complete theory of quantum transport. In this sense, it is important to be
aware of its limitations. Among them we want to emphasize two of special
interest for the scope of this book:
(i) The scattering approach gives no hints on how to compute the trans-
mission or, more generally, the scattering matrix. In particular, it does not
tell us how to determine the actual transmission of an atomic contact or
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

The scattering approach 107

a molecular circuit. In this sense, one might think that this formalism
has merely replaced a problem by another. This would be, of course, un-
fair. The scattering approach can be combined with simple models, as we
showed in section 4.4, or with more sophisticated techniques like random
matrix theory [170] to predict the transport properties of a great variety
of systems such as diffusive wires, chaotic cavities, superconducting nanos-
tructures, resonant tunneling systems, tunnel junctions, etc.
(ii) The scattering picture is an one-electron theory which is valid only
as long as inelastic scattering processes can be neglected. In this formalism
one assumes that the electron propagation is a fully quantum coherent
process over the entire sample. According to normal Fermi-liquid theory,
such a description would be strictly valid at zero temperature and only
for electrons at the Fermi energy. At finite bias the coherent propagation
may be limited by inelastic scattering processes due to electron-phonon
and electron-electron collisions. The theoretical description of transport
in situations where inelastic interactions play an important role requires
more sophisticated methods like the Green’s function techniques that will
be described in the next chapters.
Let us mention that there is a phenomenological way of describing the
effect of inelastic or phase-breaking mechanisms within the scattering ap-
proach, which is due to Büttiker [171]. In this description the inelastic scat-
tering events are simulated by the addition of voltage probes distributed
over the sample. The chemical potential on these probes is fixed by im-
posing the condition of no net current flow through them. Thus, although
the presence of the probes does not change the total current through the
sample, they introduce a randomization of the phase which tends to destroy
phase coherence. The current in such a structure will contain a coherent
component, corresponding to those electrons which go directly from one
lead to the other, and an inelastic component, corresponding to those elec-
trons which enter into at least one of the voltage probes in their travel
between the leads.

4.10 Exercises

4.1 Transmission through a potential step: Show that the transmission

probability as a function of energy, E, for the potential step shown in Fig. 4.13
is given by
4k1 k2 /(k1 + k2 )2 if E > V0

T (E) =
0 if E < V0
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108 Molecular Electronics: An Introduction to Theory and Experiment

p p
where k1 = 2mE/~2 , k2 = 2m(E − V0 )/~2 and m is the electron mass.

V(x)
V0

0 x

Fig. 4.13 Potential step of height V0 .

4.2 Penetration of a rectangular barrier: Show that the probability for an

electron to cross the rectangular barrier shown in Fig. 4.3 for energies E > V0 is
given by Eq. (4.16).
4.3 A rectangular barrier under an applied voltage: Consider the rectan-
gular barrier under an applied bias shown in Fig. 4.5(a). Show that the energy
and voltage dependence of the transmission for E < V0 is given by
˛ ˛2
˛ 4k1 k2 ˛ k3
T (E, V ) = ˛
˛ ˛ ,
2(k1 k3 − k22 ) sinh(k2 L) + 2ik2 (k1 + k3 ) cosh(k2 L) ˛ k1
√ p p
where k1 = 2mE/~, k2 = 2m(V0 − E)/~ and k3 = 2m(E + eV )/~.
Use this result and the Landauer formula [Eq. (4.11)] to compute the zero-
temperature current-voltage characteristics for a barrier of height V0 = 4 eV and
width L = 1 nm.
4.4 Penetration of an arbitrary potential barrier: Let us consider the
potential barrier shown in Fig 4.14. Here, in a region x < a (region I), V (x) =
V0 = const.; when x > a, V (x) is a positive and smooth function decreasing
monotonically from the positive value Va = V (a) to V (∞) = 0.

V(x)
Va
I II
III

E
a b x
V0

Fig. 4.14 Arbitrary potential barrier.

Use the WKB approximation to show that the transmission coefficient

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The scattering approach 109

through that barrier is given by

p Z bp
(Va − E)(E − V0 ) −2τ 2m[V (x) − E]
T (E) = 4 e , where τ = dx.
Va − V0 a ~

Hint: The WKB approximation is nicely explained, e.g., in Ref. [146].

4.5 Resonant tunneling in a finite square well: Analyze the transmission
coefficient in the case of the square well shown in Fig 4.15. In particular, show
that in the energy range E > V3 this coefficient is given by

4k1 k3 k22
T (E) = ,
k22 (k1 + k3 )2 cos (k2 L) + (k22
2 + k1 k3 )2 sin2 (k2 L)

where L = a − b and ki is the electron momentum in the region i =I,II,III.

V(x)
III II
V3 I
V1
V2

a b x

Fig. 4.15 Square well.

Show also that the transmission coefficient above exhibits resonances as a

function of energy. In particular, calculate the position of those resonances and
show that the transmission maxima are given by 4k1 k3 /(k1 + k3 )2 .
4.6 Transmission through a delta function barrier: Let us model a one-
dimensional conductor with the following Hamiltonian

~2 ∂ 2
H=− + V0 δ(x),
2m ∂x2
where V0 is the strength of the delta potential that acts at x = 0.
(a) Demonstrate that the boundary conditions for the scattering states ψk (x),
k being the electron momentum, are: (i) continuity at x = 0 and (ii) ψk′ (x =
0+ ) − ψk′ (x = 0− ) = (2mV0 /~)ψ(x = 0), where the prime symbol indicates
derivative with respect to x.
(b) Use the previous result to show that the transmission probability through
this delta potential can be expressed as: T = 1/(1 + Z 2 ), where Z ≡ mV0 /(~2 k).
4.7 Scattering matrix:
(a) Show that in the presence of a magnetic field the scattering matrix fulfills
the property of Eq. (4.40).
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110 Molecular Electronics: An Introduction to Theory and Experiment

(b) Derive the relations of Eq. (4.43).

4.8 Resonant tunneling through a symmetric double barrier: Consider
a symmetric double barrier system formed by combining two square barriers (see
Exercise 4.2) of height V0 and width L that are separated a distance d.
(a) Compute the total transmission through this system for energies smaller
than V0 . Hint: Use the idea of the combination of scattering matrices, see
Eq. (4.44) in section 4.5.2, and the results of Exercise 4.2.
(b) As in the case of the potential well of Exercise 4.5, the transmission in
this double barrier system exhibits pronounced resonances. Find the position of
those resonances and show that, in the limit in which they are well separated,
the transmission around one of those resonances can be written as
4ΓL ΓR
T (E) = ,
(E − ǫ0 )2 + (ΓL + ΓR )2

where ǫ0 is the position of the resonance and ΓL,R are the scattering rates as-
sociated to the left and right potential barriers. Find an expression for these
rates in terms of the transmissions of the barriers. Hints: (i) The resonances are
well separated when the transmissions T1 and T2 are small (R1 , R2 ≈ 1). (ii)
The round-trip phase shift that appears in Eq. (4.44) is θ = 2kd, where k is the
electron momentum in the region between the two barriers.
4.9 Conductance quantization in a 2DEG: One of the most successful ap-
plications of the Landauer formula is the explanation of the conductance quan-
tization that takes place in split-gate constrictions (or quantum-point contacts)
in a two-dimensional electron gas (2DEG). A useful model to study the occur-
rence of conductance steps is the so-called saddle point model used by Büttiker in
Ref. [172]. In this model it is assumed that near the bottleneck of the constriction
the electrostatic potential can be expressed as

1 1
V (x, y) = V0 − mωx2 x2 + mωy2 y 2 . (4.91)
2 2
Here, V0 is the electrostatic potential at the saddle, ωx characterizes the curvature
of the potential barrier in the constriction and ωy the lateral confinement. Show
that for this potential the transmission probabilities are given by

1
Tn (E) = .
exp[π(E − V0 − (n + 1/2)ωx )/ωy ] + 1

Using this expression in combination with the Landauer formula, find the criteria
for the observation of well-defined conductance steps at low temperatures.
4.10 Shot noise and thermopower in a quantum point contact: Use the
saddle point model of the previous exercise to study the shot noise [155] and the
thermopower [161, 163] in a quantum point contact as a function of the Fermi
energy (or gate voltage).
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Chapter 5

Introduction to Green’s function

techniques for systems in equilibrium

The discussion of the scattering formalism in the previous chapter has left
two basic questions open: (i) How to calculate the elastic transmission
of real systems such as atomic and molecular junctions? and (ii) how to
generalize Landauer formula to take into account correlation effects and
inelastic mechanisms? Indeed, both questions can be answered, at least
to a large extent, with the help of Green’s function techniques. For this
reason, we initiate here a series of three chapters devoted to this subject.
We are aware of the fact that at this point part of the readership will be
certainly tempted to jump to the next part of the book. The words Green’s
functions cause in many people an immediate rejection because they asso-
ciate them to some obscure theoretical techniques reserved to specialists.
We believe that this judgment is a bit unfair. The degree of difficulty of
the Green’s function techniques depends primarily on the type of prob-
lems addressed. Thus for instance, we shall show that what is required to
answer the first question posed above reduces to a standard problem of lin-
ear algebra that should be accessible to any student with a background in
quantum mechanics. The answer to the second question requires however
more elaborate methods, which will also be presented in this book. With
this distinction in mind, we shall guide you through the next three chapters
indicating the type of problems that we have in mind and we shall warn
you about the possible difficulties.
In our discussion on the Green’s function techniques we shall start in
this chapter by introducing the subject concentrating ourselves on the case
of electronic systems in equilibrium. This chapter is meant to give a first
insight into what Green’s functions in quantum mechanics are, what kind of
physical information they contain and how they can be calculated in some
simple situations. Having in mind the first question above, we shall focus

111
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112 Molecular Electronics: An Introduction to Theory and Experiment

on the analysis of noninteracting systems. Then, the next chapter will deal
with the diagrammatic theory, which provides a systematic perturbative
approach to compute the Green’s functions of many-body systems where
correlations and inelastic mechanisms in general play a fundamental role.
Finally, since our final goal is the analysis of the transport properties of
atomic-scale junctions, we shall present in Chapter 7 the Keldysh formalism
that allows us to compute the Green’s functions of nonequilibrium systems.
Then, at the end of that chapter, we shall apply this formalism to the
calculation of the transmission in some illustrative examples.
This chapter is organized as follows. First, we shall remind the reader
of the basics of the Schrödinger and Heisenberg representations of quantum
mechanics. Then, we shall introduce the retarded and advanced Green’s
functions in energy space for a noninteracting electron system and show how
they can be computed in certain simple examples. We shall then define
the general (valid also for interacting systems) time-dependent retarded,
advanced and causal Green’s functions and analyze their main analytical
properties, their relation with the observables of interest and how they can
be computed, in principle, with the so-called equation-of-motion method.
One last comment before we get started. We shall constantly make
use of the second quantization formalism in our discussion of the Green’s
functions techniques. So, if you are not very familiar with this formalism,
we strongly recommend you to read Appendix A.

5.1 The Schrödinger and Heisenberg pictures

Let us start by reviewing the two most standard pictures or representations

in quantum mechanics. The usual way to introduce quantum mechanics
makes use of the so-called Schrödinger picture, which is based on the time-
dependent Schrödinger equation
∂
i~ ΨS (t) = HΨS (t), (5.1)
∂t
where H is the time-independent Hamiltonian of the system and ΨS (t) is
the time-dependent wave function. Let us stress that in what follows, unless
said otherwise, we shall set ~ = 1 to simplify the different formulas and the
operators will be written in boldface.
The previous equation has the formal solution

ΨS (t) = e−iH(t−t0 ) ΨS (t0 ), (5.2)

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Introduction to equilibrium Green’s function techniques 113

where t0 is an arbitrary initial time. Here, the exponential of any operator

A is defined, as usual, by means of its Taylor series
∞
X 1 n
exp(A) = A . (5.3)
n=0
n!
From this result, it is obvious that the operator exp[−iH(t − t0 )] is the
time-evolution operator in the Schrödinger picture, in the sense that by
acting on the wave function at a initial time, t0 , this operator transforms
it into the wave function at the time t. If we take t0 = 0, we have
ΨS (t) = e−iHt ΨS (0). (5.4)
For the moment, since we are only interested in equilibrium situations,
we shall assume that the operators describing the observables in this rep-
resentation, OS , do not have any explicit time dependence.
Another typical representation in quantum mechanics is the so-called
Heisenberg picture, which can be defined from the Schrödinger one by means
of the following unitary transformation
ΨH (t) = eiHt ΨS (t) = ΨS (0)
OH (t) = eiHt OS e−iHt . (5.5)
Thus, in Heisenberg picture the time dependence has been transferred
from the wave functions to the operators. The wave function in this rep-
resentation is stationary and equal to the wave function in Schrödinger
picture at time zero, i.e. ΨH = ΨS (0), whereas the operators, OH (t), do
depend explictly on time. Their time evolution can be obtained by taking
the derivative with respect to time in the previous equation
∂
i OH = [OH , H] , (5.6)
∂t
which is the equation of motion of an operator in this representation (see
Exercise 5.1).
Both representations are equivalent in the sense that the expectation
values are the same, irrespective of the picture used. This is a simple
consequence of the fact that both representations are related by means of
a unitary transformation.

5.2 Green’s functions of a noninteracting electron system

Green’s functions are commonly used in traditional contexts such as clas-

sical mechanics and electromagnetism. In those cases, Green’s functions
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114 Molecular Electronics: An Introduction to Theory and Experiment

are defined as the inverse of differential operators. One can indeed pro-
ceed in a similar way with the Schrödinger equation, which is a second
order differential equation. As an illustration, let us consider the prob-
lem of an electron in an one-dimensional system, which is described by the
Schrödinger equation
H(x)Ψ(x) = EΨ(x). (5.7)
Now, we define the electron Green’s function (or propagator) as
[E − H(x)] G(x, x′ ) = δ(x − x′ ), (5.8)
where
1 ∂2
H(x) = − + V (x), (5.9)
2m ∂x2
V (x) being an external potential acting on the electron. Notice that the
Green’s function is a complex function that depends both on the spatial
coordinates and on the energy, E.
In the case of a free electron, V (x) = V0 = constant, the Green’s function
can be obtained exactly (see Exercise 5.2). Indeed, one can show that a
solution is given by
i ′
G(x − x′ , E) = − eik|x−x | , (5.10)
v
p
where k = 2m(E − V0 ), v = k/m and we have included the energy,
E, as an argument. As it will become clear later on, one can interpret the
Green’s function as the propagation amplitude of an electron. In this sense,
the previous expression corresponds to the propagation of a free electron
at energy E from the position x′ to the right (x − x′ > 0) or to the left
(x − x′ < 0).
It is important to notice that there is another solution that corresponds
to the time-reserved solution as compared with the previous one:
i −ik|x−x′ |
G(x − x′ , E) = e . (5.11)
v
This simply reflects the fact that the Green’s function is not completely de-
termined until we specify the boundary conditions for its differential equa-
tion.
Eq. (5.10) corresponds to the so-called retarded Green’s function, Gr ,
while Eq. (5.11) corresponds to the advanced Green’s function, Ga . Al-
though the time does not appear explicitly in these functions, we shall show
later that one can relate Gr [Eq. (5.10)] with the propagation of an electron
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Introduction to equilibrium Green’s function techniques 115

forwards in time, while Ga [Eq. (5.11)] is the corresponding time-reversed

function (describing the electron propagation backwards in time).
An easy way to obtain the retarded/advanced function in the previous
problem is by introducing an infinitesimal imaginary part in the energy in
the expression defining G(x−x′ ). Thus, the substitution E → E ±iη selects
the retarded Green’s function for the plus sign and the advanced one for
the minus sign. A rigorous definition of the retarded Green’s function for
this one-dimensional problem would then be

limη→0 [E + iη − H(x)] Gr (x, x′ ) = δ(x − x′ ), (5.12)

and a similar one for the advanced function.

This definition for the one-dimensional problem can be generalized to
any single-particle problem. If H is the Hamilton operator of the system,
we can define the retarded and advanced Green’s functions as
−1
Gr,a (E) = lim [(E ± iη)1 − H] , (5.13)
η→0

where we have written the equation as an operator identity in order to have

an expression that is independent of the representation. Here, 1 is the iden-
tity operator. It is possible to write the previous equation in an alternative
form in terms of the eigenfunctions and eigenvalues of H (H|ψn i = ǫn |ψn i):
X |ψn ihψn |
Gr,a (E) = , (5.14)
n
E − ǫn ± iη

where from now on the limit limη→0 is implicitly assumed in all the ex-
pressions in which the parameter η appears. Are you able to show the
equivalence of Eqs. (5.13) and (5.14)? If not, see hints in Exercise 5.3.
Eq. (5.14) shows that the Green’s functions (for a noninteracting case)
have poles precisely at the eigenenergies, ǫn , of the system. This is the first
important piece of information contained in these functions.
From the previous equations, one can deduce a number of important
properties of the functions Gr,a . Let us discuss the most useful ones for our
purposes:
Property 1. The imaginary part of the Green’s functions is related to
the density of states of the system. To demonstrate this, let us remind that
the local density of states in a given position r can be written in terms of
the eigenstates of H as follows
X
ρ(r, E) = |hr|ψn i|2 δ(E − ǫn ). (5.15)
n
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116 Molecular Electronics: An Introduction to Theory and Experiment

From Eq. (5.14) we can write

X hr|ψn ihψn |ri
Gr,a (r, E) = , (5.16)
n
E − ǫn ± iη

and comparing these last two equations, one obtains

1
ρ(r, E) = ∓ Im {Gr,a (r, E)} . (5.17)
π
Here, we have used the relation
µ ¶
1 1
=P ∓ iπδ(E), (5.18)
E ± iη E
where P denotes a Cauchy principal value.
If we use a discrete basis of atomic orbitals, we would have
1
ρi (E) = ∓ Im {Gr,a
ii (E)} , (5.19)
π
where i indicates that the density of states has been projected onto the
atom (or site) i.
Property 2. The diagonal Green’s functions satisfy in any basis that
Im{Grii (E)} ≤ 0 and Im{Gaii (E)} ≥ 0. This is obvious from Eq. (5.14).
Property 3. The real and imaginary parts of Gr,a are related through
a Hilbert transformation:
Z ∞
r,a dE ′ Im {Gr,a (E ′ )}
Re {G (E)} = ∓P . (5.20)
−∞ π E − E′
This is a consequence of the pole structure of Eq. (5.14) and it can be easily
shown with the help of Eq. (5.18). As a result of this relation, Gr,a (E) can
be written as
Z ∞
ρ(E ′ )
Gr,a (E) = dE ′ , (5.21)
−∞ E − E ′ ± iη
where we have defined the density operator ρ(E) ≡ ∓Im{Gr,a (E)}/π. This
way of writing the Green’s function in terms of the density of states is known
as spectral representation and, as we shall show below, it is also valid in the
case of interacting systems.
Property 4. An important consequence of the spectral representation
is the asymptotic form of the diagonal Green’s functions for E → ∞. As
ρi (E) is a bounded function, one has
1
lim Gr,a
ii (E) = . (5.22)
E→∞ E
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Introduction to equilibrium Green’s function techniques 117

This is a consequence of the fact that the energy integral of ρi (E) is equal
to 1, i.e.
Z ∞
1 ∞
Z
dE ρi (E) = ∓ dE Im {Gr,a
ii (E)} = 1. (5.23)
−∞ π −∞

Property 5. As one can easily see from Eq. (5.13), the following simple
relation between Gr and Ga holds:
†
Gr (E) = [Ga (E)] . (5.24)

This means in practice that we only need to compute one of these two type
of functions.
Property 6. As a last issue, let us consider the case in which the
Hamiltonian H can be written as

H = H0 + V, (5.25)

where H0 is the Hamiltonian of a problem for which the Green’s functions

are known, gr,a , and V is an arbitrary single-particle perturbation. We
want to express the Green’s functions of the full problem in terms of the
unperturbed Green’s functions. This can be easily done starting from the
definition of Eq. (5.13)
−1
Gr,a (E) = [(E ± iη)1 − H0 − V] . (5.26)

Taking into account that for the unperturbed problem we have

−1
gr,a (E) = [(E ± iη)1 − H0 ] , (5.27)

it is easy to obtain the following relation (see Exercise 5.4)

Gr,a (E) = gr,a (E) + gr,a (E)VGr,a (E), (5.28)

The previous equation is known as Dyson’s equation and it can also be

derived in the interacting case, as we shall show in the next chapter. How-
ever, in the general case the operator V is replaced by a energy-dependent
operator, Σ(E), known as self-energy. Dyson’s equation is extremely useful
to compute the Green’s functions in different situations, as we shall illus-
trate in the next section. We shall also show that it is possible to have
a energy-dependent self-energy in single-particle problems when one deals
with a subspace of the full Hilbert space of the problem.
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118 Molecular Electronics: An Introduction to Theory and Experiment

5.3 Application to tight-binding Hamiltonians

In this section we shall apply what we have learned so far to the computa-
tion of the Green’s functions of several simple electronic systems described
in terms of tight-binding Hamiltonians.1 Such Hamiltonians, as we shall
see in the next chapters, play a fundamental role in the field of molecular
electronics. A generic tight-binding Hamiltonian adopts the following form
in the language of second quantization (see Appendix A)
X †
tij c†iσ cjσ .
X
H= ǫi ciσ ciσ + (5.29)
iσ i6=j;σ

Here, the indexes i and j run over of the sites (atoms) of the system and
σ represents the electron spin (σ =↑, ↓). The different operators have the
following meaning. For instance, c†iσ is the operator that creates an elec-
tron in the site i with spin σ, while ciσ annihilates such an electron. For
the sake of simplicity, we shall assume in this discussion that there is a
single relevant orbital per site. The parameters ǫi are the on-site energies,
while the hoppings tij describe the coupling between the different sites (see
Appendix A for a precise definition of all these parameters).
Our goal is the calculation of the different Green’s functions Gr,a
ij (E) in
this local basis representation. In principle, we have three methods at our
disposal: (i) the definition of Eq. (5.13), (ii) the spectral representation of
Eq. (5.14) and (iii) Dyson’s equation, see Eq. (5.28). We shall illustrate the
use of these different approaches with the analysis of three basic examples
that will be frequently used in subsequent chapters.

5.3.1 Example 1: A hydrogen molecule

We describe a hydrogen molecule with the following two-sites tight-binding
Hamiltonian (see Fig. 5.1)
X †
(c1σ c2σ + c†2σ c1σ ).
X
H = ǫ0 (n1σ + n2σ ) + t (5.30)
σ σ

Here, niσ = c†iσ ciσ ,

ǫ0 is the 1s-level of the hydrogen atoms and t is the
hopping connecting these two levels and it is assumed to be real. Our goal
is to compute the retarded/advanced diagonal Green’s function of site 1, i.e.
1 The tight-binding approach is briefly described in Appendix A and it is explained in

detail in Chapter 9. Here, we shall use the term tight-binding to refer to models or
Hamiltonians where the electronic structure is described in terms a local (atomic-like)
basis. We shall not discuss here how the matrix elements of such a Hamiltonian are
actually computed, and we shall just use them as parameters.
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Introduction to equilibrium Green’s function techniques 119

(a) t (b) ε0+|t|

ε0 ε0
1 2
ε0 −|t|
Fig. 5.1 (a) Model for the hydrogen molecule. We consider a single orbital per site
(atom) with energy ǫ, and the coupling is described by a hopping t. (b) Level scheme
of the hydrogen molecule in which the two orbitals hybridize to form the bonding and
antibonding states with energies ǫ0 ± |t|.

Gr,a
11 (E) (since the problem has spin degeneracy, we omit the spin indexes
in the Green’s functions). For symmetry reasons, this Green’s function is
equal to Gr,a
22 (E). In order to compute this function, we shall employ the
three methods mentioned above:
Method 1: Direct definition. According to the definition of Eq. (5.13),
the matrix Green’s function can be simply calculated by inverting the
Hamiltonian of Eq. (5.30). In the basis of the atomic states localized in
the hydrogen atoms, {|1i, |2i}, this Hamiltonian adopts the following ma-
trix form
µ ¶
ǫ0 t
H= , (5.31)
t ǫ0
and therefore the matrix Green’s function is given by
µ r,a ¶−1
E − ǫ0 −t
Gr,a (E) = , (5.32)
−t E r,a − ǫ0
where E r,a ≡ E ± iη, η being the infinitesimal imaginary part of the energy
appearing in the definition of Eq. (5.13). Thus, the element (1, 1) that we
are looking for reads
E r,a − ǫ0 1/2 1/2
Gr,a
11 (E) = = r,a + . (5.33)
(E r,a − ǫ0 )2 − t2 E − (ǫ0 + t) E r,a − (ǫ0 − t)
One can show that this expression fulfills the different properties of
a Green’s function discussed in the previous section. Thus for instance,
notice that Eq. (5.33) has precisely the form of the spectral representation
of Eq. (5.14). The poles in this case are nothing else but the energies ǫ± =
ǫ0 ± t of the bonding and antibonding orbitals of the hydrogen molecule,2
2 The hooping t is indeed a negative quantity and thus ǫ+ = ǫ0 + t corresponds to the
lowest energy level (bonding state).
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120 Molecular Electronics: An Introduction to Theory and Experiment

see Fig. 5.1. Notice also that the sum of the weights (coefficients appearing
in the numerators) is equal to 1.
On the other hand, the density of states projected onto the site 1 is
given in this case by
1 1 1
ρ1 (E) = ∓ Im {Gr,a11 (E)} = δ(E − ǫ+ ) + δ(E − ǫ− ), (5.34)
π 2 2
i.e. it is a sum of delta functions evaluated at the molecular energies. This
is a consequence of the fact that we are dealing with a finite system. In a
similar way, one could demonstrate that the rest of the properties listed at
the end of the previous section are satisfied. In particular, properties 4 and
5 are rather obvious from Eq. (5.33).
Method 2: Spectral representation. Let us now use the spectral repre-
sentation of Eq. (5.14). To evaluate this expression we need both the eigen-
functions and the eigenvalues of the hydrogen molecule. For this purpose
we just need to diagonalize the Hamiltonian of Eq. (5.31). The eigenfunc-
tions are simply the bonding
√ (|ψ+ i) and antibonding (|ψ− i) states given
by: |ψ± i = (|1i ± |2i)/ 2 with the corresponding eigenvalues ǫ± . Thus,
the function Gr,a11 (E) is then given by
X h1|ψn ihψn |1i X |h1|ψn i|2
Gr,a
11 (E) = h1|G|1i = r,a
= . (5.35)
n=+,−
E − ǫn n=+,−
E r,a − ǫn
√
Using the fact that h1|ψ± i = 1/ 2, we arrive immediately at the expres-
sion of Eq. (5.33). Obviously, this method is not very practical in general
since it requires the knowledge of the eigenfunctions of the system, which
are typically unknown.
Method 3: Dyson’s equation. Now, our starting point is Eq. (5.28).
The first thing to do is to divide the Hamiltonian of Eq. (5.30) into the
unperturbed part H0 and the perturbation V. The natural choice is that
the perturbation be the coupling term between the two atoms (second term
in Eq. (5.30)). Thus, these two parts of the Hamiltonian adopt the following
matrix form
µ ¶ µ ¶
ǫ0 0 0t
H0 = ; V= . (5.36)
0 ǫ0 t0
To solve Dyson’s equation we also need the Green’s functions of the
unperturbed system, gr,a . These functions are simply given by
µ r,a ¶−1
r,a r,a −1 E − ǫ0 0 1
g = [E 1 − H0 ] = r,a = r,a 1. (5.37)
0 E − ǫ0 E − ǫ0
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Introduction to equilibrium Green’s function techniques 121

Now, we can determine the function Gr,a

11 (E) by taking the element (1, 1)
in Eq. (5.28), i.e.

Gr,a r,a r,a r,a

11 (E) = g11 (E) + g11 (E)V12 G21 (E). (5.38)

Remember that gr,a is diagonal, while V is purely off-diagonal. In order to

get a closed equation for Gr,a r,a
11 , we still need an equation for G21 . Taking
now the element (2, 1) in Eq. (5.28), we get

Gr,a r,a r,a

21 (E) = g22 (E)V21 G11 (E). (5.39)

Substituting this expression now in Eq. (5.38), we arrive at

Gr,a r,a r,a r,a r,a

11 (E) = g11 (E) + g11 (E)V12 g22 (E)V21 G11 (E). (5.40)

This equation can now be trivially inverted and using the explicit expression
of the unperturbed Green’s functions one arrives once more at the result of
Eq. (5.33).
We can use the discussion above to illustrate the concept of self-energy,
which was briefly mentioned at the end of the last section. In the previous
equation, we can identify the following energy-dependent function

Σr,a r,a 2 r,a

11 (E) ≡ V12 g22 (E)V21 = t g22 (E). (5.41)

This function describes how the properties of the atom 1 are modified via
the interaction with the second atom. This can be better seen by rewriting
Eq. (5.40) as
1
Gr,a
11 (E) = , (5.42)
E r,a − ǫ0 − Σr,a
11 (E)

where we have used the expressions of the unperturbed Green’s functions.

In this equation we see that the self-energy renormalizes dynamically (de-
pending on the energy) both the position (ǫ0 ) and the lifetime of the en-
ergy level in the atom 1 (this latter point will become clearer in the next
examples). Notice that the self-energy depends both on the coupling to
the second atom and on the electronic structure of this second atom. We
shall see in the next examples that, no matter the problem, the concept
of self-energy appears naturally and it describes the renormalization of the
properties of a finite system due to its interaction with an external sys-
tem. In particular, we shall show in the next chapter that the concept of
self-energy remains valid even in the presence of interactions.
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122 Molecular Electronics: An Introduction to Theory and Experiment

5.3.2 Example 2: Semi-infinite linear chain

As a first example of an infinite solid, we consider now a semi-infinite linear
chain with only nearest-neighbor couplings. This system, which is schemat-
ically illustrated in Fig. 5.2(a), will be sometimes used in the next chapters
as a model for a metallic electrode. The corresponding tight-binding Hamil-
tonian of this system reads
X³ † ´
ciσ ci+1σ + c†i+1σ ciσ ,
X
H = ǫ0 niσ + t (5.43)
iσ iσ
where i = 1, 2, 3, ... represents the different sites starting from the surface.
We shall carry out here the calculation of the surface Green’s function,
Gr,a
11 (E). As in the previous example, there are, in principle, three methods
avaliable. However, the first two are rather impractical. The first one would
require the inversion of an infinite matrix, while the second would need
the calculation of the eigenfunctions and eigenvalues of this infinite (non-
periodic) system. For these reasons, we shall resort to Dyson’s equation.
The first step in this method is to choose the unperturbed problem and
the corresponding perturbation. One possible choice would be to select the
uncoupled atoms as unperturbed system and the coupling between them
as the perturbation. Such a legitimate choice would lead us to an infinite
algebraic system, which is really difficult to solve (try it, just for fun!).
There is an alternative “trick” that does the job in a few steps. The idea
goes as follows. Let us consider that the unperturbed system is composed of
two uncoupled systems, namely the atom 1 and the rest of the chain. Then,
the perturbation is simply the coupling between these two subsystems, i.e.
X³ † ´
V=t c1σ c2σ + c†2σ c1σ . (5.44)
σ
This means in practice that the only two non-zero elements of the pertur-
bation are V12 = V21 = t.
Now, we can use Dyson’s equation [Eq (5.28)] to obtain the equation
for Gr,a
11 (E). Taking the element (1, 1) we have

G11 (E) = g11 (E) + g11 (E)V12 G21 (E)

G21 (E) = g22 (E)V21 G11 (E),
where the second relation is necessary to obtain a closed equation for
G11 (E). Here, we have omitted again the spin index σ since there is spin
degeneracy in this problem and we have also dropped the superindexes
r, a because the equations are valid for both retarded and advanced func-
tions. The unperturbed function g11 of the site i = 1 is simply given by
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Introduction to equilibrium Green’s function techniques 123

(a) t t t t
........
1 2 3 4 5
1

(b)
0.5

a
t Re{G 11
(E)}
-0.5 a
t Im{G 11(E)}

-1
-3 -2 -1 0 1 2 3
(E - ε0)/t

Fig. 5.2 (a) Semi-infinite linear chain with a single orbital per site and only nearest-
neighbor couplings. (b) Real and imaginary parts of the advanced surface Green’s func-
tion, Ga
11 , of the semi-infinite chain as a function of the energy, see Eq. (5.46).

g11 (E) = 1/(E − ǫ0 ). On the other hand, the unperturbed function g22
is nothing else but the surface Green’s function of a semi-infinite chain,3
which is precisely what we are looking for, i.e. g22 = G11 . This allows us
to obtain the following closed equation for G11 (E)
(E − ǫ0 )G11 (E) = 1 + t2 G211 (E). (5.45)
This is a quadratic equation that possesses two possible solutions. In order
to choose the “physical” one, it is necessary to take into account the bound-
ary condition E → E r,a = E ± iη to distinguish between the retarded and
advanced solutions. As a practical advice, remember that the imaginary
part of these functions has a well-defined sign. The final solution adopts
the following expression
 sµ 
r,a r,a − ǫ
¶2
1 E − ǫ0 E 0
Gr,a
11 (E) =
 − − 1 . (5.46)
t 2t 2t

The real and imaginary parts of the advanced function are depicted in
Fig. 5.2(b). Notice that the imaginary part, and therefore the density of
states, is only non-zero in the region |E − ǫ0 | < 2|t|, which defines the
3 The removal of an atom from the chain does not modify the fact that the remaining
chain is again a semi-infinite chain.
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124 Molecular Electronics: An Introduction to Theory and Experiment

energy band of the linear chain. In this region, the Green’s function adopts
the following form
 s 
µ ¶2
1 E − ǫ0 E − ǫ0
Gr,a
11 (E) =
 ∓i 1− . (5.47)
t 2t 2t

This expression can be written in a form that is very useful to do algebraic

manipulations (see Exercise 5.5) by defining cos(φ) ≡ (E − ǫ0 )/2t:
1
Gr,a
11 (E) = exp(∓iφ) (5.48)
t
The density of states in the surface atom of the chain can be then
expressed as
s µ ¶2
1 a 1 E − ǫ0
ρ1 (E) = Im {G11 (E)} = 1− , |E − ǫ0 | ≤ 2|t|. (5.49)
π πt 2t
and it can be seen in Fig. 5.2(b). Contrary to the example of the hydrogen
molecule, in this case there is an infinite number of states that are grouped
in an energy band of width 4t. Notice that we have not specified the actual
occupation of this band. If we had an electron per site, the band would be
half-filled (with the Fermi energy equal to ǫ0 ) and there would be electron-
hole symmetry.
It is worth mentioning that in Eq. (5.45) one can identify the self-energy
Σr,a 2 r,a
11 (E) = t G11 (E), which plays exactly the same role as in the case of
the hydrogen molecule and it has the same functional form.
Let us say to conclude this discussion that one can check that the expres-
sion of Eq. (5.46) satisfies the different properties discussed in the previous
section. The reader is encouraged to show, in particular, that
1
lim Re {Gr,a
11 (E)} = , (5.50)
E→∞ E
and that the following sum rule is fulfilled
Z ∞
dE ρ1 (E) = 1. (5.51)
−∞

5.3.3 Example 3: A single level coupled to electrodes

We consider now the case of single energy level coupled to two infinite
electrodes. This is a very important example that will teach us a couple
of important lessons for molecular electronics. The system that we are
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Introduction to equilibrium Green’s function techniques 125

0110 0110
(a) (b) 1010 1010
0000
1111 100 1010
1111
0000 0000
1111 ΓL 1 1010 ε0 1010 ΓR
0000
1111
0000
1111 0000
1111
0000
1111
t t 1010 1010
0000 L
1111 R0000
1111 1010 Γ
0000
1111
L
0000
1111 0000
1111
R
0000
1111
1010
0000 ε
1111 0000
1111 L 1010 1010 R
0000
1111
0000
1111 0 0000
1111
0000
1111 1010 1010
0000
1111
0000
1111 0000
1111
0000
1111 10 10
Fig. 5.3 (a) A single level of energy ǫ0 is coupled to two infinite electrodes via the hop-
pings tL and tR . (b) The corresponding energy scheme where one can see the continuum
of states in the electrodes filled up to the Fermi energy and the resonant level, which
has acquired a half width at half maximum equal to Γ = ΓL + ΓR due to the coupling
to the reservoirs.

interested in is schematically represented in Fig. 5.3(a), and it is described

by the following Hamiltonian
ǫ0 c†0σ c0σ +
X
H = HL + HR + (5.52)
σ
³ ´ X ³ ´
c†0σ cLσ + c†Lσ c0σ + tR c†0σ cRσ + c†Rσ c0σ .
X
tL
σ σ
Here, the Hamiltonians HL and HR describe the left and right electrodes
that are coupled to a single energy level. It will not be necessary for the
present discussion to specify anything about the shape or concrete electronic
structure of these two leads. The subindex 0 refers to the localized level,
the energy of which is denoted by ǫ0 . This level is coupled to the electrodes
via the hoppings tL and tR , which are assumed to be real. The subindexes
L and R refer here to the outermost sites of the left and right electrodes
(we have in mind again that there is a single relevant orbital per site in
these leads).
The question that we want to address is: How is this level modified
by the coupling to the electrodes? This question is very relevant for many
different contexts. We have in mind the problem of a molecule (or atom)
coupled to metallic leads, but it is also important for problems like the
chemisorption of molecules on surfaces (in this case there would be only one
electrode). In order to answer this question, we will compute the local den-
sity of states projected onto the level. This requires the calculation of the
Green’s function G00 (E) (no matter whether it is retarded or advanced).
For this purpose, we resort to Dyson’s equation. Our choice for the un-
perturbed Hamiltonian H0 is the sum of the Hamiltonians of the three
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126 Molecular Electronics: An Introduction to Theory and Experiment

uncoupled subsystems, i.e. the right hand side of the first line of Eq. (5.52).
Thus, the perturbation V is the term that describes the coupling between
the localized level and the electrodes (second line in Eq. (5.52)). Notice
that we are assuming that there is no direct coupling between the leads.
With this choice in mind, we take the element (0, 0) in Eq. (5.28) to
obtain
G00 (E) = g00 (E) + g00 (E)V0L GL0 (E) + g00 (E)V0R GR0 (E), (5.53)
where V0L = tL and V0R = tR and g00 (E) = 1/(E − ǫ0 ) is the unperturbed
Green’s function of the single-level system. As usual, to close this equation,
we have to determine the functions GL0 and GR0 . This can be done by
taking the corresponding elements in Dyson’s equation, i.e.
GL0 (E) = gLL (E)VL0 G00 (E)
GR0 (E) = gRR (E)VR0 G00 (E),
where VL/R0 = tL/R and gLL and gRR are the Green’s functions of the two
outermost sites of the left and right electrodes, respectively. Substituting
these expressions in Eq. (5.53), we obtain the following closed equation
G00 (E) = g00 (E) + g00 (E)V0L gLL (E)VL0 G00 (E) (5.54)
+ g00 (E)V0R gRR (E)VR0 G00 (E).
In this expression one can identify, as in the previous examples, the self-
energy Σ00 (E) = t2L gLL (E) + t2R gRR (E), which in this case is the sum of
two contributions associated to the two leads. In terms of the self-energy
we can express the function G00 (E) as
1
G00 (E) = , (5.55)
E − ǫ0 − Σ00 (E)
where we have used the expression of g00 (E). Here, we see once more that
the self-energy describes how the resonant level is modified by the inter-
action with the leads. In particular, its real part is responsible for the
renormalization of the level position, which becomes ǫ̃0 = ǫ0 + Re{Σ00 (E)},
while its imaginary part describes the finite energy “width” acquired by
the level via the interaction with the leads. This latter point becomes
more clear by using the following approximation. Let us assume that the
Green’s functions of the leads are imaginary for energies in the vicinity
of ǫ0 and that they do not depend significantly on energy in this region.4
r,a
Thus, we can approximate these functions by gLL,RR ≈ ∓i/WL,R , where
4 This approximation is usually known as wide-band approximation.
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Introduction to equilibrium Green’s function techniques 127

WL,R are energy scales related to the density of states of the leads at the
energy ǫ0 .5 For instance, if we modeled the electrodes by the semi-infinite
chains like in the previous example, WL,R would then be the bulk hop-
ping element of these chains. Within this approximation, the self-energy
becomes Σr,a 00 = ∓i (ΓL + ΓR ), where we have defined the scattering rates
ΓL,R ≡ t2L,R /WL,R . Obviously, with this approximation the level position
remains unchanged (see Exercise 5.9). Finally, the function G00 (E) adopts
in this case the form
1
Gr,a
00 (E) = , (5.56)
E r,a − ǫ0 ± i (ΓL + ΓR )
Thus, the local density of states that we wanted to calculate is given by
1 1 ΓL + ΓR
ρ0 (E) = ∓ Im {Gr,a 00 (E)} = , (5.57)
π π (E − ǫ0 )2 + (ΓL + ΓR )2
which is a Lorentzian function, where Γ = ΓL + ΓR is the half-width at
half-maximum (HWHM). This result shows clearly that the resonant level,
which originally had zero width (it was an eigenstate of the isolated central
system), acquires a finite width Γ via the coupling to the leads. This fact
is illustrated in Figs. 5.3(b). It is worth stressing that the width depends
both on the strength of the coupling to the electrodes (via t2L,R ) and on
the local electronic structure of the leads (via WL,R or, more generally, via
gLL,RR ). The time scale ~/Γ that can be interpreted as the finite lifetime of
the resonant level due to the interaction with the leads, or in other words,
as the time that an electron spends in the resonant level.
Thus, the take-home message of this example is that when an isolated
molecule (or an atom) is coupled to a continuum of states, its levels are,
in general, shifted and they acquire a width that depends on the strength of
the coupling and on the local electronic structure of the leads.
Let us finally say that we hope that the reader has realized that all the
calculations of this section involved simple algebraic manipulations. Indeed,
we shall show in the next chapters that, as long as we deal with systems
with only elastic interactions (described by mean-field Hamiltonians), the
evaluation of the Green’s functions, both in equilibrium and out of equilib-
rium, reduces to straightforward exercises of linear algebra. So maybe, this
Green’s function stuff is not so scary after all, don’t you think?
For more detailed discussion of Green’s functions in the framework of
tight-binding models, we recommend the book of Ref. [181], as well as the
exercises 5-9 at the end of this chapter.
5 This energy scales are simply given by WL,R = 1/[πρL,R (E = ǫ0 )], where ρL,R are
the local densities of states of the two outermost sites of the leads.
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128 Molecular Electronics: An Introduction to Theory and Experiment

5.4 Green’s functions in time domain

The energy-dependent retarded and advanced Green’s functions introduced

in the previous section for single-particle problems can be considered as
Fourier transforms of time-dependent Green’s functions, the definition of
which is much more general and they are still valid in the case of interacting
systems. The utility of these new definitions will become apparent in the
next chapter when we deal with the perturbation theory. Moreover, it will
be clear that we need to introduce a new kind of function known as the
causal Green’s function.
Using the second quantization formalism and an arbitrary representa-
tion (or basis), the retarded Green’s function that depends on two time
arguments can be defined as follows
n o
Grij (t, t′ ) = −iθ(t − t′ )hΨH | ciσ (t), c†jσ (t′ ) |ΨH i, (5.58)

where |ΨH i = |ΨS (0)i is the wave function of the ground state of the system
(that can include interactions) and the operators are in Heisenberg picture.
We shall only include explicitly the spin index σ in Grij in those problems
where the spin symmetry is broken. In this definition, the step function, θ,
ensures that t > t′ and the symbol { , } stands for the anticommutator.
The Green’s functions are often defined using the basis {|ri} formed by
the eigenfunctions of the position operator. The corresponding creation and
annihilation operators in this representation are known as field operators
and they are denoted by Ψ†σ (r) and Ψσ (r), These operators are simply
related to c†iσ and ciσ by the basis transformation

φ∗i (r)c†iσ ,
X X
Ψσ (r) = φi (r)ciσ and Ψ†σ (r) = (5.59)
i i

where φi (r) are the basis wave functions of the discrete representation.
These field operators satisfy the standard type of anticommutation rela-
tions, i.e.
{Ψσ (r), Ψ†σ′ (r′ )} = δ(r − r′ )δσ,σ′ ; etc. (5.60)
In terms of these field operators, the retarded Green’s function is defined
as
Gr (rt, r′ t′ ) = −iθ(t − t′ )hΨH | Ψσ (r, t), Ψ†σ (r′ , t′ ) |ΨH i,
© ª
(5.61)
which is a complex function that depends on two spatial arguments and
two time arguments.
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Introduction to equilibrium Green’s function techniques 129

The advanced Green’s function has a similar definition, the only differ-
ence being that the propagation takes place backwards in time
n o
Gaij (t, t′ ) = iθ(t′ − t)hΨH | ciσ (t), c†jσ (t′ ) |ΨH i. (5.62)
Finally, it is convenient to define an additional Green’s function, namely
the one known as causal Green’s function, which is defined as follows
h i
Gcij (t, t′ ) = −ihΨH |T ciσ (t)c†jσ (t′ ) |ΨH i, (5.63)
where T is the time-ordering operator. It acts on a product of time-
dependent operators by ordering them chronologically from right to left.
Thus for instance, the previous function has the following explicit form
(
c ′ −ihΨH |ciσ (t)c†jσ (t′ )|ΨH i t > t′
Gij (t, t ) = (5.64)
ihΨH |c†jσ (t′ )ciσ (t)|ΨH i t′ > t.
Notice the sign change for t′ > t due to the anticommutation of fermion
operators.
So far, our discussion in this section has been a bit mathematical and
there are questions that arise naturally. The first one is: What is the
physical meaning of the Green’s functions? To answer this question no-
tice that these functions contain factors like hΨH |ciσ (t)c†jσ (t′ )|ΨH i. Here,
c†jσ (t′ )|ΨH i describes the creation (or injection) in the ground state of an
electron at time t′ in the state j. Then, the previous expectation value
yields the probability amplitude of finding such an electron at a later time
t in the state i. In other words, the Green’s functions simply describe the
probability amplitude of the occurrence of certain processes. The type of
processes described depends on the arguments of these functions. Thus for
instance, they can describe the propagation of electrons in time domain or
in energy space, propagation in real space, in momentum space or simply
in an atomic lattice.6
Another natural question is: What is the relation between this definition
of the Green’s functions and the one put forward in the previous section?
At a first glance, it seems that there is no relation at all. However, we
shall show below that if the system is noninteracting, the Fourier transform
with respect to the time arguments of these new Green’s functions fulfill
Eqs. (5.13) and (5.14), i.e. these two type of functions are equivalent.
Simple example: degenerate electron gas. To illustrate the previ-
ous definitions, we consider now the example of a free electron gas at zero
6 Inthis sense, it is not surprising that the elastic transmission of any real system can
be naturally expressed in terms of these functions.
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130 Molecular Electronics: An Introduction to Theory and Experiment

temperature, which is discussed in Exercise 5.1. As we know, the ground

state of this noninteracting system is a Fermi sea, where the single-particle
states are occupied up to the Fermi energy, EF (or chemical potential µ).
These states, |kσi, are plane waves characterized by an energy ǫk = k 2 /2m,
where k is the electron momentum. In this case, it is easy to compute both
the exact time evolution of the Heisenberg operators (see Exercise 5.1)
and the expectation values over this ground state (Fermi sea). Thus for
instance,

hΨH |c†kσ ck′ σ |ΨH i = δk,k′ θ(kF − k), (5.65)

where kF is the Fermi momentum.

Bearing these ideas in mind, it is easy to show that the retarded and
advanced Green’s functions defined in Eqs. (5.58) and (5.62) can be written
in the k-basis (momentum space) as
′
Gr (k, t − t′ ) = −iθ(t − t′ )e−iǫk (t−t ) (5.66)
a ′ ′ −iǫk (t−t′ )
G (k, t − t ) = +iθ(t − t)e ,

while the causal function can be written as

( ′
c ′ −iθ(k − kF )e−iǫk (t−t ) t > t′
G (k, t − t ) = ′ (5.67)
iθ(kF − k)e−iǫk (t−t ) t < t′ .

Notice first that these functions depend on the difference of the time ar-
guments, which is a general property for equilibrium systems. Notice also
that they are diagonal in k-space. Having in mind the physical meaning of
the Green’s functions, it is easy to understand why they have such a simple
time dependence. Since we are injecting electrons in a state |kσi, which
is an eigenstate of the system, the probability of finding it at a later time
in such state must be equal to one. This is precisely what the previous
expressions illustrate.
It is instructive to make contact with the results of the previous section.
For this purpose we must now Fourier transform the previous functions with
respect to the time difference, i.e.
Z ∞
′
Gr,a,c (k, E) = dt Gr,a,c (k, t)e−iE(t−t ) . (5.68)
−∞

In the course of doing the Fourier transformations, one gets the impression
that the time integrals diverge. This can be cured by introducing a small
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Introduction to equilibrium Green’s function techniques 131

imaginary part in the energy (E → E ± iη).7 So finally, the retarded and

advanced Green’s functions in energy space are given by
1
Gr,a (k, E) = . (5.69)
E − ǫk ± iη
This is exactly the result that one would have obtained directly from
Eq. (5.13) in this plane wave basis.
On the other hand, the causal function adopts the form
θ(k − kF ) θ(kF − k) 1
Gc (k, E) = + = . (5.70)
E − ǫk + iη E − ǫk − iη E − ǫk + isgn(k − kF )η
Therefore, for the free electron gas, the causal Green’s function is equal to
the retarded one for E > µ and equal to the advanced one for E < µ. This
relation is true in general, as we shall show below.

5.4.1 The Lehmann representation

The goal is now to get an insight into the energy dependence of the Green’s
functions introduced above for a general interacting system. For this pur-
pose, we shall derive here the spectral representation of a Green’s function,
which for the noninteracting case reduces to Eq. (5.14). We shall focus our
analysis on the causal function defined in Eq. (5.63). In equilibrium, this
function depends only on the difference of the time arguments. Choosing
t′ = 0 we have h i
†
Gcij (t) = −ihΨN N
0 |T ciσ (t)cjσ (0) |Ψ0 i, (5.71)
where we have added the superindex N in the ground state wave function,
|ΨN
0 i = |ΨH i, to indicate the total number of electrons in the system.
Writing explicitly the time-evolution of Heisenberg operators (see Eq. (5.5))
one has
Gcij (t) = −iθ(t)hΨN
0 |e
iHt
ciσ e−iHt c†jσ |ΨN
0 i (5.72)
† iHt
+iθ(−t)hΨN 0 |cjσ e ciσ e−iHt |ΨN0 i.
We now use the fact that H|ΨN 0 i = E 0
N
|Ψ N
0 i, where E N
0 is the ground state
energy of the system with N electrons, to arrive at
−iHt † iE0N t
Gcij (t) = −iθ(t)hΨN 0 |ciσ e cjσ |ΨN0 ie (5.73)
† iHt N
+iθ(−t)hΨN
0 |cjσ e ciσ |ΨN
0 ie
−iE0 t
.
7 A more rigorous way of solving this problem involves the introduction of the integral

representation of the step function:

Z ∞ ′
dE e−iE(t−t )
θ(t − t′ ) = − .
−∞ 2πi E + iη
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132 Molecular Electronics: An Introduction to Theory and Experiment

N +1
ihΨN +1
P
We now insert m |Ψm m | in the part for t > 0 and
N −1 N −1
< 0, where |ΨN +1
i and |ΨN −1
P
m |Ψ m ihΨm | in the part for t m m i are
the eigenfunctions of the system with one more and one less electrons, re-
spectively. The resulting expression reads
N +1
+1 † −E0N )t
X
Gcij (t) = −iθ(t) hΨN
0 |ciσ |Ψm
N +1
ihΨN
m |cjσ |ΨN
0 ie
−i(Em

m
X † N N −1
+iθ(−t) hΨN N −1
0 |cjσ |Ψm ihΨN
m
−1
|ciσ |ΨN
0 ie
−i(E0 −Em )t
.
m

We now Fourier transform with respect to the time argument to obtain

the expression of the Green’s function in energy space
X hΨN N +1 +1 †
0 |ciσ |Ψm ihΨN
m |cjσ |ΨN
0 i
Gcij (E) = N +1
(5.74)
m
E − (Em − E0N ) + iη
X hΨN † N −1
0 |cjσ |Ψm ihΨN
m
−1
|ciσ |ΨN
0 i
+ N −1
,
m
E + (Em − E0N ) − iη
which in the diagonal case adopts the form
+1 †
X |hΨNm |ciσ |ΨN
0 i|
2 X |hΨNm
−1
|ciσ |ΨN
0 i|
2
Gcii (E) = N +1
+ N −1
.
m
E − (Em − E0N ) + iη m
E + (Em − E0N ) − iη
(5.75)
This expression, referred to as Lehmann or spectral representation,
shows clearly the pole structure of the Green’s functions of a general elec-
tron system. The poles appear at the energy of the quasi-particles of the
system, that is, at the energies that are necessary to add or remove an elec-
tron in the ground state of the system.8 Before analyzing in more detail
the properties of Gc (E), let us see how the spectral representation of the
retarded/advanced function looks like. One can repeat the process above
to arrive at
X hΨN N +1 +1 †
0 |ciσ |Ψm ihΨNm |cjσ |ΨN0 i
Gr,a
ij (E) = N +1 N
+ (5.76)
m
E − (Em − E0 ) ± iη
X hΨN † N −1
0 |cjσ |Ψm ihΨN
m
−1
|ciσ |ΨN
0 i
N −1
.
m
E + (Em − E0N ) ± iη
The previous expressions of the Green’s functions in energy space can be
written in a slightly different way in the thermodynamical limit (N → ∞).
Let us focus on the expressions of the denominators. Considering first the
8 Due to the factors ±iη, the poles appear slightly shifted with respect to the real axis
in the complex plane.
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Introduction to equilibrium Green’s function techniques 133

part of electrons, we can add and subtract the energy of the ground state
with N + 1 electrons:
N +1
E − (Em − E0N ) = E − (E0N +1 − E0N ) − (Em
N +1
− E0N +1 ). (5.77)
The energy difference E0N +1− E0N in the limit N → ∞ is the chemical po-
tential µ of the system, N +1
while Em −E0N +1 is the energy of the excited state
of the system with N + 1 electrons. Repeating the same operations for the
hole part, one can finally write the Green’s functions in the thermodynamic
limit as (we only consider diagonal elements)
X |hΨN +1 |c† |ΨN i|2 X |hΨN −1 |ciσ |ΨN i|2
m 0 m 0
Gcii (E) = iσ
N +1
+ N −1
(5.78)
m
E − µ − ǫm + iη m
E − µ + ǫm − iη
X |hΨN +1 |c† |ΨN i|2 X |hΨN −1 |ciσ |ΨN i|2
Gr,a
ii (E) =
m iσ
N +1
0
+ m
N −1
0
, (5.79)
m
E − µ − ǫm ± iη m
E − µ + ǫm ± iη
where ǫNm
+1
= Em N +1
− E0N +1 and ǫN
m
−1
= EmN −1
− E0N −1 are the excitation
energies of the system with N + 1 and N − 1 electrons, respectively.
From the previous expressions one can show that the spectral represen-
tation reduces to Eq. (5.14) in the noninteracting case (this exercise is left
to the reader). This is one way to establish the connection between the
definitions introduced in this section and those of section 5.2.
From the general spectral representation, it is possible to derive the
following important properties of the exact Green’s functions of an arbitrary
electronic system, which are practically identical to those of section 5.2:
Property 1. It is possible to define a spectral density related to the
imaginary part of the Green’s functions as (we only write the diagonal
elements)
+1 †
X
ρi (E) = |hΨN
m |ciσ |ΨN 2
0 i| δ(E − µ − ǫm
N +1
)+ (5.80)
m
X
|hΨN
m
−1
|ciσ |ΨN 2 N −1
0 i| δ(E − µ + ǫm ).
m
In a case in which i stands for a site index in a tight-binding problem,
the previous expression represents the quasiparticle density of states of the
system projected onto that site. The relation of the previous function to the
imaginary part of the Green’s functions is obvious. Comparing Eq. (5.80)
with Eqs. (5.78) and (5.79), one obtains
1
ρi (E) = ∓ Im {Gr,aii (E)} (5.81)
π
1
ρi (E) = −sgn(E − µ) Im {Gcii (E)} . (5.82)
π
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134 Molecular Electronics: An Introduction to Theory and Experiment

Property 2. The diagonal Green’s functions satisfy in any basis that

Im{Grii (E)} ≤ 0 and Im{Gaii (E)} ≥ 0.
Property 3. Due to the pole structure of the Green’s functions in
energy space, their real and imaginary parts are related through a Hilbert
transformation:
dE ′ Im {Gr,a
Z ∞ ′
r,a ii (E )}
Re {Gii (E)} = ∓P (5.83)
−∞ π E − E′
Z ∞
dE ′ Im {Gcii (E ′ )} sgn(E ′ − µ)
Re {Gcii (E)} = −P . (5.84)
−∞ π E − E′
As in the single-particle case, it is possible to write the Green’s functions
in terms of the spectral density as
Z ∞
r,a ρi (E ′ )
Gii (E) = dE ′ (5.85)
−∞ E − E ′ ± iη
Z ∞
ρi (E ′ )
Gcii (E) = dE ′ . (5.86)
−∞ E − E ′ + sgn(E ′ − µ)iη
Property 4. The previous expressions imply that
1
lim Gr,a c
ii (E) = lim Gii (E) = , (5.87)
E→∞ E→∞ E
where we have used the fact that the spectral density is normalized to 1.
Property 5. From the spectral representations, one can easily deduce
the following relations
½ r
¤∗ Gij (E), if E > µ
Gaij (E) = Grji (E) and Gcij (E) =
£
.
Gaij (E), if E < µ

5.4.2 Relation to observables

So far, we have seen that the Green’s functions provide important informa-
tion such as the density of states of states (or the excitation spectrum). But
the main reason for studying the Green’s functions is that the expectation
value of any one-electron operator in the ground state of the system can be
expressed in terms of the functions that we have just introduced. Thus for
instance, the electronic density n(r) in the ground state is given by
X
n(r) = hn(r)i = hΨ†σ (r)Ψσ (r)i, (5.88)
σ

which is directly related to the causal Green’s function

Gcσ (rt, r′ t′ ) = −ihΨH |T Ψσ (rt)Ψ†σ (r′ t′ ) |ΨH i,
£ ¤
(5.89)
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Introduction to equilibrium Green’s function techniques 135

by means of
X
n(r) = −i Gcσ (rt, rt+ ), (5.90)
σ

where t+ is an abbreviation that means that t′ tends t from above.

Analogously, if we use a discrete basis {|ii}, the occupation of the state
i will be given by
hniσ i = −iGciiσ (t, t+ ). (5.91)
For instance, for the free electron gas, the time-dependent Green’s func-
tion is given by Eq. (5.67) and thus, the occupation of a state with wave
vector k in the ground state (Fermi sphere) is
hnk i = θ(kF − k). (5.92)
Let us now demonstrate the general statement made above. One-
electron operators can be expressed generically in second quantized form
as
Vij c†iσ cjσ ,
X
V= (5.93)
ijσ

where Vij = hi|V (r)|ji.

Now, we want to compute the expectation value of this operator in the
ground state, i.e.
Vij hΨH |c†iσ cjσ |ΨH i.
X
hVi = (5.94)
i,j,σ

The expectation values appearing in the previous expression can be related

to the Green’s functions. For instance, if we recall the definition of the
causal Green’s functions in the time representation, we have
Gcij (t) = −ihΨH |T[ciσ (t)c†jσ (0)]|ΨH i. (5.95)
If we evaluate this function at t = 0−
Gcij (0− ) = −ihΨH |c†jσ ciσ |ΨH i, (5.96)
and therefore
hΨH |c†jσ ciσ |ΨH i = −iGcij (0− ). (5.97)
On the other hand,
∞
dE c
Z
+
Gcij (0− ) = Gij (E)eiE0 . (5.98)
−∞ 2π
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136 Molecular Electronics: An Introduction to Theory and Experiment

Making use of the spectral representation for Gcij (E), we obtain

1 1 µ
I Z
† c
dE Im Gcij (E) . (5.99)
© ª
hΨH |cjσ ciσ |ΨH i = dE Gij (E) =
2πi π −∞
Similar expressions can also be found in terms of the retarded and ad-
vanced functions.
Let us consider as an example the case in which the index i stands for a
site in a tight-binding model. The average occupation per spin of this site
is
1 µ
Z
hniσ i = hΨH |c†iσ ciσ |ΨH i = dE Im {Gcii (E)} , (5.100)
π −∞
as it should be, since Im{Gcii (E)}/π is nothing else than the local density
of states projected onto the state i.
To conclude this subsection, let us say that in general the expectation in
the ground state of two-electron operators, i.e. those containing two creation
and two annihilation operators (see Appendix A), cannot be expressed in
terms of the one-particle Green’s functions that we have introduced in this
chapter. However, a notable exception is the total energy of the system
(for a discussion of this issue, see e.g. Ref. [173]).

5.4.3 Equation of motion method

So far we have discussed some of the properties of the “new” Green’s func-
tions and we have seen that they contain very important information. Now,
let us discuss how they can be computed. In particular, we shall describe
in this section a method referred to as equation of motion. Let us illus-
trate it in an example that is already familiar to us, namely in the case of
an electron system described by a simple tight-binding Hamiltonian of the
form
tij c†iσ cjσ .
X
H= (5.101)
ijσ
Here, the diagonal matrix elements ti i correspond to the on-site energies,
ǫi , in the notation used in previous sections.
Our goal is the calculation of, for instance, the retarded Green’s function
Grij,σ (t) = −iθ(t)hΨH |ciσ (t)c†jσ (0) + c†jσ (0)ciσ (t)|ΨH i. (5.102)
For this purpose, let us calculate its time derivative
∂ r
G (t) = −iδ(t)hΨH |ciσ (t)c†jσ (0) + c†jσ (0)ciσ (t)|ΨH i (5.103)
∂t ij,σ
∂ ∂
−iθ(t)hΨH | ciσ (t)c†jσ (0) + c†jσ (0) ciσ (t)|ΨH i,
∂t ∂t
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Introduction to equilibrium Green’s function techniques 137

where we have used the fact that the derivative of the step function is a
δ-function.
Now, in order to compute the time derivative of the annihilation oper-
ator appearing in the previous equation, we make use of the equation of
motion for operators in the Heisenberg picture, see Eq. (5.6). Thus,
∂ X
i ciσ = [ciσ , H] = i tik ckσ , (5.104)
∂t
k

where we have used Eq. (5.101) to obtain the last result. Substituting this
expression in Eq. (5.103), we arrive at
∂ X
i Grij,σ (t) = δ(t)δij + tik Grkj,σ (t). (5.105)
∂t
k
It is now convenient to Fourier transform to energy space to convert this
differential equation into an algebraic one. Thus, introducing
Z ∞ Z ∞
1 1
Grij,σ (t) = dE e−iEt Grij,σ (E) ; δ(t) = dE e−iEt (5.106)
2π −∞ 2π −∞
in Eq. (5.105), we obtain the following algebraic equation of the Green’s
function in energy space
X
EGrij,σ (E) = δij + tik Grkj,σ (E). (5.107)
k

This is nothing else but the element (i, j) of the matrix equation
−1
Gr (E) = [E1 − H] , (5.108)
which is precisely the expression that we used as a definition in section 5.2
[see Eq. (5.13)]. Thus, we have shown again the equivalence of the two type
of definitions for the case of noninteracting electron systems.
It is important to emphasize that the equation-of-motion method
illustrated above is by no means restricted to noninteracting system.
However, if the Hamiltonian contains two-electron terms (with four cre-
ation/annihilation operators), in general there is no straightforward way to
get a closed system of equations, as in the previous example. The problem
is that the equation of motion for the one-particle Green’s function couples
this function to higher-order ones containing an increasing number of oper-
ators and the resulting algebraic system has, strictly speaking, an infinite
dimension. In practice, one has to find an appropriate way of truncating
the system, which is not an easy task in general.
In order to illustrate what we meant in the previous paragraph, let us
consider the Anderson model that describes the interaction of a single level
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138 Molecular Electronics: An Introduction to Theory and Experiment

(including the electron-electron interaction in this level) with a continuum

of states. This model can describe, for instance, a magnetic impurity in a
metal or a quantum dot (or a molecule) coupled to metallic reservoirs. The
Hamiltonian of this model adopts the form (see Appendix A)
X³ ´ X
Vk0 c†kσ c0σ + V0k c†0σ ckσ +
X
H= ǫk nkσ + ǫ0 n0σ + U n0↑ n0↓ ,
k,σ k,σ σ
(5.109)
where the subindex 0 refers to the correlated level and k to the metallic
states in the reservoirs. Our goal is to compute the (retarded or advanced)
Green’s function G00,σ (E) in the impurity. For this purpose, we proceed as
above and determine the time derivative of this function. This calculation
requires the evaluation of the time derivative of the operator c0σ (t), which
in turn requires the determination of the commutator of this operator with
the Hamiltonian. The novel term, as compared with the tight-binding
example above, is U n0↑ n0↓ and the corresponding commutator with it is
[c0σ , U n0↑ n0↓ ] = U c0σ n0σ̄ , (5.110)
where we have used the notation σ̄ = −σ. Inserting this term in the
equation of motion, it is straightforward to show that one arrives at (after
Fourier transforming to energy space)
X
(E − ǫ0 )G00σ (E) = 1 + V0k Gk0 (E) (5.111)
k
−iU θ(t)hΨH | {c0σ (t)n0σ̄ (t), c0σ } |ΨH i,
where { } stands for the anticommutator. Here, the novelty with respect to
Eq. (5.107) is the appearance of the term in the second line that contains
four operators. To close the equation, we need now an equation for this new
expectation value. The reader can convince himself, that such an equation
would generate terms containing expectation values of six operators. Then,
the equation for these functions would involve terms with eight operators
and so on and so forth. So, the only way to solve these equations in practice
is to truncate the system with sensible arguments, but in most cases it is not
clear how to do it. In the next chapter we shall discuss a more systematic
approach to obtain the Green’s functions in interacting problems.
There is one limit in which it is possible to obtain the exact Green’s
function, namely in the limit where the coupling to the reservoirs tends to
zero (V0k → 0 with U finite). In this case the equation of motion can be
truncated and one obtains (see Problem 5.11)
1 − hn0σ̄ i hn0σ̄ i
G00σ (E) = + , (5.112)
E − ǫ0 E − ǫ0 − U
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Introduction to equilibrium Green’s function techniques 139

where hn0σ i is the occupation of the level ǫ0 for spin σ, which in turn has
to be calculated with the full Green’s function of Eq. (5.112). Thus, in this
limit the Green’s functions exhibit poles at energies equal to ǫ0 and ǫ0 + U .
This tells us in particular that U is the energy that one has to supply to
accommodate a second electron in the level. The expression of Eq. (5.112)
can be used as an starting point to analyze the so-called Coulomb blockade
in quantum dots or molecular transistors (see Exercise 8.9).
Let us conclude this section by recommending Chapter 9 of Ref. [185]
for a more detailed discussion about the equation-of-motion method.

5.5 Exercises

5.1 Time evolution of the operators in Heisenberg picture:

(a) Let us consider a free electron gas described by the Hamiltonian
X
H= ǫk c†kσ ckσ .
k,σ

Show that the time evolution of the operators c†kσ and ckσ in Heisenberg
picture is given by

c†kσ (t) = c†kσ (0)eiǫk t and ckσ (t) = ckσ (0)e−iǫk t .

(b) Let us consider a diatomic molecule described by the following two-sites

tight-binding Hamiltonian
X X †
H = ǫ0 (n1σ + n2σ ) + t (c1σ c2σ + c†2σ c1σ ).
σ σ

Obtain the temporal evolution of the operators c1σ and c2σ in Heisenberg
picture.
5.2 Green’s function of a free electron in 1D: Let us consider the
Schrödinger equation of a free electron in a 1D potential

1 ∂2
» –
− + V 0 Ψ(x) = EΨ(x),
2m ∂x2

where V0 is a spatially constant potential. Show that the electron Green’s function
is given by the expressions detailed in section 5.2.
5.3 Equivalence of expressions (5.13) and (5.14): Show the equivalence
of Eq. (5.13) and Eq. (5.14). Hints: (i) Multiply bothPsides of Eq. (5.13) by
[(E ± iη) − H]. (ii) Introduce then the closure relation n |ψn ihψn | = 1, where
ψn are the eigenfunctions of H. (iii) Use H|ψn i = ǫn |ψn i and (iv) multiply by
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140 Molecular Electronics: An Introduction to Theory and Experiment

the inverse of the operator on the left hand side of the Green’s function to obtain
Eq. (5.14).
5.4 Dyson’s equation: Starting from Eq. (5.26), show that the Green’s func-
tions fulfill the Dyson’s equation (5.28).
5.5 Semi-infinite tight-binding chain: Let us consider the Hamiltonian of
Eq. (5.43) for a semi-infinite chain. Calculate the off-diagonal retarded Green’s
functions Grn1 of the chain (where 1 is the first site and n an arbitrary one) and
demonstrate that it is given by the following expression for |E − ǫ0 | < 2|t|:

e−inφ
Grn1 (E) = where cos φ = (E − ǫ0 )/2t.
t

5.6 Infinite tight-binding chain: Let us consider an infinite chain of identical

atoms with only nearest-neighbor hoppings, t.
(a) Making use of the eigenvalues of this problem, ǫk = ǫ0 + 2t cos(ka), where
a is the lattice constant, and the corresponding eigenfunctions, demonstrate that
the advanced Green’s functions Gaij (E) are given by

i −iφ|i−j|
Gaij (E) = e cos φ for |E − ǫ0 | < 2|t|.
|t|

(b) An infinite chain can be viewed as two coupled semi-infinite chains. In this
sense, consider the coupling between the semi-infinite chains as a perturbation
and use Dyson’s equation to obtain the diagonal advanced Green’s functions in a
site of the chain and demonstrate that it coincides with the result derived in (a)
for i = j.
5.7 Tight-binding chain with a defect: Let us consider an infinite chain as
in the previous problem in which a diagonal perturbation is introduced in one
of the sites, let us say in site i, such that its on-site energy becomes ǫ0 + ∆.
Calculate the local density of states in the site i and, in particular, investigate
the possibility of having a localized state outside the band. Study also the spatial
extension of such a state by calculating the occupation of this state in different
sites away from the one in which the defect is located.
5.8 Finite tight-binding chain: Let us consider a finite chain with N sites
and only nearest-neighbor interactions. Calculate the advanced Green’s function
Gan1 (E), where 1 refers to the atom in one of the extremes of the chain and n to
an arbitrary site. Demonstrate in particular that for |E − ǫ0 | < 2|t|

1 sin[(N − n + 1)φ]
Gan1 (E) = .
t sin[(N + 1)φ]

5.9 Resonant level coupled to metallic electrodes: In the example 3 of

section 5.3 we considered a single site with energy ǫ0 connected to two electron
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Introduction to equilibrium Green’s function techniques 141

reservoirs. We computed the local density of states in the wide-band approx-

imation, see Eq. (5.57). Assume now that the electrodes are modeled by the
semi-infinite linear chain of the example 2 of section 5.3 with on-site energy equal
to zero and a hopping integral t. Study the local density of states in the central
site as a function of the values of ǫ0 and the coupling elements tL and tR . Discuss
in particular how the level position is renormalized.
5.10 Time-dependent Green’s functions: Make use of the expressions of
the time dependence of the creation and annihilation operators of the two-sites
problem of Exercise 5.1.(b) to compute the time-dependent retarded Green’s func-
tions. Show that the energy-dependent Green’s functions that can be obtained
from the previous solution coincide with the result of Eq. (5.13).
5.11 Equation of motion: Atomic limit of the Anderson’s model: Let us
consider the Anderson’s Hamiltonian given in Eq. (5.109). Use the equation-of-
motion method to show that in the atomic limit (V0k → 0) the Green’s function
of the level can indeed be written as in Eq. (5.112).
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142 Molecular Electronics: An Introduction to Theory and Experiment

January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Chapter 6

Green’s functions and Feynman

diagrams

In the previous chapter we have seen that the calculation of the zero-
temperature Green’s functions of a non-interacting system in equilibrium
reduces to solve an algebraic linear system, summarized in Dyson’s equa-
tion. This is practically all we need to tackle the problem of the determi-
nation of the elastic transmission of realistic systems. However, if we want
to go beyond and treat systems where the electron correlations or inelastic
interactions play a major role, we need many-body techniques. For this
reason, we present in this chapter a systematic perturbative approach for
the calculation of zero-temperature equilibrium Green’s functions.1 This
formalism is valid for any type of system and interaction and constitutes
the most general method for the computation of Green’s functions. More-
over, the nonequilibrium formalism introduced in the next chapter follows
closely the perturbative approach that we are about to describe.
The perturbative (or diagrammatic) approach is nicely explained in dif-
ferent many-body textbooks (see e.g. Refs. [173–175, 182–185]) and for this
reason, our description here will be rather brief.2 This approach is concep-
tually rather simple, but it contains several technical points that usually
make it rather obscure. In the spirit of this monograph, we shall avoid
very formal discussions and we shall provide instead simple plausibility ar-
guments or we shall simply refer the reader to the adequate literature.
Before the trees do not let us see the forest, let us give a brief overview
of what we are about to see. First, we shall learn how to write down a
perturbative series for the Green’s functions, i.e. how to express systemat-
ically the corrections to the Green’s function due to a perturbation such

1 In some sense, this approach is simply a generalization of the perturbation theory for

the wave functions that one studies in elementary courses of quantum mechanics.
2 This chapter is mainly based on Chapter 3 of Ref. [173]

143
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144 Molecular Electronics: An Introduction to Theory and Experiment

as an external potential, electron-electron interaction, etc. Then, we shall

discuss how these contributions can be “visualized” with the help of the so-
called Feynman diagrams. These diagrams will in turn help us to organize
and simplify the perturbative series. Finally, we shall show that this series
can be formally resumed and cast in the Dyson’s equation, which we have
already introduced for case of non-interacting systems. Dyson’s equation
is expressed in terms of the concept of self-energy. This concept was also
introduced in the previous chapter and in this one its precise meaning will
be clarified.
So, it is time get started. The general problem that we want to tackle
in this chapter is the analysis of an electron system in equilibrium that is
described by a Hamiltonian of the following form
H = H0 + V, (6.1)
where H0 is a single-particle Hamiltonian and V is a perturbation that may
contain an external potential and any type of interaction. Our goal is the
compute the Green’s functions of the system in terms of the unperturbed
Green’s functions, i.e. those associated with the Hamiltonian H0 , which
are supposed to be known. For this purpose, we shall develop a system-
atic perturbation theory, but before doing that we shall now introduce a
convenient representation of quantum mechanics, known as the interaction
picture, that will be very useful in what follows.

6.1 The interaction picture

Let us consider a system described by the Hamiltonian of Eq. (6.1). We

define the interaction picture starting from the Schrödinger one by means
of the following unitary transformation3
ΨI (t) = eiH0 t ΨS (t) and OI (t) = eiH0 t OS (t)e−iH0 t . (6.2)
Notice that, contrary to the case of the Schrödinger and Heisenberg pic-
tures, in the interaction picture both wave functions and operators depend
explicitly on time.
Let us analyze first the time evolution of the operators. It is obvious
from Eq. (6.2) that the operators in this representation are the Heisenberg
operators of the unperturbed system. Taking the derivative with respect to
time in the definition of an operator in the interaction picture, one obtains
∂
i OI = [OI , H0 ] . (6.3)
∂t
3 In this chapter we shall also set ~ = 1.
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Green’s functions and Feynman diagrams 145

Therefore, the dynamics of the operators in this representation is governed

by H0 and it is thus known.
Turning to the wave functions, we can make use of the evolution of the
wave function in Schrödinger picture to obtain
ΨI (t) = eiH0 t ΨS (t) = eiH0 t e−iHt ΨS (0). (6.4)
Let us remind that
eiH0 t e−iHt 6= e−iVt ,
since, in general, [H0 , H] 6= 0.
In order to find the equation that describes the time evolution of the
wave function in this picture, we now take the derivative with respect to
time in Eq. (6.2)
∂ ∂
i ΨI (t) = −H0 eiH0 t ΨS (t) + ieiH0 t ΨS (t), (6.5)
∂t ∂t
and making use of the Schrödinger equation on the right hand side of the
previous expression, one obtains
∂
i ΨI (t) = eiH0 t (H − H0 )ΨS (t) = eiH0 t Ve−iH0 t eiH0 t ΨS (t), (6.6)
∂t
which can be simply written as
∂
i ΨI (t) = VI (t)ΨI (t). (6.7)
∂t
This equation plays the role of the standard Schrödinger equation in this
new picture. Notice that the dynamics of the wave functions is governed
by the perturbation. This is very important because it makes possible, by
means of an adiabatic hypothesis in which the perturbation is adiabatically
switched on, to relate the perturbed and unperturbed ground states of the
system by means of the evolution of the wave function in this picture. Due
to this fact, the operator that describes the time evolution of the wave
functions is of special interest and it will be discussed in detail in the next
section.
To end this section, let us discuss now the relation between the Heisen-
berg picture and the interaction picture. Using the definitions of Eq. (6.2),
one can easily show that
ΨI (t) = eiH0 t e−iHt ΨH (6.8)
iH0 t −iHt iHt −iH0 t
OI (t) = e e OH (t)e e .
The inverse transformation is obviously given by
ΨH (t) = eiHt e−iH0 t ΨI (t) (6.9)
iHt −iH0 t iH0 t −iHt
OH (t) = e e OI (t)e e .
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146 Molecular Electronics: An Introduction to Theory and Experiment

6.2 The time-evolution operator

We define the time-evolution operator in the interaction picture as

ΨI (t) = S(t, t0 )ΨI (t0 ). (6.10)
It is easy to find a formal expression for the operator S in terms of the
system Hamiltonian. From the definition of the interaction picture one has
ΨI (t) = eiH0 t ΨS (t). (6.11)
Making use of the expression of the time evolution of the wave function
in the Schrödinger picture we can write
ΨI (t) = eiH0 t e−iH(t−t0 ) ΨS (t0 ). (6.12)
Transforming the wave function ΨS (t0 ) to the interaction picture, one
has finally
ΨI (t) = eiH0 t e−iH(t−t0 ) e−iH0 t0 ΨI (t0 ). (6.13)
Comparing this expression with the definition of Eq (6.10), we can iden-
tify
S(t, t0 ) = eiH0 t e−iH(t−t0 ) e−iH0 t0 . (6.14)
From the definition of the time-evolution operator or from its formal
expression, one can easily show the following properties:

• The operator S is unitary, i.e. S−1 = S† .

• S(t, t) = 1.
• S(t, t′ )S(t′ , t′′ ) = S(t, t′′ ).
• S(t, t′ ) = S† (t′ , t) .

The time-evolution operator is also related to the unitary transformation

that relates Heisenberg and interaction pictures. From Eq. (6.14) one has
S(0, t) = eiHt e−iH0 t . (6.15)
Comparing now with Eq. (6.9), we can write
ΨH = S(0, t)ΨI (t) (6.16)
OH (t) = S(0, t)OI (t)S(t, 0).
The operator S satisfies its own equation of motion, which is very similar
to the equation for the wave functions in this representation. Taking the
derivative with respect to time in Eq. (6.14) one has
∂
i S(t, t0 ) = VI (t)S(t, t0 ). (6.17)
∂t
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Green’s functions and Feynman diagrams 147

Finally, the time-evolution operator can be expressed as a perturbative

series in the interaction VI (t). This can be shown either by solving itera-
tively the previous equation or by using the equation for the wave function
ΨI (t). We choose the second option and write Eq. (6.7) as an integral
equation
Z t
ΨI (t) = ΨI (t0 ) − i dt′ VI (t′ )ΨI (t′ ). (6.18)
t0

This equation can now be solved iteratively. To zero order we have

ΨI (t) = ΨI (t0 ). (6.19)

Substituting this zero-order result in Eq. (6.18) we obtain the first-order

result
· Z t ¸
ΨI (t) = 1 − i dt1 VI (t1 ) ΨI (t0 ). (6.20)
t0

Iterating we can arrive at

"
X Z t
ΨI (t) = 1 + (−i)n dt1 VI (t1 )× (6.21)
n t0
Z t1 Z tn−1 ¸
dt2 VI (t2 ) · · · dtn VI (tn ) ΨI (t0 ).
t0 t0

The expression inside the brackets is just the time-evolution operator

S(t, t0 ) expanded as a power series in the operator VI (t). This expres-
sion is not very inconvenient because the upper and lower limits of the
time integrals are different. It is possible to rewrite the previous expres-
sion in more adequate manner by noticing that the integration variables
fulfill t > t1 > t2 > · · · > tn > t0 . This makes possible to rewrite the
time-evolution operator in the interaction picture as
∞
(−i)n t
X Z Z t Z t
S(t, t0 ) = dt1 dt2 · · · dtn T [VI (t1 )VI (t2 ) · · · VI (tn )] ,
n=0
n! t0 t0 t0
(6.22)
where the n = 0 term is the unit operator and T is the time-ordering
operator that we introduced in the last chapter. The demonstration of this
last step is left to the reader as an exercise.
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148 Molecular Electronics: An Introduction to Theory and Experiment

6.3 Perturbative expansion of causal Green’s functions

Our goal now is the calculation of a generic causal Green’s function, which
in a discrete basis is given by
h i
hΨH |T ciσ (t)c†jσ (t′ ) |ΨH i
Gij (t, t′ ) = . (6.23)
hΨH |ΨH i
Here, the expectation value is evaluated in the ground state of the system
described by the Hamiltonian of Eq. (6.1) and the operators are written in
Heisenberg picture. Notice that we omit the superindex c to abbreviate the
notation and we introduce the denominator for normalization reasons that
will become clear later on.
As explained in the previous section, it is convenient to use the interac-
tion picture. We first transform the operators:
h i
(0) (0)†
hΨH |T S(0, t)ciσ (t)S(t, t′ )cjσ (t′ )S(t′ , 0) |ΨH i
Gij (t, t′ ) = . (6.24)
hΨH |ΨH i
Here, we have used the superindex (0) to emphasize that the operators in
the interaction picture correspond to Heisenberg operators of the unper-
turbed system. We now transform the wave function by using
|ΨH i = S(0, t)|ΨI (t)i, (6.25)
where t is an arbitrary time. Now, we want to relate the state |ΨI (t)i with
the unperturbed ground state (for V = 0), |φ0 i. This can be done using
the so-called adiabatic hypothesis. In this hypothesis, one assumes that
if the perturbation is switched on at an initial time, let us say t = −∞,
and grows slowly to its actual value at t = 0, the physics is not modified.
This adiabatic switch on is achieved by replacing the perturbation V by
Ve−ǫ|t| , where ǫ is an infinitesimally small positive parameter. In this
way, at t = ±∞ the perturbation vanishes and the system tends to the
unperturbed ground state
|ΨH i = S(0, −∞)|φ0 i. (6.26)
This procedure is not completely well-defined and one can show that
during the evolution of the ground state from t = −∞ to t = 0 with the
operator S, the wave function acquires a phase that diverges as ǫ tends to
zero. These phase factors are finally canceled by the terms in the denomi-
nator of the expectation value. The rigorous statement of this fact is known
as the Gell-Mann and Low theorem and for more information we refer the
reader to the book of Fetter and Walecka [173].
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Green’s functions and Feynman diagrams 149

We now make use of Eq. (6.26) to write the causal Green’s function as
follows
h i
(0) (0)†
hφ0 |S(∞, 0)T S(0, t)ciσ (t)S(t, t′ )cjσ (t′ )S(t′ , 0) S(0, −∞)|φ0 i
Gij (t, t′ ) = .
hφ0 |S(∞, −∞)|φ0 i
(6.27)
Here, we have used the time symmetry of the problem that implies in par-
ticular that the ground state wave function is recovered at t = +∞ (apart
from a phase factor). On the other hand, it is obvious that in the previous
expression we can introduce the time-evolution operators appearing next to
the wave functions inside the time-ordered products. Thus, the expectation
value now reads
h i
(0) (0)†
hφ0 |T ciσ (t)cjσ (t′ )S(∞, −∞) |φ0 i
Gij (t, t′ ) = , (6.28)
hφ0 |S(∞, −∞)|φ0 i
where we have grouped all the pieces of the operator S since the operator
T ensures the proper ordering. Now, we use the expansion of Eq. (6.22) for
the operator S to write the expectation value as a perturbative expansion
"∞
1 X (−i)n Z ∞
′
Gij (t, t ) = dt1 ... dtn × (6.29)
hφ0 |S(∞, −∞)|φ0 i n=0 n! −∞
h i i
(0) (0)†
hφ0 |T ciσ (t)cjσ (t′ )V(0) (t1 ) · · · V(0) (tn ) |φ0 i ,

where the zero-order term (n = 0) corresponds to the unperturbed Green’s

(0)
function, which we shall denote as Gij (t, t′ ). The previous expression is
the central result of this section.
The perturbative expansion adopts the same form, irrespectively of the
basis used. Thus for instance, if one uses a spatial representation, the
previous expression becomes
"∞
1 X (−i)n Z ∞
′ ′
G(rt, r t ) = dt1 · · · dtn × (6.30)
hφ0 |S(∞, −∞)|φ0 i n=0 n! −∞
h i i
hφ0 |T Ψ(0)σ (rt)Ψ (0)† ′ ′
σ (r t )V (0)
(t 1 ) · · · V (0)
(t n ) |φ 0 i .

6.4 Wick’s theorem

With the perturbative formalism that we have developed so far, the problem
of calculating a Green’s function or any expectation value of an operator
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150 Molecular Electronics: An Introduction to Theory and Experiment

in the ground state reduces to the calculation of expectation values in the

unperturbed ground state of the following type
h i
(0) (0)†
hφ0 |T ciσ (t)cjσ (t′ )V(0) (t1 ) · · · V(0) (tn ) |φ0 i. (6.31)
This is something that we can, in principle, calculate in an exact manner
because we know the evolution of the operators in the unperturbed problem.
However, in practice, the direct calculation of expectation values like the
one in Eq. (6.31) is extremely cumbersome. Fortunately, Wick’s theorem
simplifies enormously this task.
Wick’s theorem is the mathematical expression of the fact that the
electrons in the unperturbed problem are uncorrelated. Before stating the
theorem, let us illustrate it with a simple example. Let us consider the
following two-sites tight-binding Hamiltonian
X³ † ´
c1σ c2σ + c†2σ c1σ .
X
H= ǫ0 (n1σ + n2σ ) + t (6.32)
σ σ
Let us also assume that we have two electrons in total. If |φ0 i is the wave
function of the noninteracting problem, it seems natural that
hφ0 |n1↑ n1↓ |φ0 i = hφ0 |n1↑ |φ0 ihφ0 |n1↓ |φ0 i, (6.33)
since in the absence of interactions the probability of finding two electrons
simultaneously in |1 ↓i and in |1 ↑i must be equal to the product of the
probabilities (see Exercise 6.1).
Wick’s theorem generalizes this result to the expectation value in a non-
interacting ground state of a product of an arbitrary number of operators.
Without many-body interactions, an average like the one in Eq. (6.31) look
like h i
(0)† (0)† (0)† (0)
hφ0 |T ciσ (t)cjσ (t′ ) · · · ckσ (t1 ) · · · clσ (tn ) |φ0 i. (6.34)
Wick’s theorem establishes that such an expectation value is equal to the
sum of all possible factorizations of averages of two operators. Since in
our case the operators are fermionic and therefore anticommute, one has
to follow the usual criterion, i.e. the factorization that respects the origi-
nal order does not contain any minus sign, whereas the factorization that
differs by an odd number of permutations from the original configuration
introduces a minus sign. Thus for instance, the following expectation value
of the product of four operators can be decomposed as follows
h i
(0) (0)† (0) (0)†
hφ0 |T ciσ (t)cjσ (t′ )ckσ (t1 )clσ (t2 ) |φ0 i = (6.35)
h i h i
(0) (0)† (0) (0)†
hφ0 |T ciσ (t)cjσ (t′ ) |φ0 ihφ0 |T ckσ (t1 )clσ (t2 ) |φ0 i
h i h i
(0) (0)† (0) (0)†
−hφ0 |T ciσ (t)clσ (t2 ) |φ0 ihφ0 |T ckσ (t1 )cjσ (t′ ) |φ0 i.
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Green’s functions and Feynman diagrams 151

Notice that in the previous factorization one could have had additional
terms containing expectation values like for instance
h i h i
(0) (0) (0)† (0)†
hφ0 |T ciσ (t)ckσ (t1 ) |φ0 i, hφ0 |T cjσ (t′ )clσ (t2 ) |φ0 i
h i
(0) (0)†
or hφ0 |T ciσ (t)cj,σ̄ (t′ ) |φ0 i.

However, they usually vanish for different reasons. In the first two cases,
the combinations of operators do not conserve the number of electrons.
The third expectation value vanishes, unless the ground state is magnetic.
Thus, husually the onlyi terms that survive are those with the following form:
(0) (0)†
hφ0 |T ciσ (t)cjσ (t′ ) |φ0 i, i.e. those with a combination of a creation and
an annihilation operator. As a convention, we shall always place the cre-
ation operator on the right hand side in these factors.
To end this section, notice that the basic factor appearing in the de-
composition that results from Wick’s theorem is closely related to a single-
particle Green’s function of the unperturbed system
h i
(0) (0)† (0)
hφ0 |T ciσ (t)cjσ (t′ ) |φ0 i = iGijσ (t, t′ ). (6.36)

Thus for instance, the expectation value of the previous example can be
written as
h i
(0) (0)† (0) (0)†
hφ0 |T ciσ (t)cjσ (t′ )ckσ (t1 )clσ (t2 ) |φ0 i = (6.37)
(0) (0) (0) (0)
−Gijσ (t, t′ )Gklσ (t1 , t2 ) + Gilσ (t, t2 )Gkjσ (t1 , t′ ).

6.5 Feynman diagrams

Feynman diagrams are a graphical representation of the different contribu-

tions of the perturbative expansion of a Green’s function, which result from
the application of Wick’s theorem. Let us recall that Green’s functions can
be interpreted as the propagation amplitude of an electron from one state
to another. In this sense, the Feynman diagrams turn out to have a simple
interpretation in terms of processes that contribute to the total amplitude
of propagation of an electron. Moreover, apart from the physical insight
that these diagrams provide, they also help in classifying and identifying
the contributions resulting from the application of Wick’s theorem.
Before describing the Feynman diagrams, we need a “dictionary” that
assigns a convenient graphical representation to the different functions
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152 Molecular Electronics: An Introduction to Theory and Experiment

that appear in the perturbation theory. Thus for instance, the unper-
turbed causal Green’s functions, which appear in the perturbative expan-
sion through the application of Wick’s theorem, will be represented by a
solid line. This is shown in Fig. 6.1(a) for the function G(0) (rt, r′ t′ ) in real
space. For this case, the arrow points from the second set of arguments
(or event) to the first one (indicating the propagation of an electron from
r′ t′ to rt). If the problem depends explicitly on the spin, we would have to
label the different events with the corresponding spin. If we use a discrete
basis, the corresponding line will look like in Fig. 6.1(b).

(a) r t (b) i t (c) rt (d)

rt r’ t’
(e)
X
r’ t’ j t’ r’ t’ rt

Fig. 6.1 Basic elements of Feynman diagrams. (a) Propagator line between the events
r′ t′ to rt. (b) Propagator line between the states jσ ′ and iσ. (c) Full propagator line.
(d) Interaction line between the events r′ t′ to rt. (e) Interaction line for an external
potential.

The full (or dressed) Green’s function that corresponds to the total
amplitude for the electron propagation will be represented as a double
line, as shown in Fig. 6.1(c). On the other hand, the electron-electron
interaction between two events will be represented by a wavy line, as in
Fig. 6.1(d). Notice that, in general, the interaction is instantaneous and
therefore U (rt, r′ t′ ) ∝ δ(t − t′ ). In the case in which the perturbation is an
external potential, V (r), this will then be represented by a dashed line, see
Fig. 6.1(e).
The structure of perturbative series and the corresponding Feynman
diagrams depends on the type of perturbation under study. In what follows,
we shall illustrate the diagrammatic approach with the analysis of two
examples where the perturbation is (i) the electron-electron interaction and
(ii) an external static potential.

6.5.1 Feynman diagrams for the electron-electron interac-

tion
Let us analyze the case of an electron system in which the electron-electron
interaction is considered to be the perturbation. In this case the Hamilto-
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Green’s functions and Feynman diagrams 153

nian has the following generic form in first quantization

N N
X 1X
H = H0 + V = h(ri ) + U (ri , rj ), (6.38)
n=1
2
i6=j

where h(r) is single-electron Hamiltonian and U (r, r′ ) is the electron-

electron (Coulomb) potential. Using the second quantization language and
the basis of the eigenfunctions of the position operator {|ri}, the previous
Hamiltonian can be expressed in terms of the field operators as follows
XZ
H= dr Ψ†σ (r)h(r)Ψσ (r) (6.39)
σ
1X
Z Z
+ dr dr′ Ψ†σ (r)Ψ†σ′ (r′ )U (r, r′ )Ψσ′ (r′ )Ψσ (r).
2 ′
σσ

Thus, the perturbation V appearing in the perturbative expansion of the

causal Green’s function of Eqs. (6.30) is given by
1X
Z Z
(0) (0)† ′ (0) ′
V (t) = dr dr′ Ψ(0)† ′ (0)
σ (rt)Ψσ ′ (r t)U (r, r )Ψσ ′ (r t)Ψσ (rt).
2 ′
σσ
(6.40)
Using this expression in Eq. (6.30) and applying Wick’s theorem, we
arrive at the following expression for the first-order correction for the causal
Green’s function4
1
Z Z
δG(1) (x, x′ ) = dx1 dx′1 U (x1 , x′1 ) { (6.41)
2
n(0) (r′1 )G(0) (x, x1 )G(0) (x1 , x′ ) + iG(0) (x, x1 )G(0) (x1 , x′1 )G(0) (x′1 , x′ )
+iG(0) (x, x′1 )G(0) (x′1 , x1 )G(0) (x1 , x′ ) + n(0) (r1 )G(0) (x, x′1 )G(0) (x′1 , x′ )
o
−iG(0) (x, x′ )G(0) (x′1 , x1 )G(0) (x1 , x′1 ) − n(0) (r1 )n(0) (r′1 )G(0) (x, x′ ) ,

where we have used the shorthand x ≡ rt to simplify the notation. In

Eq. (6.41) it was necessary to write the causal Green’s function with equal
time arguments, i.e. G(0) (t, t), which has an ambiguous mathematical ex-
pression. We have used the following criterion that provides the correct
result: G(0) (t, t+ ), i.e. in Eq. (6.41) we have used
G(0) (x, x) = G(0) (rt, rt+ ) (6.42)
= ihφ0 |Ψ(0)† (0)
σ (rt)Ψσ (rt)|φ0 i = in (0)
(r). (6.43)
Now, we can use the graphical conventions introduced in Fig. 6.1 to
represent the six different contributions to the first-order correction of the
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154 Molecular Electronics: An Introduction to Theory and Experiment

x x x
(1) (2) (3)
x’1 x’1
x1 x1
x1
x’1

x’ x’ x’
x (4) x (5) x (6)

x1
x’1
x1 x’1 x1 x’1

x’ x’ x’

Fig. 6.2 First-order Feynman diagrams for the electron-electron interaction.

causal Green’s function. This can be seen in Fig. 6.2, where we have num-
bered the terms from 1 to 6 following the order of Eq. (6.41).
Let us summarize some of the main features of these diagrams, which
are also found in higher-order contributions:

• The only thing that matters in the diagrams is their topology, i.e.
the way in which the different events are connected.
• The Green’s functions with equal time arguments are represented
by a closed loop and their value is equal to in(0) (r). If we used a
local representation {|ii}, then we would have
(0) (0)
Gii (t, t+ ) = ihniσ i. (6.44)
• Notice that all the intermediate events are linked by an interaction
line and they have an incoming and an outgoing propagator, which
correspond to the scattering process that the electron undergoes
due to the electron-electron interaction. These intermediate events
are known as vertexes (see Fig. 6.3).
• In Fig. 6.2 there are diagrams that have parts that are not con-
nected to the the rest of the diagram and, in particular, to the
initial and final events. Since there is an integration over the in-
termediate arguments appearing in these disconnected parts, they
4 We assume here that there is spin symmetry in the unperturbed problem. Thus, all

the Green’s functions are diagonal in spin space and we will not write explicitly their
spin index to abbreviate the notation.
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Green’s functions and Feynman diagrams 155

Fig. 6.3 Vertex: point where two propagator lines and an interaction line meet.

simply give a constant that multiplies the contribution of the rest

of the diagram. More importantly, one can show that these type of
diagrams do not contribute to the final expansion because they are
exactly canceled by the denominator of the full Green’s functions.
For a demonstration of this fact we refer the reader to the Exercise
6.3.
• As we can see in Fig. 6.2, several diagrams are topologically equiva-
lent (e.g. diagrams 1 and 3 or 2 and 4) and the only difference is the
order in which the arguments appear. However, since there are in-
tegrations over such intermediate variables, see Eq. (6.41), all these
equivalent diagrams give exactly the same contribution. This hap-
pens indeed at any order of the perturbative expansion. Thus, at
order n, any topologically connected diagram appears 2n n! times.
The factor 1/2 in the expression of V (0) together with the factor
1/n! in the perturbative expansion (see Eq. (6.30)) cancel exactly
this multiplicity. Therefore, we need to consider the topologically
connected diagrams only once.

Summarizing, the series of diagrams that contribute to the expansion

of the causal Green’s function are formed by the topologically distinct con-
nected diagrams. Moreover, the denominator in Eq. (6.30) drops. There-
fore, we can finally write the diagrammatic series of Eq. (6.30) as
∞
X Z ∞ Z ∞
G(rt, r′ t′ ) = G(0) (rt, r′ t′ ) + (−i)n+1 dt1 · · · dtn × (6.45)
n=1 −∞ −∞
h i
hφ0 |T Ψ(0)
σ (rt)V (0)
(t 1 ) · · · V (0)
(t n )Ψ (0)† ′ ′
(r t ) |φ0 i connected ,
where only the contribution of the topologically distinct connected diagrams
is considered. Of course, there would be a similar expression for the Green’s
functions in a discrete representation (or basis).
It is a very useful exercise to find the 10 topologically distinct connected
Feynman diagrams that contribute to the second-order correction of the
causal Green’s function (see Exercise 6.4). In Fig. 6.4 we show some of
these diagrams.
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156 Molecular Electronics: An Introduction to Theory and Experiment

Fig. 6.4 Some of the 10 second-order topologically distinct connected Feynman dia-
grams for the electron-electron interaction.

The Feynman diagrams provide a very intuitive way of evaluating the

different contributions to the perturbative expansion of a causal Green’s
function. In this sense, one proceeds sometimes by identifying directly
the relevant diagrams rather than calculating the systematic perturbative
series. Indeed, one can derive simple rules to quantify the contribution of
the different diagrams. For the sake of completeness, we state here these
rules for obtaining diagrammatically the contribution at a given order n to
the causal Green’s function in the case of the electron-electron interaction:

(1) Draw all the topologically distinct connected diagrams containing n

interaction lines and 2n + 1 propagator lines between the initial and
the final events.
(2) Every event must be labeled with its corresponding space-time coordi-
nate rt (or it, if one works with a discrete basis |ii). All the events,
apart from the initial and final ones, contain a vertex as the one of
Fig. 6.3.
(3) Every propagator line connecting the events x2 = r2 t2 and x1 = r1 t1
contributes with a factor G0 (x1 , x2 ).
(4) Every interaction line connecting the events x2 = r2 t2 and x1 = r1 t1
introduces a factor U (x1 , x2 ) = U (r1 , r2 )δ(t1 − t2 ). In the case of a
discrete basis, this factor would be Uijkl (corresponding matrix element
of the Coulomb potential).
(5) One has to include integrals over all intermediate variables.
(6) Every diagram of order n contains a pre-factor in .
(7) Finally, there is a sign (−1)F , where F is the number of closed loops
in the diagram. The closed loop can be formed either by a single
propagator or by a combination of several of them. Moreover, a Green’s
function with equal time variables must be interpreted as G(0) (xt, x′ t+ ).

As an illustration of these rules, let us compute the contribution corre-

sponding to the last diagram in Fig. 6.4. This second-order contribution is
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Green’s functions and Feynman diagrams 157

equal to
Z Z Z Z
2
−i ′
dx1 dx1 dx2 dx′2 G(0) (x, x1 )U (x1 , x′1 )G(0) (x′1 , x′2 )G(0) (x′2 , x′1 )

G(0) (x1 , x2 )U (x2 , x′2 )G(0) (x2 , x′ ).

6.5.2 Feynman diagrams for an external potential

Now, we assume that the electrons are subjected to an external time-
independent perturbation of the form
N
X
V= V (ri ), (6.46)
i=1

which in second quantization can be written as (in the interaction picture)

XZ
V(0) = dr Ψ(0)† (0)
σ (rt)V (r)Ψσ (rt). (6.47)
σ

x
+ x + + .......
x

Fig. 6.5 Diagrammatic series for the propagator in the case of an external potential.

For the sake of simplicity, we have assumed that the potential does
not depend on the electron spin. In this case, the diagrammatic series
is very simple. Applying Wick’s theorem to Eq. (6.30), one obtains the
diagrammatic series shown in Fig. 6.5. This means that in the propagation
of the electron from the initial instance to the final one, one simply has a
series of sequential scattering events with the external potential. The rules
for computing the contribution to the nth-order correction of the causal
Green’s functions are very simple in this case:

(1) Draw the sequential diagrams like in Fig. 6.5 with n + 1 propagators
and n interaction lines.
(2) Associate the corresponding Green’s function to every propagator line.
(3) Assign the corresponding external potential to every interaction line.
(4) Integrate over the intermediate variables.
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158 Molecular Electronics: An Introduction to Theory and Experiment

(5) The prefactor is 1.

Due to the simplicity of the diagrammatic series in this case, it is often

possible to sum up all the contributions up infinite order (notice that the
diagrammatic expansion leads to a geometrical series). As an illustration of
the previous rules, the second-order diagram in Fig. 6.5 gives a contribution
equal to
Z Z
dx1 dx2 G(0) (x, x1 )V (r1 )G(0) (x1 , x2 )V (r2 )G(0) (x2 , x′ ). (6.48)

6.6 Feynman diagrams in energy space

In spite of all the simplifications that we have introduced in the last section,
it is still very difficult to compute the different terms of the perturbative
series. This is due to the presence of the integrals over the intermediate
arguments. Thus for instance, a diagram of order 1 for the electron-electron
interaction contains up to six integrals.
The problem can be simplified by noticing first that in an equilibrium
situation the Green’s functions depend exclusively on the difference of the
time arguments. Thus, we can Fourier transform with respect to time and
work in the energy space. The introduction of the Fourier transformation
modifies the Feynman diagrams and we now study how this occurs in detail.
On the other hand, if the system is spatially homogeneous, the prob-
lem can be simplified even further since then the Green’s functions de-
pend only on the difference of the space coordinates. We shall first discuss
this case and later on, we shall generalize the results to an arbitrary non-
homogeneous system.
As we have just said, if the system is spatially homogeneous and in
equilibrium, the Green’s functions satisfy

G(rt, r′ t′ ) = G(r − r′ , t − t′ ), (6.49)

or, using the four-dimensional notation (x ≡ rt), G(x, x′ ) = G(x − x′ ). If

we assume that the interaction potential also satisfies U (x, x′ ) = U (x − x′ ),
we can then Fourier transform
dk dE i(k·r−Et)
Z Z
G(rt) = e G(k, t). (6.50)
(2π)3 2π
In what follows, we shall use the following simplified notation: p ≡
(k, E) and p · x = kr − Et. With this notation, the different Fourier
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Green’s functions and Feynman diagrams 159

transforms read
dp ipx dp ipx
Z Z
G(x) = e G(p); U (x) = e U (p), (6.51)
(2π)4 (2π)4
where dp ≡ d3 kdE is the volume element in (k, E)-space.
In order to illustrate how the diagrams are modified in energy space, we
choose a first-order diagram for the electron-electron interaction, namely
diagram 2 in Fig. 6.2. The contribution of this diagram, which we shall
denote as D(x − x′ ), is given by
Z Z
D(x − x′ ) = i dx1 dx′1 G(0) (x − x1 )U (x1 − x′1 ) (6.52)

G(0) (x1 − x′1 )G(0) (x′1 − x′ ).

Substituting in the right hand side of this expression the Fourier trans-
form of G(0) and U , one has
dp dq dp′ dq′
Z Z Z Z Z Z
D(x − x′ ) = i dx1 dx′1 4 4 4
(6.53)
(2π) (2π) (2π) (2π)4
′ ′ ′ ′ ′ ′
G(0) (p)U (q)G(0) (q′ )G(0) (p′ )eip(x−x1 ) eiq(x1 −x1 ) eiq (x1 −x1 ) eip (x1 −x ) .
This expression can be greatly simplified in the following way. First, we
regroup the exponential terms as follows
′ ′ ′ ′ ′ ′
eipx eix1 (−p+q+q ) eix1 (p −q−q ) e−ip x . (6.54)
Now, we integrate over the variables x1 and x′1 :
Z
′
dx1 eix1 (−p+q+q ) = (2π)4 δ(p − q − q′ ) ⇒ q′ = p − q (6.55)
Z
′ ′ ′
dx′1 eix1 (p −q−q ) = (2π)4 δ(p′ − q − q′ ) ⇒ p′ = q + q′ = p.

The previous equations simply express the conservation of the four-

dimensional moment (momentum and energy) in every vertex, as we illus-
trate in Fig. 6.6, where the momentum lost by the electron in the scattering
process is carried by the interaction line. If we now substitute Eq. (6.55)
in Eq. (6.52), we obtain

dp ip(x−x′ )dq
Z Z
D(x − x′ ) = i e U (q)G(0) (p)G(0) (p − q)G(0) (p).
(2π)4 (2π)4
(6.56)
This implies that the Fourier transform of the diagram can be written as
dq
Z
D(p) = i U (q)G(0) (p)G(0) (p − q)G(0) (p). (6.57)
(2π)4
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160 Molecular Electronics: An Introduction to Theory and Experiment

p−q
q

Fig. 6.6 Energy and momentum conservation in a vertex.

The previous derivation would be similar for any diagram. The key idea
is that the energy and the momentum are conserved in every vertex. Thus,
one can view the diagrams as flow diagrams in which the propagator lines
and the interaction lines carry momentum and energy. The momentum k
and the energy E carried by the initial propagator are also carried by the
final one, due to the conservation of momentum and energy in every vertex
of the diagram. This is illustrated in Fig. 6.7 with two first-order diagrams
and a second-order one. Notice that, since the interaction lines carry both
momentum and energy, one has to assign to them a direction, which is
indicated by an arrow in the diagram.

kE kE q E’’
kE
q E’
k−q k’+q
k’E’
E−E’’ E’+E’’
kE k−q,E−E’
kE q E’’
kE

Fig. 6.7 Feynman diagrams in momentum and energy space.

As in the case of real space, it is possible to establish the diagrammatic

rules for computing the perturbative expansion of the causal Green’s func-
tion in energy space. Those rules for the nth-order correction now read:

(1) Draw all the topologically distinct connected diagrams with n interac-
tion lines and 2n + 1 propagator lines. These diagrams are the same as
in the ones in (r, t)-space.
(2) Assign the flow direction (arrows) of the momentum and energy to
every interaction and propagator line.
(3) The momentum and the energy must be conserved in every vertex.
(4) Every propagator with momentum k and energy E contributes with a
factor that is equal to the unperturbed causal Green’s function, which
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Green’s functions and Feynman diagrams 161

for a homogeneous electron gas has the form

1
G(0) (k, E) = . (6.58)
E − ǫk − iηsgn(k − kF )
(5) Every interaction line with momentum k introduces an interaction po-
tential in momentum space. For the homogenous system and for the
Coulomb potential, it has the form
4πe2
U (k) = . (6.59)
k2
(6) We have to integrate over all intermediate momenta and energies (for
a non-homogeneous systems only over the energies).
(7) As a consequence of the previous rule, there is a factor for a diagram
of order n equal to 1/(2π)4n (equal to 1/(2π)n , if one only needs to
integrate over the energies). Moreover, there is a factor in , as in the
case of real space.
(8) As in the case of real space, there is a sign (−1)F , where F is the
number of closed loops.
(9) Finally, let us remind that for the diagrams in real space, there was
an ambiguity that occurs when the time arguments of the causal
Green’s function are equal. This problem was solved with the crite-
rion G(0) (t, t) = G(0) (t, t+ ). The consequence of this choice when we
Fourier transform is the introduction of a convergence factor exp(iEη),
which must appear associated to every propagator that forms a closed
loop and to those that are connected by an interaction line (if the in-
teraction is instantaneous).

As an example, let us write the contribution of the second-order diagram

in Fig. 6.7. The result is
dq dk′ dE ′′ dE ′ 2
Z Z Z Z
U (q)G(0) (k, E)G(0) (k − q, E − E ′′ )×
(2π)3 (2π)3 2π 2π
G(0) (k′ , E ′ )G(0) (k′ + q, E ′ + E ′′ )G(0) (k, E).
To conclude this section, it is convenient to generalize the results ob-
tained so far to the case of non-homogeneous systems. Indeed, this gen-
eralization is quite simple. Since the momentum is not a good quantum
number, it makes no sense to Fourier transform with respect to the spatial
coordinates. However, since the system is in equilibrium, one can still in-
troduce the Fourier transform with respect to the time arguments. This is
done exactly in the way explained above for the homogeneous system.
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162 Molecular Electronics: An Introduction to Theory and Experiment

0,σ
E E’’ _
0,σ
0,σ
E−E’’ E’ E’+E’’
0,σ _
E E’’ 0,σ
0,σ

Fig. 6.8 Second-order Feynman diagrams in energy space for the Anderson model.

As an example, let us calculate the contribution of second-order diagram

of Fig. 6.8 for the Anderson model that we discussed in section 5.4.3:
dE ′′ dE ′ (0)
Z Z
2 (0) (0) (0) (0)
U G (E)G00σ (E − E ′′ )G00σ̄ (E ′ )G00σ̄ (E ′ + E ′′ )G00σ (E).
2π 2π 00σ
Here, the index 0 refers to the impurity level.

6.7 Electronic self-energy and Dyson’s equation

In the previous sections we have analyzed the structure of the diagrammatic

series of an electronic Green’s function. In this section we shall show that
it is possible to sum formally the diagrams up to infinite order, leading to
the Dyson’s equation. But before describing this further simplification of
the perturbative expansion, let us introduce the concept of self-energy.
In Fig. 6.9 we show again the diagrammatic expansion for the Green’s
function in the cases in which the perturbation is an external potential and
the electron-electron interaction. Notice that in both cases the diagrams
have the same type of structure in the following sense. They are formed by
an initial and a final Green’s function (the same in all diagrams) and by
a central part where one can find all the scattering processes. Obviously,
this latter part is the interesting one. This structure of the diagrammatic
series allows us to define the (improper) electronic self-energy as the sum
of the central part of the diagrams to all orders (ΣI in Fig. 6.10). Thus,
the diagrammatic series for the self-energy insertion has the form shown
in Fig. 6.11 for the cases of an external potential and the electron-electron
interaction.
Notice that in the previous discussion we have neither specified the rep-
resentation nor the space (time/energy). In this sense, the result discussed
in the previous paragraphs is quite general. The diagrammatic expansion
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Green’s functions and Feynman diagrams 163

(a)
X
= + X + + .......
X

(b)

= + + + .......

Fig. 6.9 Diagrammatic expansion for the propagator for (a) an external potential and
(b) the electron-electron interaction.

of Fig. 6.9 can be summarized in the following equation in real space (r-
representation)
Z Z
G(x, x′ ) = G(0) (x, x′ ) + dx1 dx2 G(0) (x, x1 )ΣI (x1 , x2 )G(0) (x2 , x′ ).
(6.60)
The equation in momentum-energy space (for a homogeneous case) reads
as follows
G(k, E) = G(0) (k, E) + G(0) (k, E)ΣI (k, E)G(0) (k, E). (6.61)
In the case of a localized basis (like in a tight-binding model), the previous
equation adopts the form:
(0)
X (0) (0)
Gij (E) = Gij (E) + Gik (E)ΣI,kl (E)Glj (E). (6.62)
kl
To avoid explicit reference to any particular representation or space, we
shall write the previous equation in matrix form:
G = G(0) + G(0) ΣI G(0) , (6.63)

111
000
000
111 000
111
000
111
000
111 000
111
000
111
000
111
000
111 000
111
000
111
000
111
000
111
000
111 111
000
000
111
000
111
000
111 000
111

ΣΙ
000
111
000
111 000
111
000
111
000
111
000
111 000
111
000
111
000
111
000
111
000
111
000
111
000
111
= 111
000
111
000
111
000
000
111
000
111
000
111
000
111 000
111
111
000
000
111
000
111
000
111 000
111
000
111
000
111
000
111 000
111
000
111
000
111

Fig. 6.10 Self-energy insertion.

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164 Molecular Electronics: An Introduction to Theory and Experiment

(a) X
111
000
000
111
000
111
000
111
000
111
000
111
000
111
000
111
X
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111 = X + + X + .......
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
X
X

111
000
000
111
(b)
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
= + + + .......
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111

Fig. 6.11 Diagrammatic expansion for the self-energy insertion. (a) External potential.
(b) Electron-electron interaction.

where the internal integrals and sums are implicitly assumed. It is possi-
ble to write this equation in a more convenient way by inspection of the
perturbative series of G or ΣI . Let us illustrate this fact first with the
example of an external potential. As we explained in previous sections, the
diagrammatic expansion has in this case the form of a geometrical series
where the diagram of order n is simply the repetition of n identical pieces.
If we define in this case the proper self-energy, Σ, as the part of the diagram
that includes only a single scattering process, which in this case is simply
the external potential, we have the following identity

ΣI G(0) = ΣG. (6.64)

This is evident when it is expressed diagrammatically as in Fig. 6.12.

X
000
111
000
111
111
000
000
111
000
111
000
111
000
111
000
111
X X
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
= X + + X + ....... =
000
111
000
111
000
111
000
111
000
111
000
111
X
X

Fig. 6.12 Relation between the self-energy insertion, ΣI and the proper self-energy, Σ.

The proper self-energy, or from now on just self-energy, does not con-
tain repetitions of the same process, but only one scattering event. Then,
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Green’s functions and Feynman diagrams 165

Eq. (6.63) can be written in terms of the self-energy as

G = G(0) + G(0) ΣG, (6.65)
which constitutes the so-called Dyson’s equation and was first obtained by
F. Dyson in 1949 in the context of the quantum electrodynamics.
Let us now discuss the derivation of this result in the case of the electron-
electron interaction. Notice first that in this case the diagrams that con-
tribute to the self-energy insertion to all orders can be classified in two
different ways. On the one hand, we have diagrams that cannot be sep-
arated in two parts by cutting a propagator line, i.e. they do not contain
repetitions of the same elementary process. These diagrams are called ir-
reducible [see Fig. 6.13(a)]. On the other hand, we have diagrams that can
be divided into parts of lower order by cutting a propagator line, these are
called reducible diagrams [see Fig. 6.13(b)].

(a)

(b)

Fig. 6.13 (a) Examples of irreducible self-energy diagrams for the electron-electron
interaction. (b) Reducible diagrams.

We define the proper self-energy (or simply self-energy) in this case as

the sum of all the irreducible self-energy diagrams. With this definition, the
Dyson’s equation is also verified in this case. The proof is more complicated
than in the case of an external potential and it will not be detailed here.
The Dyson’s equation can be represented graphically as shown in
Fig. 6.14. Notice that the due to the symmetry of the diagrammatic series,
we could have chosen to close the Dyson’s equation in an alternative way:
G = G(0) + GΣG(0) . (6.66)
On the other hand, notice that the Dyson’s equation obtained in the
previous chapter for single-electron problems, see Eq. (5.28), is just a par-
ticular example of Eq. (6.65), which is valid for any electronic system.
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166 Molecular Electronics: An Introduction to Theory and Experiment

= + Σ

Fig. 6.14 Pictorial representation of the Dyson’s equation.

For systems in equilibrium it is convenient to write the Dyson’s equation

in energy space
G(E) = G(0) (E) + G(0) (E)Σ(E)G(E), (6.67)
which will be our starting point for the description of the equilibrium prop-
erties of any system.
Taking into account the definition of the single-particle Green’s function
in energy space introduced in the previous chapter, we can rewrite the
previous Dyson’s equation as
h i−1
G(0) (E) G(E) = 1 + Σ(E)G(E) (6.68)
[E1 − H0 ] G(E) = 1 + Σ(E)G(E),
which allows us to write the Green’s function matrix of the full system as
−1
G(E) = [E1 − H0 − Σ(E)] . (6.69)
From this expression, one can interpret the self-energy as the matrix
whose elements renormalize dynamically the matrix elements of the unper-
turbed system. Thus for instance, for the homogeneous electron gas with
electron-electron interaction, the problem is diagonal in the plane wave
basis that diagonalizes H0 and the previous Dyson’s equation becomes
1
G(k, E) = . (6.70)
E − ǫk − Σ(k, E)
In summary, the perturbative analysis reduces to the evaluation of the
proper self-energy (or just self-energy) of the electronic system. For the
two cases considered in the last sections, namely external potential and
electron-electron interaction, this implies to calculate the diagrammatic
series depicted in Fig. 6.15.
Finally, let us conclude this section with some comments and the main
analytical properties of the electronic self-energy:
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Green’s functions and Feynman diagrams 167

(a) Σ= X

(b) Σ= + + + .......

Fig. 6.15 Diagrammatic expansion for the proper self-energy. (a) External potential
and (b) electron-electron interaction.

• The Dyson’s equation relates directly the self-energy with the full
Green’s function. Therefore, the analytical properties of Σ(E) can
be derived from those of G(E).
• One can interpret Eq. (6.69) as a definition of Σ(E) in terms of G(E).
Thus, it is also possible to define a retarded and advanced self-energy.
• From Lehmann’s representation of the Green’s functions, one can de-
duce the following properties that we state here without any proof:
Im {Σrii (E)} ≤ 0 ; Im {Σaii (E)} ≥ 0 (6.71)
Im {Σcii (E)} ≥ 0 if E < µ ; Im {Σcii (E)} ≤ 0, if E > µ.
• ImΣii (E) and ReΣii (E) are related through a Hilbert transformation:
dE ′ Im {Σr,a
ii (E )}
′
Z
r,a
Re {Σii (E)} = ∓P (6.72)
π E − E′
dE ′ Im {Σcii (E ′ )} sgn(E ′ − µ)
Z
Re {Σcii (E)} = −P .
π E − E′

6.8 Self-consistent diagrammatic theory: The Hartree-Fock

approximation

Apart from the Dyson’s equation, there exist other ways to include certain
diagrams in the expansion of the self-energy up to infinite order. By inspec-
tion of the set of diagrams that contribute to the self-energy, it is possible
to distinguish two types of diagrams. On the one hand, there are diagrams,
like the one shown in Fig. 6.16, in which in one of the propagators there
is a self-energy insertion. On the other hand, there exist diagrams that do
not contain insertions and they are called skeleton diagrams. An example
of a second-order skeleton diagram is shown in Fig. 6.15(b).
Analyzing the diagrammatic series of the self-energy, one realizes that if
we consider any skeleton diagram, there appear diagrams at higher orders
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168 Molecular Electronics: An Introduction to Theory and Experiment

Fig. 6.16 Example of diagram with a self-energy insertion in one of the propagators.

with the same structure (or skeleton), but with all possible self-energy in-
sertions in their propagators. This fact makes possible to sum up to infinite
order all the diagrams that share the same skeleton, which leads to effective
diagrams like the one depicted in Fig. 6.17. Here, we have taken into ac-
count the fact that by adding all the diagrams with the same structure, the
propagator in the skeleton diagram can be replaced by the full (dressed)
propagator.

Fig. 6.17 Second-order skeleton diagram.

The previous result implies that it is possible to write the self-energy

as an expansion that contains exclusively skeleton diagrams, where the
propagators are the full ones (they are sometimes referred to as dressed or
renormalized propagators). This is illustrated in Fig. 6.18.

Σ= + + + ......

Fig. 6.18 Expansion of the self-energy in terms of skeleton diagrams.

It is worth stressing that the propagators that appear in these skeleton

diagrams are the perturbed ones, which are unknown and they have to be
determined by solving the Dyson’s equation. This means that the expansion
of Fig. 6.18, together with the corresponding Dyson’s equation provide two
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Green’s functions and Feynman diagrams 169

equations that have to be solved in a self-consistent manner. The most

common practice is to include just a few diagrams in the expansion of
Fig. 6.18. An interesting example that illustrates this procedure is the
Hartree-Fock approximation, which from a diagrammatic point of view, is
given by the approximation for the self-energy schematized in Fig. 6.19.

Σ HF = +

Fig. 6.19 Hartree-Fock approximation for the self-energy.

Let us show now this approximation is indeed equivalent to the well-

known Hartree-Fock approximation in the more standard wavefunction-
based language (see section 10.1.3). The diagram that contains the bubble
(Hartree diagram) has the following expression in the representation |ri
XZ
ΣHσ (r) = dr′ U (r − r′ )Gσ′ (r′ t′ , r′ t′+ ) (6.73)
σ′
XZ XZ e2 nσ′ (r′ )
= dr′ U (r − r′ )nσ′ (r′ ) = dr′ ,
|r − r′ |
σ′ σ′

which is nothing else but the Hartree potential, where nσ (r) is the perturbed
electron density with spin σ that has to be determined self-consistently.
Analogously, the second diagram in Fig. 6.19 is given by (in the repre-
sentation |ri)
ΣX ′ ′ ′ +
σ (r, r ) = iU (r − r )Gσ (rt, r t ). (6.74)
One can show that this expression leads to the known nonlocal (Fock)
exchange potential. For this purpose, one just needs to expand the field
operators in the previous expression in terms of an arbitrary single-electron
basis and take into account that the ground state is noninteracting. This
leads to
X e2 φiσ (r′ )φiσ (r)
ΣX ′
σ (r, r ) = − . (6.75)
i
|r − r′ |
As an additional illustration of the Hartree-Fock approximation, we
discuss now the calculation of the energy bands in this approximation of a
homogeneous electron gas (see Exercise 6.5). In this case, it is not neces-
sary to do the self-consistency because it is automatically guaranteed due
homogeneity of the system with a constant density n = N/V . Instead of
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170 Molecular Electronics: An Introduction to Theory and Experiment

using the expressions derived above, we compute now the self-energy in this
approximation in the (k, E)-space. Evaluating the Hartree-Fock diagrams
in this space, one arrives at
X Z dk′ Z dE ′ ′
ΣHσ = U (q = 0) 3
Gσ′ (k′ , E ′ )eiE η . (6.76)
(2π) 2π
′ σ

Since the Fourier transform of the Coulomb potential, U (q) = 4πe2 /q 2 ,

diverges at q = 0, we replace the potential U (q) by limµ→0 4πe2 /(q 2 + µ2 ),
which allows us to control the divergence. This new expression is simply
the Fourier transform of a Yukawa-like potential exp(−µr)/r. Thus, if one
computes the integral in the expression of ΣHσ , one obtains

4πe2
ΣH
σ = n. (6.77)
µ2
Although this result diverges when µ → 0, it is exactly canceled in the jel-
lium model by the potential created by the uniform background of positive
charge. Thus, the only remaining contribution is the exchange one that can
be expressed as
dq dν dk′ 4πe2
Z Z Z
ΣXσ (k) = i 3
U (q)Gσ (k−q, E −ν) = − hnk′ σ i.
(2π) 2π (2π)3 |k − k′ |
(6.78)
Now using the Dyson’s equation in this representation, G(k, E) =
−1
[E − ǫk − Σ(k, E)] , we see that the energy bands in the Hartree-Fock
approximation are given by ǫk,HF = ǫk + ΣX (k). The explicit expression of
the dispersion relation is computed in Exercise 6.5.

6.9 The Anderson model and the Kondo effect

The goal of this section is two-fold. On the one hand, we shall use the
Anderson model, already discussed in section 5.4.3 and Appendix A, to
illustrate the perturbative approach described in this chapter. On the other
hand, we shall use this model to get a flavor of the Kondo effect. This is
a many-body phenomenon which can appear in molecular junctions and it
will be described in much more detail in Chapter 15.
The Anderson model describes the interaction of a localized level with
electron-electron interaction with the continuum of states of a metallic sys-
tem. It was introduced by Anderson to describe a magnetic impurity in
a metal host, but it can also be used to describe a metal-molecule-metal
junction, which is the problem that we are interested in. In this model, the
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Green’s functions and Feynman diagrams 171

Hamiltonian is given by Eq. (5.109), where in particular, the U -term de-

scribes the electron-electron interaction in this level. In the absence of this
interaction, this model reduces to the resonant tunneling model of section
5.3.3.
Our goal now is to study the influence of the electron-electron inter-
action in the equilibrium properties of a molecular junction, with special
attention to the local density of states. For this purpose, we shall make use
of the perturbative approach described in this chapter. In this approach we
shall consider the entire system without electron-electron interaction as the
unperturbed system and this interaction, i.e. the last term in Eq. (5.109),
will be considered as the perturbation. The unperturbed Green’s functions
projected onto the localized level were already obtained in section 5.3.3, see
Eq. (5.56). In particular, the causal function adopts the following form in
the wide-band approximation5
(0) 1
G00 (E) = , (6.79)
E − ǫ0 − isgn(E − µ)Γ
where µ is the chemical potential of the system and Γ = ΓL + ΓR is the
total broadening of the level acquired via the interaction with the metal
electrodes. In what follows, we shall only consider symmetric situations
(ΓL = ΓR ). As we saw in section 5.3.3, in this approximation the density
of states in the localized level is a Lorentzian with Γ as its half width at
half maximum.
In the rest of this section, and in order to study the effect of the electron-
electron interaction, we shall first discuss the so-called Friedel sum rule,
which is an exact result that relates the local density of states at the Fermi
energy to the occupation of the level, and then we shall do a perturbative
analysis up to second order in the interaction U .

6.9.1 Friedel sum rule

We discuss now an important exact result, known as Friedel’ sum rule,
which is a consequence of the Fermi liquid properties of the system described
by the Anderson model.6 This sum rule can be derived as follows. The effect
of the electron-electron interaction in the localized level can be included via
the exact self-energy of the problem, Σ00,σ (E).7 The (retarded) full Green
5 Notice that this function is independent of the spin.
6 Although we have not discussed the Fermi liquid theory in this book, we find important
to introduce this discussion about Friedel sum rule because it provides a simple way to
understand the appearance of the Kondo effect.
7 Notice that we have now included the spin index σ in the self-energy.
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172 Molecular Electronics: An Introduction to Theory and Experiment

function projected onto the level can written in terms of the self-energy as
1
Gr00,σ (E) = . (6.80)
E − ǫ0 + iΓ − Σr00,σ (E)
Taking now into account that the density of states in the level is given
by ρ0σ (E) = −(1/π)ImGr00,σ (E), the corresponding occupation can be ex-
pressed as
Z µ
1 µ 1
Z
hn0σ i = dE ρ0σ (E) = − dE . (6.81)
−∞ π −∞ E − ǫ0 + iΓ − Σr00,σ (E)
We can now use the relation
1 ∂
ln E − ǫ0 + iΓ − Σr00,σ (E) +
£ ¤
r =
E − ǫ0 + iΓ − Σ00,σ (E) ∂E
∂Σr00,σ (E)/∂E
(6.82)
E − ǫ0 + iΓ − Σr00,σ (E)
together with the Ward identity (see Exercise 6.6)
∂Σr00,σ (E)
Z µ
dE Gr00,σ (E) = 0, (6.83)
−∞ ∂E
to write the occupation as
Z µ
1 ∂
ln E − ǫ0 + iΓ − Σr00,σ (E) .
£ ¤
hn0σ i = − Im dE (6.84)
π −∞ ∂E
Integrating this expression we arrive at
ǫ0 − µ − ReΣr00,σ (µ)
· ¸
1 1
hn0σ i = − tan−1 . (6.85)
2 π Γ
Here, we have used the fact that in a Fermi liquid ImΣr00,σ (µ) = 0, which
physically means that the quasiparticles have an infinite lifetime at the
Fermi energy.
Thus, we can write the local density of states as
1 Γ + ImΣr00,σ (E)
ρ0σ (E) = £ ¤ . (6.86)
π E − ǫ0 − ReΣr (µ) 2 + Γ + ImΣr (E) 2
¤ £
00,σ 00,σ

Using Eq. (6.85), we can relate the exact density of states at the Fermi
energy with the occupation of the level as follows
1
ρ0σ (µ) = sin2 [πhn0σ i] , (6.87)
πΓ
which is known as Friedel sum rule. In a case with electron-hole symmetry
and hn0σ i = 1/2, the previous expression reduces to
1
ρ0σ (µ) = . (6.88)
πΓ
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Green’s functions and Feynman diagrams 173

0,σ E’’ −
0,σ
(a) (b)
E’ E’ E’+E’’
E−E’’
0,σ −
0,σ
0,σ −
E’’ 0,σ
Fig. 6.20 First (a) and second (b) order self-energy diagrams in the Anderson model.

Notice that this equation implies that in the symmetric case, the density
of states at the Fermi energy coincides with the corresponding one in the
(0)
unperturbed problem, i.e. ρ0σ (µ) = ρ0σ (µ).
Friedel sum rule implies the appearance of a narrow peak in the density
of states in the limit U/Γ → 0. Let us discuss how this comes about. In
section 5.4.3 we saw that the level Green’s function in the limit U/Γ → 0
(atomic limit) is given by Eq. (5.112). This equation suggests that when
U ≫ Γ, the density of states consists mainly of two subbands (of width
∼ Γ) around ǫ0 and ǫ0 + U , which have most of the total spectral weight.
However, Eq. (6.88) tells us that there is a finite density at the Fermi
energy. Therefore, the exact density of states must exhibit a narrow peak
at the Fermi energy, known as Kondo peak or Kondo resonance, the width
of which tends to zero in the limit U/Γ → 0. Indeed, it can be shown that
this weight decays exponentially in this limit.

6.9.2 Perturbative analysis

We now want to calculate the properties of the system via a perturbative
expansion of the Green’s functions. For this purpose, we need an approx-
imation for the self-energy, which can be obtained from the lowest-order
diagrams. Expanding up to second order in U , one finds only two self-
energy diagrams that give a finite contribution, namely those depicted in
Fig. 6.20. The first-order diagram, see Fig. 6.20(a), is the Hartree diagram
and it yields the following contribution
Z ∞
(1) dE ′ (0) ′
Σ00,σ (E) = U G00,σ (E ′ )eiE η = U hn0σ̄ i. (6.89)
−∞ 2π
The standard Hartree approximation requires to determine the occupation
hn0σ̄ i in a self-consistent manner, i.e. by dressing the Green’s function line
in the Hartree diagram.
The level Green’s function can then be written within this approxima-
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174 Molecular Electronics: An Introduction to Theory and Experiment

tion as
1
G00,σ (E) = , (6.90)
E − ǫ0 + iΓsgn(E) − U hn0σ̄ i
where we have set µ = 0. Notice that the role of the interaction is to
shift the position of the resonant level, which moves to ǫ0 + U hn0σ̄ i. In
the special case in which ǫ0 = −U/2, known as the symmetric case, the
self-consistent solution, assuming that there is no magnetic solution, is
hn0σ i = hn0σ̄ i = 1/2. The problem exhibits in this case electron-hole
symmetry around µ = 0 and the density of states is still described by a
Lorentzian of width Γ.
Let us now analyze the contribution of the second-order diagram, see
Fig. 6.20(b). Such contribution is given by
Z ∞
dE ′′ ∞ dE ′ (0)
Z
(2) 2 (0) (0)
Σ00,σ (E) = U G00σ (E − E ′′ )G00σ̄ (E ′ )G00σ̄ (E ′ + E ′′ ).
−∞ 2π −∞ 2π
(6.91)
This expression is not easy to evaluate, but the main features of this self-
energy can be reproduced in a simple analytical calculation in which one
assumes a constant density of states for the unperturbed problem (see Ex-
ercise 6.7).
If in the diagram of Fig. 6.20(b) the Green’s function line is dressed with
the Hartree diagram and one considers the symmetric case (ǫ0 = −U/2), the
second-order approximation preserves the electron-hole symmetry around
(0)
µ = 0 and one has hn0σ i = hn0σ i. Moreover, in this case one can show that
(2) (2) (0)
ReΣ00,σ (µ) = ImΣ00,σ (µ) = 0. This implies that ρ0σ = ρ0σ and therefore
the Friedel sum rule is satisfied. This is one of the reasons why this second-
order approximation gives an excellent description in the symmetric case,
even if U is not too small in comparison with Γ.
In order to illustrate the effect of the electron-electron interaction in the
density of states, we have computed it numerically in the symmetric case
using the second-order self-energy of Eq. (6.91). The results for different
values of the ratio U/Γ are shown in Fig. 6.21.8 As one can see, as the U/Γ
increases, the density of states exhibits two subbands around ǫ0 and ǫ0 + U
and a narrow peak at the Fermi energy (the Kondo peak). Notice that the
height of this peak remains constant and it is equal to 1/(πΓ), as in the
case without electron-electron interaction. The appearance of this peak at
8 In this figure we explore cases in which U is considerably larger than Γ, which in
principle should be out of the scope of this second-order approximation. However, as
stated above, this approximation works nicely in the symmetric case and it reproduces
the main features of the exact solution [651].
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Green’s functions and Feynman diagrams 175

1
1
U/Γ = 0.0
U/Γ = 5.0 0.8
0.8
U/Γ = 10.0 0.6
U/Γ = 15.0
DOS (1/πΓ)

0.4
0.6 0.2
0
-1 -0.5 0 0.5 1
0.4

0.2

0
-10 -8 -6 -4 -2 0 2 4 6 8 10
(E-µ)/Γ
Fig. 6.21 Density of states projected onto the localized level as a function of the energy
in the Anderson model for ǫ0 = −U/2 and different values of the ratio U/Γ. The
calculation has been done including the self-energy diagrams up to second order. The
inset shows a blow-up of the energy region close to the Fermi energy.

the Fermi energy has very important consequences for the low-temperature
transport properties of molecular junctions. This will be discussed in detail
in section 15.6.2.

6.10 Final remarks

In this chapter we have presented a systematic perturbative approach to

compute zero-temperature Green’s functions of an electronic system. The
next natural step in most textbooks is to discuss the generalization of this
approach to finite temperatures. However, we shall skip this extension and
jump in the next chapter to the nonequilibrium formalism in which the
temperature will enter in a natural manner. Anyway, the reader is now in
position to study the finite-temperature formalism, which can be found in
different textbooks, see e.g. Refs. [173, 174, 182, 185].
It is worth stressing that in this chapter we have focused on the de-
scription of electronic systems, but a similar perturbative approach can be
extended to other types of systems. For instance, in nanoscale junctions
phonons or local vibrations play an important role both in the electronic
and thermal transport properties. In this sense, it is interesting to learn
how the diagrammatic formalism described in this chapter can be applied
to phonons and other bosonic degrees of freedom. This subject will not be
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176 Molecular Electronics: An Introduction to Theory and Experiment

address in this monograph and for those readers interested in this topic we
recommend Refs. [173, 174, 182, 185].
Finally, we would like to emphasize that at this stage the reader is ready
to study many important topics in solid state physics which are out of the
scope of this book. For instance, the formalism detailed in this chapter
is the starting point to understand the Fermi liquid theory, which is very
important to get a deeper insight into the physics of metals. The reader is
now also prepared to study the physics of the homogeneous electron gas,
which is a model system where one can learn many important lessons related
to the relevance of electronic correlations. Again, Refs. [173, 174, 182, 185]
are very recommendable for studying these topics.

6.11 Exercises

6.1 Wick’s theorem I: Let us consider the two-sites tight-binding Hamiltonian

of Exercise 5.1(b). Compute the ground state wave function, |φ0 i, for the case in
which there are 2 electrons in the system. Then, show that the following relations
hold:

hφ0 |n1↑ n1↓ |φ0 i = hφ0 |n1↑ |φ0 ihφ0 |n1↓ |φ0 i
hφ0 |n2↑ n2↓ |φ0 i = hφ0 |n2↑ |φ0 ihφ0 |n2↓ |φ0 i.

6.2 Wick’s theorem II: Starting from the results of Exercise 5.1(b) about the
time evolution of the creation and annihilation operators of the two-sites system,
show without applying Wick’s theorem that
h i
(0) (0)
hφ0 |T c1σ (t)c†2σ̄ (t)c†2σ (t′ )c1σ̄ (t′ ) |φ0 i = −G12σ (t − t′ )G12σ̄ (t′ − t),

which is the result that one obtains using Wick’s theorem.

6.3 Cancellation of the disconnected diagrams: Compute the denomina-
tor of the Green’s function, hφ0 |S|φ0 i up to first order for the electron-electron
interaction and show that it exactly cancels the contribution of the disconnected
diagrams that appear in the numerator of the Green’s function (see Fig. 6.2).
Hint: Show that hφ0 |S|φ0 i has the following diagrammatic expansion up to first
order:

1 + +
Fig. 6.22 Diagrammatic expansion of the denominator of the Green’s function up to
first order in the electron-electron interaction.
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Green’s functions and Feynman diagrams 177

6.4 Feynman diagrams for the electron-electron interaction: Let us con-

sider a system of interacting electrons with the electron-electron interaction as a
perturbation. Use Wick’s theorem to compute the different contributions of the
10 second-order topologically distinct diagrams. Check that the rules presented
in section 6.5.1 reproduce these results.
6.5 Hartree-fock approximation for the homogeneous electron gas: De-
rive the expression for the exchange potential of an interacting electron gas and
demonstrate that the energy dispersion relation in this case is equal to

~k2 2e2 kF 1
» ˛ ˛–
1 − k0 ˛˛ 1 + k0 ˛˛
ǫk,HF = − − ln ˛ ,
2m π 2 4k0 1 − k0 ˛

where k0 ≡ k/kF . Show also that the derivative of the dispersion relation exhibits
a logarithmic divergence at k = kF .
6.6 Ward identity: Demonstrate the Ward identity of Eq. (6.83).
6.7 Density of states and Kondo resonance in the Anderson model:
Compute the second-order contribution to the retarded self-energy in the Ander-
son model, see Eq. (6.91), in the symmetric ǫ0 = −U/2 by assuming that the
unperturbed density of states adopts the form

(0) 1/W, −W/2 < E < W/2
ρ0σ (E) =
0, |E| > W/2

where W is a constant. Use this result to plot the density of states in the level
as a function of energy for different values of the ratio U/Γ. Hint: Use first the
spectral representation to write the unperturbed Green’s function appearing in
(0)
Eq. (6.91) in terms of the density of states ρ0σ .
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178 Molecular Electronics: An Introduction to Theory and Experiment

January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Chapter 7

Nonequilibrium Green’s functions

formalism

So far we have shown how the Green’s function techniques can help us to
understand the physics of systems in equilibrium. Since our goal is the
analysis of the transport properties of different nanocontacts, we have to
generalize those techniques to deal with situations in which the systems
are driven out of equilibrium. This is precisely the goal of this chapter
in which we shall discuss the so-called nonequilibrium Green’s function for-
malism (NEGF). This formalism was developed independently by Kadanoff
and Baym [186] and Keldysh [187] in the early 1960’s. Here we shall follow
Keldysh formulation of this approach and we shall refer to it as the Keldysh
formalism. This formalism is a natural extension of the diagrammatic the-
ory that we have presented in the previous chapter. The importance of
the Keldysh formalism lies in the fact that it allows us to go beyond the
usual linear response in a systematic manner. Since its appearance, it has
been used in a great variety of topics (see Refs. [188, 189] and references
therein). In particular, it has been applied to the study of electronic trans-
port in many types of nanoscale devices and it constitutes a basic tool that
will be used throughout the rest of the book.
Apart from the original paper [187], there exist a number of excellent
reviews devoted to the Keldysh formalism in the literature [188–191]. We
try to explain it here in a didactic manner, concentrating ourselves on its
application to the problems of molecular electronics that we have in mind,
rather than entering into very technical discussions about its foundation.
Bearing this in mind, we have organized this chapter as follows. We first
present the general ideas of the Keldysh formalism. Then, we shall briefly
discuss how to perform the diagrammatic expansion within this formalism.
We shall finish the formal discussion by reviewing both the main properties
of the functions appearing in this nonequilibrium formalism and the main

179
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180 Molecular Electronics: An Introduction to Theory and Experiment

practical equations. Finally, the last part of this chapter is devoted to the
application of the Keldysh formalism to some simple transport problems.

7.1 The Keldysh formalism

In an out-of-equilibrium situation the perturbative approach detailed in the

previous chapter is not applicable. However, its generalization to nonequi-
librium situations is straightforward. Let us consider an electron system
that is described by the following Hamiltonian
H = H0 + V(t), (7.1)
where H0 is a noninteracting Hamiltonian and V(t) is a time-dependent
perturbation that can contain external potentials and interaction terms.
As in the equilibrium case, we are interested in the calculation of ex-
pectation values of operators like the following one
hΨH |AH (t)|ΨH i
hAi = , (7.2)
hΨH |ΨH i
where, for the sake of clarity, we consider the expectation value of a single
operator rather than the usual product of two of them.
We now change to the interaction picture, where this expectation value
becomes
hΨI |AI (t)|ΨI i
hAi = . (7.3)
hΨI |ΨI i
Although the perturbation in this case may depend on time, one can still
assume that the interaction is adiabatically switched on and off at t = −∞
and t = ∞, respectively. As usual, this can be done by the replacement
V(t) → exp(−ǫ|t|)V(t), where ǫ is an infinitesimally small positive param-
eter. In the equilibrium case, the time symmetry is preserved and at time
t = ∞ we recover the same noninteracting state |φ0 i that we had at t = −∞
(apart from a phase factor). However, out of equilibrium this symmetry
is in general broken and the starting point for the perturbative expansion
must be the following one
hφ0 |S(−∞, t)AI (t)S(t, −∞)|φ0 i
hAi = . (7.4)
hφ0 |S(−∞, t)S(t, −∞)|φ0 i
At a first glance, one might think that now the perturbative expansion
becomes very cumbersome because we cannot group all the pieces of the
time-evolution operator into a single one. Keldysh showed that one can
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Nonequilibrium Green’s functions formalism 181

− 8 upper branch (+)

+

8
−
8

lower branch (−)

Fig. 7.1 The Keldysh contour.

still order the time arguments along a modified time contour. This contour
is referred to as the Keldysh contour and it is depicted in Fig. 7.1.
On this contour, the time runs from −∞ to +∞ in the upper branch,
whereas it does it backwards in the lower one, i.e. from +∞ to −∞. In
order to indicate in which branch the time arguments lie, we introduce
a subindex that will be equal to + for the upper branch and − for the
lower one. With this notation, we can write now the expectation value of
Eq. (7.4) as
hφ0 |S− (−∞, ∞)S+ (∞, t)AI (t)S+ (t, −∞)|φ0 i
hAi = , (7.5)
hφ0 |S− (−∞, ∞)S+ (∞, t)S+ (t, −∞)|φ0 i
if t lies in the upper branch or
hφ0 |S− (−∞, t)AI (t)S− (t, ∞)S+ (∞, −∞)|φ0 i
hAi = , (7.6)
hφ0 |S− (−∞, t)S− (t, ∞)S+ (∞, −∞)|φ0 i
if t lies in the lower one. Defining the operator Tc that orders the time
arguments along the Keldysh contour, we can rewrite the expectation value
as
hφ0 |Tc [AI (t)S− (−∞, ∞)S+ (∞, −∞)] |φ0 i
hAi = . (7.7)
hφ0 |S− (−∞, ∞)S+ (∞, −∞)|φ0 i
This expression can be in turn rewritten in a more familiar way by defining
the operator that describes the time-evolution along the Keldysh contour
Sc (∞, −∞) ≡ S− (−∞, ∞)S+ (∞, −∞). (7.8)
With this definition we can finally write the expectation value hAi as
follows
hφ0 |Tc [AI (t)Sc (∞, −∞)] |φ0 i
hAi = . (7.9)
hφ0 |Sc (∞, −∞)|φ0 i
Analogously, one can express the expectation value of any operator product.
The expectation value of Eq. (7.9) has formally the same structure as
in an equilibrium situation. The main difference is the fact that one has
to keep track of the branch in which the time arguments lie (t+ and t− ).
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182 Molecular Electronics: An Introduction to Theory and Experiment

This implies that when defining the propagators in this formalism, there
are four different possibilities depending on the two time arguments. These
definitions are analogous to those of the causal function in the equilibrium
formalism
h i
hΨH |Tc ciσ (tα )c†jσ (t′β ) |ΨH i
Gij (tα , t′β ) = −i (7.10)
hΨH |ΨH i
h i
hΨH |Tc Ψσ (rtα )Ψ†σ (r′ t′β ) |ΨH i
G(rtα , r′ t′β ) = −i , (7.11)
hΨH |ΨH i
depending on whether we use the representation |ii or |ri. The subindexes
α and β take the values + and − and indicate in which branch the time
arguments lie. Let us now discuss in detail the expression for the four
possible functions:

(1) t = t+ and t′ = t′+ :

In this case both time arguments lie in the upper branch and the cor-
responding Green’s function reads (for a discrete representation)
h i
†
G++ ′ ′
ij (t, t ) = −ihT ciσ (t)cjσ (t ) i, (7.12)

where, from now on, the subindexes α, β = +, − will appear as su-

perindexes of the Green’s functions. Moreover, in order to simplify the
notation, we shall drop the wave functions in the expectation values and
we shall not include the denominator hΨH |ΨH i, which indeed turns out
to be equal to 1 (see discussion below). Notice that this function is
nothing else but the causal Green’s function.
(2) t = t+ and t′ = t′− :
In this case, since any time in the lower branch of the Keldysh contour
is “larger” than any time in the upper branch, one has
†
G+− ′ ′
ij (t, t ) = ihcjσ (t )ciσ (t)i. (7.13)
This function plays a fundamental role in the nonequilibrium Green’s
functions theory and, as we shall see later, it contains information about
the distribution function of the electrons.
(3) t = t− and t′ = t′+ :
In this case we have
†
G−+ ′ ′
ij (t, t ) = −ihciσ (t)cjσ (t )i. (7.14)
This function contains essentially the same information as G+− ′
ij (t, t ).
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Nonequilibrium Green’s functions formalism 183

(4) t = t− and t′ = t′−

In this last possibility, both time arguments lie in the lower branch,
where the arguments are ordered in an antichronological way. There-
fore, this new function reads
h i
†
G−−
ij (t, t ′
) = −ihT̄ c iσ (t)c jσ (t ′
) i, (7.15)

where the operator T̄ orders the time arguments in the opposite way as
compared with the usual time-ordering operator T, i.e. in a antichrono-
logical order.

The four Green’s functions defined above can be grouped in a matrix

as follows
µ ++ +− ¶
G G
Ǧ = , (7.16)
G−+ G−−

where the check symbol (ˇ) indicates that we are dealing with a 2 × 2
matrix in Keldysh space. The perturbative expansion couples the different
components of this matrix, which effectively leads to an enlargement of the
propagator space in a factor of 2. This enlargement is indeed quite natural
since in an out-of-equilibrium situation we have to determine not only the
states, the information of which is contained in the causal function, but also
the distribution function that describes how such states are occupied. This
latter information is provided by the off-diagonal functions in Eq. (7.16).
Formally speaking, the perturbative expansion is very similar to the
equilibrium one, and one has only to keep track of the matrix structure. A
additional complication is that in time-dependent problems, the products
are replaced by convolutions over intermediate arguments, which makes the
calculations considerably more complicated. Fortunately, transport prob-
lems often admit a stationary solution and then, the application of the
nonequilibrium formalism is not more complicated than the equilibrium
one.
As stated above, apart from the matrix structure introduced by the
Keldysh formalism, the rest of the perturbative approach is very similar to
the equilibrium one. To derive the perturbative expansion of the matrix
propagator of Eq. (7.16), one can use the expression of Eq. (7.9) and expand
the operator Sc . Let us recall that Sc (∞, −∞) ≡ S− (−∞, ∞)S+ (∞, −∞)
and the perturbative expansions of both time-evolution operators are given
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184 Molecular Electronics: An Introduction to Theory and Experiment

by
∞
(−i)n ∞
X Z Z ∞
S+ (∞, −∞) = dt1 · · · dtn T [VI (t1 ) · · · VI (tn )] (7.17)
n=0
n! −∞ −∞
∞
(−i)n −∞
X Z Z −∞
S− (−∞, ∞) = dt1 · · · dtn T̄ [VI (t1 ) · · · VI (tn )] .
n=0
n! ∞ ∞

After expanding the operators S+ and S− , one applies the Wick’s theo-
rem in the standard way. Therefore, the resulting diagrammatic structure
is analogous to the one in equilibrium, the main difference being the en-
largement of the space that is encoded in the indexes α and β. We shall
discuss the peculiarities of the nonequilibrium diagrammatic expansion in
the next section.
Finally, since the structure of the diagrammatic expansion is identical to
the equilibrium one, such an expansion can be also summarized in a Dyson’s
equation, which in the nonequilibrium case has the following matrix form
Z Z
Ǧ(t, t′ ) = ǧ(t, t′ ) + dt1 dt2 ǧ(t, t1 )Σ̌(t1 , t2 )Ǧ(t2 , t′ ). (7.18)

Here, we have denoted the unperturbed propagators by ǧ instead of Ǧ(0) to

simplify the notation. Here, the self-energy has a 2 × 2 matrix structure in
Keldysh space analogous to Eq. (7.16). In general, the functions appearing
in Eq. (7.18) depend on two time arguments and the Dyson’s equation is
an integral equation. However, in many stationary situations, both the
propagators and the self-energies depend on the time difference and, after
Fourier transforming, Eq. (7.18) recovers its standard equilibrium form of
an algebraic equation with the frequency as the argument, i.e.
Ǧ(E) = ǧ(E) + ǧ(E)Σ̌(E)Ǧ(E). (7.19)

7.2 Diagrammatic expansion in the Keldysh formalism

Let us discuss now some of the peculiarities of the diagrammatic expansion

in the Keldysh formalism. One of them is the fact that in this formalism
the denominator of the Green’s functions does not play any role (indeed
hφ0 |Sc |φ0 i = 1, see Exercise 7.1). One can show that in the expansion of Sc
the terms of order higher than zero cancel each other order by order. One
might think that this fact creates a problem related to the cancellation of
the disconnected diagrams. However, this is not the case because, as it is
easy to show by applying Wick’s theorem, these diagrams also cancel each
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Nonequilibrium Green’s functions formalism 185

other. Therefore, as in equilibrium, one needs to consider the topologically

distinct diagrams only once.
Let us discuss the diagrammatic structure in two situations of interest:

• Case 1: Time-dependent external potential.

Let us consider a system with N noninteracting electrons subjected to
an external potential that can be time-dependent. The Hamiltonian in
first quantization reads in this case
H = H0 + V(t), (7.20)
where
N
X
V(t) = V (ri , t). (7.21)
i=1

The diagrams in this case are trivial because, as in the case of a static
potential, they consist of the repetition of identical scattering events.
The matrix self-energy is therfore given by (see Exercise 7.2)
µ ¶
V (r, t) 0
Σ̌(r, t) = . (7.22)
0 −V (r, t)
It is interesting to note that for this single-electron perturbation the
components Σ+− and Σ−+ vanish. The existence of off-diagonals com-
ponents of the self-energies in the Keldysh space is only possible in
the case of inelastic mechanisms such as electron-electron interaction
or electron-phonon interaction (see next case).
• Case 2: Electron-electron interaction.
Let us consider an electronic system where the electron-electron in-
teraction is assumed to be the perturbation. The system might be
out of equilibrium due to, for instance, the presence of a current. For
the sake of concreteness, let us assume that the unperturbed system
can be described by a tight-binding Hamiltonian and the interaction is
Hubbard-like (see Appendix A)
X
H = H0 + U ni↑ ni↓ . (7.23)
i

The diagrams are topologically identical to the equilibrium ones and the
only difference is the fact that one has to indicate where the time argu-
ments reside on the Keldysh contour. In this respect, every equilibrium
diagram gives rise to several diagrams for the different components of
the self-energy in Keldysh space. We illustrate this fact in Fig. 7.2,
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186 Molecular Electronics: An Introduction to Theory and Experiment

+ + − +
iσ i−σ iσ i−σ

iσ i−σ iσ
+ + + + i−σ
Fig. 7.2 Examples of second-order self-energy diagrams in the Keldysh space for the
electron-electron interaction. The indexes + and − indicate in which branch the time
arguments lie.

where we show the self-energy diagrams of second order in U for the

components Σ++ and Σ+− . The expression of the self-energy Σ+−
ii , for
instance, would be (ignoring the spin dependence)
¤2 −+ ′
Σ+− ′ 2
£ +−
ii (t, t ) = U gii (t, t′ ) gii (t , t). (7.24)

7.3 Basic relations and equations in the Keldysh formalism

In the previous section we have seen that the Dyson’s equation has acquired
an additional 2 × 2 matrix structure, which gives the impression that one
has to solve four times more equations than in the equilibrium case. Indeed,
one can show that the different functions in the 2 × 2 matrix of Eq. (7.16)
are not independent and the number of equations that one has to solve in
practice can be reduced to only two. In this sense, the goal of this section
is to derive those equations and to discuss the general properties of the
Keldysh-Green’s functions.

7.3.1 Relations between the Green’s functions

Let us explore the different relations between the functions appearing in the
Keldysh formalism. We start by showing that the four Green’s functions
G++ , G+− , G−+ and G−− are not independent, but satisfy
G++ + G−− = G+− + G−+ . (7.25)
This is a direct consequence of the definition of these functions. Thus for
instance,
† †
G++ ′ ′ ′ ′ ′
ij (t, t ) = −iθ(t − t )hciσ (t)cjσ (t )i + iθ(t − t)hcjσ (t )ciσ (t)i
= θ(t − t′ )G−+ ′ ′ +− ′
ij (t, t ) + θ(t − t)Gij (t, t ). (7.26)
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Nonequilibrium Green’s functions formalism 187

Analogously,
G−− ′ ′ +− ′ ′ −+ ′
ij (t, t ) = θ(t − t )Gij (t, t ) + θ(t − t)Gij (t, t ). (7.27)
Adding these two equations, we obtain the relation stated above.
On the other hand, from this relation and using the Dyson’s equation
in Keldysh space, see Eq. (7.18), one can show the following relation be-
tween the different elements of the self-energy matrix in Keldysh space (see
Exercise 7.3)
Σ++ + Σ−− = − Σ+− + Σ−+ .
¡ ¢
(7.28)
Other important relations are those between the Keldysh-Green’s func-
tions and the advanced and retarded functions Ga and Gr . Such relations
can be found as follows. Using the expression of Eq. (7.26), one obtains
G++ +−
′ ′ ′
£ +− ′ −+ ′
¤
ij (t, t ) − Gij (t, t ) = −θ(t − t ) Gij (t, t ) − Gij (t, t ) , (7.29)
and using the definitions of G+− and G+− , we arrive at
† †
G++ ′ +− ′ ′ ′ ′
ij (t, t ) − Gij (t, t ) = −iθ(t − t )hciσ (t)cjσ (t ) + cjσ (t )ciσ (t)i
= Grij (t, t′ ) (7.30)
Proceeding in an analogous way, one can show the following relations
Gr = G++ − G+− = G−+ − G−− (7.31)
a ++ −+ +− −−
G =G −G =G −G . (7.32)
These relations are crucial for the discussion of next section.

7.3.2 The triangular representation

As we have seen above, there are redundancies in the Green’s functions and
in that sense it is natural to try to get rid of them to simplify the equations
as much as possible. In what follows, we shall try to eliminate G++ and
G−− in favor of Gr and Ga . For this purpose, we will apply a unitary
transformation to perform the following change
µ ++ +− ¶
0 Ga
µ ¶
G G
−→ , (7.33)
G−+ G−− Gr GK
where GK = G++ +G−− = G+− +G−+ is known as the Keldysh function.

It is easy to show that the unitary transformation has the form

µ ¶
1 1 −1 1
Ř = √ = √ (1̌ − iσ̌y ), (7.34)
2 1 1 2
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188 Molecular Electronics: An Introduction to Theory and Experiment

where σ̌y is the corresponding Pauli matrix. The representation above

is known as the triangular representation and it is important from the
practical point of view. Let us now denote the standard Keldysh matrix
by Ǧ and the corresponding matrix in the triangular representation as G̃.
They are related by G̃ = ŘǦŘ−1 . Applying the transformation Ř to the
Dyson’s equation1
Ǧ = ǧ + ǧΣ̌Ǧ, (7.35)
we obtain the corresponding Dyson’s equation in the triangular represen-
tation
G̃ = g̃ + g̃Σ̃G̃, (7.36)
where the self-energy in this representation has the form
µ K r¶
Σ Σ
Σ̃ = . (7.37)
Σa 0
Here, the new self-energy components are expressed in terms of those of
the original representation as follows
ΣK = Σ++ + Σ−− = − Σ+− + Σ−+
¡ ¢
(7.38)
r ++ +−
¡ −− −+
¢
Σ =Σ +Σ =− Σ +Σ (7.39)
a ++ −+
¡ −− +−
¢
Σ =Σ +Σ =− Σ +Σ . (7.40)
From Eqs. (7.36) and (7.37) one can show that the advanced and re-
tarded Green’s functions satisfy independent Dyson’s equations, i.e.
Gr,a = gr,a + gr,a Σr,a Gr,a . (7.41)
Notice that this equation is formally identical to the equilibrium one. In
the case in which the perturbation is an external potential, as we showed
in the previous section, the corresponding self-energies reduce to Σa (r, t) =
Σr (r, t) = V (r, t), i.e. like in equilibrium.
On the other hand, the Keldysh function GK fulfills the following equa-
tion
GK = gK + gK Σa Ga + gr Σr GK + gr ΣK Ga . (7.42)
Notice now that GK is coupled to Gr,a and this equation requires to solve
first Dyson’s equation for these latter functions. Let us recall that the
retarded and advanced functions are related, which in practice means that
1 Inthis equation, as in the next ones, the integrations over the intermediate arguments
are implicitly assumed.
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Nonequilibrium Green’s functions formalism 189

there are only two functions to be determined, as we stated at the beginning

of this section.
The previous equation can be written in a more symmetric way as fol-
lows. We first group on the left hand side all the terms containing GK
−1
and then we multiply from the left by (1 − gr Σr ) on both sides of the
equation to arrive at
−1 −1
GK = (1 − gr Σr ) gK (1 + Σa Ga ) + (1 − gr Σr ) gr ΣK Ga . (7.43)
Then, using the Dyson’s equation for the retarded function, we finally ob-
tain
GK = (1 + Gr Σr ) gK (1 + Σa Ga ) + Gr ΣK Ga . (7.44)
In this book, we shall mainly use the function G+− , rather than the
Keldysh function GK . For this reason, we now proceed to derive the corre-
sponding equation for G+− . We first take the element +− in the Dyson’s
equation, i.e.
+−
G+− = g+− + (gΣG) . (7.45)
Then, we make use of the relations derived above between the different
functions to arrive at (see Exercise 7.3)
G+− = g+− + g+− Σa Ga + gr Σr G+− − gr Σ+− Ga . (7.46)
The function G−+ fulfills a similar equation that can be obtained from the
previous one by exchanging + by − and vice versa. Eq. (7.46) for G+− can
be written in a more symmetric way, in analogy with what we did for the
function GK . Thus, we obtain finally
G+− = (1 + Gr Σr ) g+− (1 + Σa Ga ) − Gr Σ+− Ga . (7.47)
−+
The function G satisfies a similar equation given by
G−+ = (1 + Gr Σr ) g−+ (1 + Σa Ga ) − Gr Σ−+ Ga . (7.48)

7.3.3 Unperturbed Keldysh-Green’s functions

In the Keldysh formalism the time dependence is introduced through the
perturbation and the unperturbed Hamiltonian H0 must correspond to
a noninteracting electron system in equilibrium. Thus, all unperturbed
Green’s functions depend only on the time difference and they are easy
to obtain in energy space. The form and properties of the unperturbed
retarded, advanced and causal functions in energy space were studied in
detail in Chapter 5, whereas the properties of the functions g−− (E) can
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190 Molecular Electronics: An Introduction to Theory and Experiment

be easily deduced from those of g++ (E). Thus, we concentrate now on the
analysis of the functions g+− (E) and g−+ (E). From its definition in the
time domain (and in a discrete basis)
†
G+−
ij (t) = ihcjσ (0)ciσ (t)i, (7.49)

it is obvious that this function is related to the electron distribution in

equilibrium. Although the temperature does not appear explicitly in the
Keldysh formalism, one uses the previous fact to introduce it. Thus, the
previous expression for t = 0 and i = j reads
Z ∞
dE +−
G+−
ii (0) = ihniσ i = Gii (E). (7.50)
−∞ 2π

This implies that G+− ii (E) = 2πiρi (E)f (E), where f (E) is the Fermi func-
tion and ρi (E) is the local density of states in the site i. In the same way,
one can show that G−+ ii (E) = −2πiρi (E)[1 − f (E)]. Taking into account
this result, it is clear that G+− ∝ f (E) and G−+ ∝ 1 − f (E). This fact
together with the general relation

Ga (t) − Gr (t) = G+− (t) − G−+ (t), (7.51)

leads to the following relations

G+− (E) = [Ga (E) − Gr (E)] f (E) (7.52)

−+ a r
G (E) = − [G (E) − G (E)] [1 − f (E)] . (7.53)

It is worth stressing that we have written the previous expressions using

capital letters to indicate that these expressions are always valid in equi-
librium, even in an interacting case. In the Keldysh formalism the unper-
turbed system is moreover non-interacting, which implies that in a basis |ii
one has
+−
£ a r
¤
gij (E) = gij (E) − gij (E) f (E) (7.54)
−+
£ a r
¤
gij (E) = − gij (E) − gij (E) [1 − f (E)] .

As a consequence, these functions are proportional to the spectral den-

sities and to the thermal distribution function. The way in which we have
introduced the temperature in the Keldysh formalism is certainly not very
satisfactory. However, one can show that a rigorous derivation leads exactly
to the result that we have just described (see for instance Ref. [192]).
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Nonequilibrium Green’s functions formalism 191

7.3.4 Some comments on the notation

The notation used here for the different Keldysh-Green’s functions is not
shared by all the authors. In this sense, it is important to devote a few lines
to make contact with other texts where the Keldysh formalism is described.
Frequently, the functions G+− and G−+ are denoted by G< and G> ,
respectively. Sometimes, the Keldysh function GK is denoted by GF or
simply by F . On the other hand, the triangular representation is often
written in a slightly different way. One first defines a new matrix function
as Ḡ = σz Ǧ, where σz is the Pauli matrix, and then the unitary trans-
formation of Eq. (7.34) is applied. This leads to a 2 × 2 matrix with the
form
µ r K¶
G G
, (7.55)
0 Ga

which is often used in the field of superconductivity [193].

7.4 Application of Keldysh formalism to simple transport

problems

In this section we shall illustrate the utility of the Keldysh formalism by

applying it to the description of the electronic transport in some simple
situations of special interest. Our goal is two fold. First, we want to
illustrate how this formalism is used in practice and second, we want to
show how the elastic transmission can be computed from an atomistic point
of view.
Most of the systems that we have in mind (atomic contacts, molecular
junctions, etc.) are conveniently described by a tight-binding Hamiltonian
of the following form
X ³ † ´
tij ciσ cjσ + c†jσ ciσ ,
X
H= ǫi niσ + (7.56)
iσ ijσ

where we have assumed, without loss of generality, that the hopping ele-
ments tij are real. Our first task is to derive an expression for the electrical
current operator in this local basis. For this purpose, we first consider the
simple case of a tight-binding chain with only nearest-neighbor hoppings,
denoted by t. Such a chain is schematically represented in Fig. 7.3. Let us
compute now the current between the sites k and k + 1. Without doing
any calculation, one can guess that the operator must adopt somehow the
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192 Molecular Electronics: An Introduction to Theory and Experiment

following form2
Xh i
I∝t c†kσ (t)ck+1σ (t) − c†k+1σ (t)ckσ (t) , (7.57)
σ
where the first term in the sum represents the current flowing in one direc-
tion and second one corresponds to the current flowing in the opposite one.
Let us see if a rigorous calculation confirms our intuition.

B A
t
........ ........
k−1 k k+1

Fig. 7.3 Schematic representation of a linear chain with only nearest-neighbor hoppings.

The current operator must be obtained from the continuity equation

that describes the charge conservation. Such equation can be written in a
discrete representation as
∂ρk
IA − IB + = 0, (7.58)
∂t
where A represents a point between the sites k and k + 1 and B a point
between k − 1 and k, see Fig. 7.3. Here, ρk is the operator that describes
the charge in the site k
X †
ρk = e ckσ ckσ (7.59)
σ
and satisfies the equation of motion of Heisenberg operators
∂ρk i
= − [ρk , H] . (7.60)
∂t ~
Notice that we have reintroduced ~, and we shall write it explicitly from
now on. Using the expression of Eq. (7.56) for the homogeneous chain that
we are considering, it is straightforward to compute the commutator that
appears in the previous equation of motion and thus, one arrives at
∂ρk −iet X n † o
= ckσ ck+1σ − c†k+1σ ckσ + c†kσ ck−1σ − c†k−1σ ckσ .
∂t ~ σ
Rewriting this expression in the form of the continuity equation, see
Eq. (7.58), we can identify the current operator, which at point A takes the
form
iet X n † o
IA (t) = ckσ (t)ck+1σ (t) − c†k+1σ (t)ckσ (t) . (7.61)
~ σ
2 We believe that no confusion can arise between the hopping t and the time appearing
as an argument in the creation and annihilation operators.
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Nonequilibrium Green’s functions formalism 193

Notice that this has exactly the intuitive form that we had anticipated
above.
This expression can be easily generalized to any 3D system described by
a tight-binding Hamiltonian as in Eq. (7.56). The electrical current through
an arbitrary surface that separates two regions A and B is given by
ie X X n † o
I(t) = tij ciσ (t)cjσ (t) − c†jσ (t)ciσ (t) . (7.62)
~ σ
i∈A;j∈B

Let us now compute the expectation value of the current operator, for
instance, for the case of the chain. According to Eq. (7.61), one can write
(dropping the subindex A)
iet X n † o
hI(t)i = hckσ (t)ck+1σ (t)i − hc†k+1σ (t)ckσ (t)i . (7.63)
~ σ
The expectation values appearing in the previous equation can be expressed
in terms of the Keldysh functions G+− as follows
e X n +− o
hI(t)i = t Gk+1,k (t, t) − G+−
k,k+1 (t, t) , (7.64)
~ σ

and there is a similar expression for the most general case of Eq. (7.62).
In many situations, for instance when there is a constant voltage applied
in a junction, the problem admits a stationary solution and the Green’s
functions depend exclusively on the difference of the time arguments. In
those cases, Eq. (7.64) can be written in terms of the Green’s functions in
energy space as
e X ∞ dE n +−
Z o
hIi = t Gk+1,k (E) − G+−
k,k+1 (E) . (7.65)
~ σ −∞ 2π
We are now in position to discuss the electronic transport in some simple
examples of special interest.

7.4.1 Electrical current through a metallic atomic contact

As a first example, we consider an atomic constriction. As we learned
in the first part of this book, such contacts can nowadays be fabricated
with the scanning tunneling microscope or with the mechanically control-
lable break-junctions. For the sake of simplicity, we consider the case of
a metal described by a tight-binding Hamiltonian with a single relevant
atomic orbital per site. We assume that the two electrodes forming the
atomic junction are only coupled through their outermost atoms, denoted
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194 Molecular Electronics: An Introduction to Theory and Experiment

Fig. 7.4 Schematic representation of a single-channel atomic contact. The electrodes

are coupled via the hopping element t that describes the coupling between the two
outermost atoms of both leads, denoted by L and R. There is a bias voltage applied
across the system giving rise to a difference in the chemical potential of the electrodes:
eV = µL − µR .

as L and R, via a single hopping element t. This situation is schematically

represented in Fig. 7.4. Here, the specific shape of the electrodes is irrele-
vant for our discussion. As it will become clear later, this is a model for a
contact with a single conduction channel and if everything is consistent, we
should arrive at the Landauer formula. However, contrary to the scattering
approach, we will now be able to obtain a microscopic expression for the
transmission coefficient in terms of the coupling element t and the local
electronic structure of the electrodes.
This model system is described by the following tight-binding Hamilto-
nian
X ³ † ´
H = HL + HR + t cLσ cRσ + c†Rσ cLσ , (7.66)
σ

where HL and HR are the Hamiltonians describing the left and right elec-
trodes, respectively. We assume that there is a bias voltage V applied across
the contact and that the potential drops abruptly in the interface region.
The task in this example is to compute the current-voltage characteris-
tics. According to Eqs. (7.63-7.65), the current evaluated at the interface
between the electrodes is given by3
2et ∞
Z
dE G+− +−
£ ¤
I = hIi = RL (E) − GLR (E) , (7.67)
h −∞
3 We assume that the voltage is time-independent and therefore the problem admits a

stationary solution. This allows us to write the current in terms of the Fourier transform
of the Green’s functions with respect to the difference of the time arguments.
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Nonequilibrium Green’s functions formalism 195

where the factor 2 is due to the spin degeneracy in this problem. At

this stage the problem is to determine the Green’s functions appearing
in Eq. (7.67). For this purpose, we employ the perturbative method that
we have just described in the previous sections. Therefore, the first thing
that we need to do is to choose the perturbation. Let us remind that in
the Keldysh formalism the unperturbed system has to be in equilibrium.
One possibility would be to introduce the voltage as a perturbation, but
this is not very convenient because such a perturbation is extended over
the whole system and the calculation would be rather cumbersome. The
most convenient choice is to treat the coupling term in Eq. (7.66) as the
perturbation and include the voltage in the unperturbed Hamiltonians by
shifting the corresponding chemical potential (e.g. µL = eV and µR = 0).4
With this choice, the retarded and advanced self-energies associated to
this single-particle perturbation adopt the form
Σr,a r,a
LR = ΣRL = t, (7.68)
while the Keldysh self-energies vanish: Σ+− = Σ−+ = 0 (there are no in-
elastic interactions). Now, the functions G+− +−
LR and GRL appearing in the
expression of the current can be determined in terms of the Green’s func-
tions of the uncoupled electrodes (unperturbed functions) using Eq. (7.47).
But before doing so, we can simplify the algebra by writing the current in
terms of the diagonal Green’s functions of both electrodes. For this pur-
pose, we compute G+− LR making use of Eq. (7.46) by writing it as (remember
that Σ+− = 0 in this problem)
G+− = g+− + g+− Σa Ga + gr Σr G+− , (7.69)
while we compute G+−
RL using this equation, but written in the following
alternative form:
G+− = g+− + G+− Σa ga + Gr Σr g+− . (7.70)
It is important to emphasize that these equations are algebraic equations
in energy space and we shall often omit, as we have just done, the energy
argument of the Green’s functions, E, to abbreviate the notation.
Using the last two equations, we can write G+− +−
LR and GRL as

G+− +− a a r r +−
LR = gLL ΣLR GRR + gLL ΣLR GRR , (7.71)
G+−
RL = G+− a a
RR ΣRL gLL + +−
GrRR ΣrRL gLL . (7.72)
4 Thisdoes not mean that the unperturbed system is out of equilibrium since in the
absence of coupling, there is no current and the electron distributions in both leads are
the equilibrium one.
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196 Molecular Electronics: An Introduction to Theory and Experiment

Substituting now G+− +−

LR and GRL in Eq. (7.67) and using the general
a r +− −+
relation G − G = G − G , one arrives at
2e 2 ∞
Z
dE gLL (E)G−+
£ +− −+ +−
¤
I= t RR (E) − gLL (E)GRR (E) . (7.73)
h −∞
We now compute the functions G+− −+
RR and GRR by using Eqs. (7.47) and
(7.48)
+−/−+ +−/−+
GRR = (1 + GrRL ΣrLR ) gRR (1 + ΣaRL GaLR ) + (7.74)
+−/−+ a
GrRR ΣrRL gLL ΣLR GaRR . (7.75)
Introducing these expressions in Eq. (7.73) we obtain
2e 2 ∞
Z
£ +− −+ −+ +−
dE |1 + tGrRL (E)|2 gLL
¤
I= t (E)gRR (E) − gLL (E)gRR (E) .
h −∞
(7.76)
Here, we have used the explicit expression of the self-energies, see Eq. (7.68),
and the fact that Ga (E) = [Gr (E)]† (thus e.g., GaLR (E) = [GrRL (E)]∗ ).
To complete the calculation we still have to determine the retarded
function GrRL (E). This can be done, very much like in equilibrium, using
its Dyson’s equation, see Eq. (7.41). Taking the element (R, L) we arrive
at
GrRL = gRR
r
ΣrRL GrLL . (7.77)
To close this equation, we need now an equation for GrLL ,
which is obtained
by taking the element (L, L) in the Dyson’s equation, i.e.
GrLL = gLL
r r
+ gLL ΣLR GrRL . (7.78)
Substituting back into the equation for GrRL , we obtain finally
r r
tgRR gLL 1
GrRL = r r and 1 + tGrRL = r gr . (7.79)
1 − t2 gRR gLL 1 − t2 gRR LL
Before coming back to the expression of current, let us remind that the
unperturbed Keldysh functions g+−/−+ can be expressed in terms of the
retarded and advanced ones using Eq. (7.54). Thus, the functions appearing
in Eq. (7.76) can be written as
+− a r
gLL (E) = [gLL (E − eV ) − gLL (E − eV )] f (E − eV ) (7.80)
= 2πiρL (E − eV )f (E − eV )
−+ a r
gLL (E) = − [gLL (E − eV ) − gLL (E − eV )] [1 − f (E − eV )]
= −2πiρL (E − eV ) [1 − f (E − eV )]
+− a r
gRR (E) = [gRR (E) − gRR (E)] f (E) = 2πiρR (E)f (E)
−+ a r
gRR (E) = − [gRR (E) − gRR (E)] [1 − f (E)] = −2πiρR (E) [1 − f (E)] ,
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Nonequilibrium Green’s functions formalism 197

where f (E) is the Fermi function and ρL/R is the local density of states of
the leads projected onto the sites L and R. Notice that we have already
taken into account the relative shift of the chemical potentials due to the
bias voltage V .
Using Eqs. (7.79) and (7.80), we can finally write the current as follows5
2e ∞ 4πt2 ρL (E − eV )ρR (E)
Z
I= dE [f (E − eV ) − f (E)] .
h −∞ |1 − t2 gLL (E − eV )gRR (E)|2
(7.81)
Notice that Eq. (7.81) has exactly the form of the Landauer formula,
i.e.
2e ∞
Z
I= dE T (E, V ) [f (E − eV ) − f (E)] , (7.82)
h −∞
where we can identify T (E, V ) as an energy and voltage-dependent trans-
mission probability given by
4πt2 ρL (E − eV )ρR (E)
T (E, V ) = . (7.83)
|1 − t2 gLL (E − eV )gRR (E)|2
As it can be seen, the transmission depends primarily on the coupling
element t and the local electronic structure of the leads.
For sufficiently low voltages, there is a linear regime where the current is
proportional to the voltage. In this limit, the conductance is given by G =
(2e2 /h)T (EF , V = 0), where T (EF , V = 0) is the zero-bias transmission at
the Fermi energy given by
4πt2 ρL (EF )ρR (EF )
T (EF , V = 0) = . (7.84)
|1 − t2 gLL (EF )gRR (EF )|2
One can often consider that the Green’s functions are constant around the
Fermi energy and one can also neglect their real part (this is the wide-band
approximation introduced in Chapter 5). This means that the lead Green’s
functions can be approximated by
i
gLL ≈
, (7.85)
W
where W = 1/πρL/R (EF ) (we are assuming a symmetric contact (gLL =
gRR ) for simplicity). Within this approximation, one obtains the following
expression for the transmission
4t2 /W 2
T = . (7.86)
(1 + t2 /W 2 )2
5 This expression for the current was first derived in Ref. [194] for a more realistic model.
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198 Molecular Electronics: An Introduction to Theory and Experiment

This expression illustrates the transition from the tunnel regime, when
the electrodes are separated by a large distance, to the contact regime at
small distances. In the former limit, the transmission given in Eq. (7.86)
can be approximated by 4t2 /W 2 . This means that the dependence of the
transmission on the distance between the electrodes, and therefore that of
the linear conductance, is determined by t2 . At large distances, a hopping
element is roughly proportional to the overlap of the atomic orbitals and
decays exponentially with the distance between the corresponding atoms.
This is how the exponential length dependence, which we already discussed
in section 4.4, comes about from an atomistic point of view. From the scat-
tering approach, see section 4.4, we concluded that the length dependence
of a metallic tunnel junction is determined by the metal work function.
However, with this simple model, we get the impression that such a de-
pendence is governed by a local property, namely the coupling between the
outermost orbitals of the electrodes. These two pictures, which at first
glance look contradictory, can indeed be reconciled. This is, however, a
subtle issue that is out of the scope of this book and we refer the reader to
Ref. [195] for a discussion of this question.
When the electrodes approach each other the hopping t becomes of the
same order as the energy scale W and the transmission can reach unity
and in turn the conductance approaches the quantum of conductance G0 =
2e2 /h. The transition from tunnel to contact was first discussed within
this type of atomistic models in Ref. [194] in connection with the first
experiment that explored such a transition [55]. For an overview on recent
experiments exploring the tunnel-to-contact transition both in single atoms
and molecules, see Refs. [196, 197].
Let us now study in more detail the tunnel limit (t → 0). In this case,
the non-linear current of Eq. (7.81) can be approximated by
8πe 2 ∞
Z
I= t dE ρL (E − eV )ρR (E) [f (E − eV ) − f (E)] , (7.87)
h −∞

which tell us that the current in this limit is determined by the convolution
of the local density of states of both electrodes. This well-known expression
is a fundamental result for the theory of STM and provides a simple inter-
pretation of the STM images. Assuming that the left electrode represents
a STM tip with a constant density of states around the Fermi energy, the
differential conductance at low temperatures is simply given by
dI 2e2
G(V ) = = 4πt2 ρL (EF )ρR (EF + eV ), (7.88)
dV h
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Nonequilibrium Green’s functions formalism 199

i.e. the conductance is a measure of the local density of states of the sample
(or right electrode in our case).

7.4.2 Shot noise in an atomic contact

Another interesting transport property that can easily be calculated with
the Keldysh formalism is the shot noise (or nonequilibrium current fluc-
tuations), which was introduced in Chapter 4. Let us consider the model
for an atomic contact discussed in the previous subsection. Our goal now
is the calculation of the current fluctuations in the zero-temperature limit
and finite bias (shot noise).
The noise is characterized by the fluctuation spectral density that is
defined as
Z ∞
P (ω) = ~ dt eiωt hδI(t)δI(0) + δI(0)δI(t)i, (7.89)
−∞

where δI(t) = I(t) − hI(t)i.

We are specially interested in the zero-frequency noise, P (0),
Z ∞
P (0) = ~ dt hδI(t)δI(0) + δI(0)δI(t)i. (7.90)
−∞

If we now substitute the expressions for I(t) and hI(t)i for an atomic
contact and we write the result in terms of the Green’s functions, we obtain
2e2 ∞
Z
dE G+− −+ +− −+
£
P (0) = LR (E)GRL (E) + GRL (E)GLR (E)−
h −∞
G+− −+ +− −+
¤
LL (E)GRR (E) − GRR (E)GLL (E) . (7.91)

Here, in order to obtain this expression, we have made use of Wick’s theo-
rem to decouple the averages of four operators (let us remind that this is
valid since our electron system is noninteracting).
At this stage the calculation of the shot noise has been reduced to
the computation of the different Keldysh-Green’s functions that appear in
Eq. (7.91). These functions can be calculated following exactly the same
procedure detailed in the previous subsection. If we now assume zero tem-
perature and use the wide-band approximation of Eq. (7.85) for the un-
perturbed Green’s functions, we can obtain the following expression (see
Exercise 7.5)
4e2
P (0) = T (1 − T )V, (7.92)
h
which is the result derived in section 4.7 using the scattering approach.
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200 Molecular Electronics: An Introduction to Theory and Experiment

7.4.3 Current through a resonant level

Let us now discuss the calculation of the current for the resonant level
model discussed in section 5.3.3. Let us remind that in this model a single
quantum level is coupled to two metallic electrodes and the corresponding
Hamiltonian is given by
X
H = HL + HR + ǫ0 n0σ + (7.93)
σ
³ ´ X ³ ´
c†Lσ c0σ + c†0σ cLσ + tR c†Rσ c0σ + c†0σ cRσ ,
X
tL
σ σ

where ǫ0 is the position of the resonant level, which in principle can also
depend on the bias voltage, and tL,R are the matrix elements describing
the coupling to the reservoirs. Here, L and R denote the outermost sites
of the left and right electrodes, respectively. On the other hand, we now
assume that there is a constant bias voltage across the system and our task
is to compute the current-voltage characteristics.
We start by evaluating the current at the interface between the left
electrode and the level, which in terms of the Green’s functions G+− can
be written as follows
2etL ∞
Z
dE G+− +−
£ ¤
I= L0 (E) − G0L (E) . (7.94)
h −∞
In order to determine the Green’s functions in the previous expression,
we use again the Keldysh formalism and we treat the coupling terms be-
tween the level and the electrodes, i.e. the second line in Eq. (7.93), as
a perturbation. With this choice the only non-vanishing elements of the
self-energy are: Σr,a r,a r,a r,a
L0 = Σ0L = tL and ΣR0 = Σ0R = tR .
Following now the same steps as in section 7.4.1, we can write the current
in terms of diagonal elements of the Green’s functions as
2etL ∞
Z
£ +−
(E)G−+ −+ +−
¤
I= dE gLL 00 (E) − gLL (E)G00 (E) . (7.95)
h −∞
Now, to determine the full Green’s functions, we use the Dyson’s equa-
tion, Eq. (7.47), to write
+−/−+ +−/−+
G00 = (1 + Gr Σr )00 g00 (1 + Σa Ga )00 + (7.96)
+−/−+ a +−/−+ a
Gr00 Σr0L gLL ΣL0 Ga00 + Gr00 Σr0R gRR ΣR0 Ga00 .
If we now substitute this expression into the current formula, the term
+−/−+ +−/−+
containing gLL is canceled. Moreover, the term proportional to g00
+−/−+
does not contribute either. The reason is that g00 (E) ∝ δ(E − ǫ0 ) and
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Nonequilibrium Green’s functions formalism 201

the prefactor of this term vanishes at E = ǫ0 .6 Thus, the current can now
be expressed as Z
∞
2e
I = 4π 2 t2L t2R dE ρL (E)ρR (E)|Gr00 (E)|2 [fL (E) − fR (E)] , (7.97)
h −∞
where it is implicitly assumed that the density of states (and distribution
function) of the left electrode is shifted by eV . Notice that we have already
used the expression of the lead Green’s functions in terms of the local
density of states and Fermi functions.
At this point, the only remaining task is the calculation of Gr00 (E), but
this is something that we have already done in section 5.3.3 and we just
recall here the result
1
Gr00 (E) = r (E) − t2 g r (E) . (7.98)
E − ǫ0 − t2L gL R R
Therefore, the currentZ adopts again the form of the Landauer formula
2e ∞
I= dE T (E, V ) [f (E − eV ) − f (E)] , (7.99)
h −∞
where this time the transmission T (E, V ) is given by
4π 2 t2L t2R ρL (E − eV )ρR (E)
T (E, V ) = r (E − eV ) − t2 g r (E)|2 . (7.100)
|E − ǫ0 − t2L gL R R
To simplify this expression, we use now as in section 5.3.3 the wide-
band approximation and neglect the energy dependence introduced by the
r
leads. This way, gL/R ≈ −iπρL/R (EF ) and we define the scattering rates
2
ΓL/R = πtL/R ρL/R (EF ). In this approximation the transmission can be
written as
4ΓL ΓR
T (E, V ) = . (7.101)
(E − ǫ0 )2 + (ΓL + ΓR )2
In this case, the voltage dependence of the transmission may only stem from
the eventual voltage dependence of the level position. This expression is
the well-known Breit-Wigner formula that was derived in Chapter 4 within
the scattering approach (see Exercises 4.5 and 4.8) and it will be used
extensively in later chapters.
Again, in the linear regime the low-temperature conductance is simply
given by G = (2e2 /h)T (EF , 0). This expression shows that the maximum
conductance is reached when EF = ǫ0 , which is the resonant condition.
In the symmetric case (ΓL = ΓR ), this maximum is equal to G0 = 2e2 /h,
irrespectively of the value of the scattering rates. These facts are illustrated
in Fig. 7.5. The non-linear current-voltage characteristics of this model will
be discussed in detail in Chapter 15.
6 Physically speaking, it is quite reasonable that this term does not contribute to the

current. It makes no sense that the current depends on the occupation of the level before
being coupled to the electrodes.
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202 Molecular Electronics: An Introduction to Theory and Experiment

1 1
0.8 (a) 0.8
(b)
G/G0

0.6 0.6
0.4 0.4
0.2 0.2
0 0
-10 -5 0 5 10 0 0.2 0.4 0.6 0.8 1
(EF - ε0)/Γ ΓL/ΓR

Fig. 7.5 Zero-temperature linear conductance in the resonant tunneling model. (a)
Linear conductance (normalized by G0 = 2e2 /h) as a function of the level position, ǫ0
for a symmetric contact ΓL = ΓR = Γ. (b) Linear conductance at resonance (ǫ0 = EF )
as a function of the ratio between the scattering rates.

7.5 Exercises

7.1 Diagrammatic expansion in the Keldysh formalism: Show explicitly

that hφ0 |Sc |φ0 i = 1 by using the expansion of the operator Sc . For this purpose,
expand Sc up to second order and show that the contributions of order higher
than zero cancel.
7.2 Time-dependent external potential: Let us consider a system with N
noninteracting electrons subjected to a time-dependent external potential:

N
X
V(t) = V (ri , t). (7.102)
i=1

Apply Wick’s theorem to demonstrate that the self-energy is given by Eq. (7.22).
7.3 Properties of the Keldysh-Green’s functions:
(a) Demonstrate the property of Eq. (7.28). Hint: Use the property of
Eq. (7.25) and the Dyson’s equation in Keldysh space.
(b) Demonstrate Eq. (7.46).
7.4 Shot noise in a single-channel point contact:
Derive the expression of the zero-frequency shot noise of a single-channel point
contact following the discussion of the example of section 7.4.2 and demonstrate
that it is given by
4e2
P (0) = T (1 − T )eV,
h
where T is the energy-independent transmission coefficient of the contact given
by Eq. (7.86).
7.5 Electrical current through a linear chain: Consider the electronic trans-
port in a finite one-dimensional system formed by a tight-binding chain with N
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Nonequilibrium Green’s functions formalism 203

sites such that the site 1 is connected to the left electrode through a hopping tL
and the site N is connected to the right electrode with a hopping tR . Show that
the current formula in this case is given by

2e 2 2 2 ∞
Z
I= 4π tL tR dE ρL (E − eV )ρR (E)|Gr1N (E)|2 [f (E − eV ) − f (E)] .
h −∞

For the sake of simplicity, consider that in the chain there are only hoppings
between nearest-neighbor atoms, t, and that the on-site energy is given by ǫ0 .
Study the linear conductance of this system as a function of the number of sites
N in the chain and show that it may exhibit parity oscillations, depending on
whether N is even or odd.
7.6 Thermopower of a single-channel point contact: Using the model
of section 7.4.1, derive the expression for the thermopower for a single-channel
contact and show that it coincides with the result obtained with the scattering
approach in section 4.8.
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204 Molecular Electronics: An Introduction to Theory and Experiment

January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Chapter 8

Formulas of the electrical current:

Exploiting the Keldysh formalism

In the previous chapter we showed how the Keldysh formalism can be com-
bined with simple Hamiltonians to compute the current in model systems.
In this chapter we shall exploit this technique and derive some general
expressions for the electrical current that can be combined with realistic
methods for the determination of the electronic structure. To be precise,
we shall address three basic issues:

(1) Derivation of Landauer formula in the framework of the non-

equilibrium Green’s function techniques. Here, the goal is the determi-
nation of the microscopic expression for the elastic transmission valid
for any atomic and molecular junction.
(2) Generalization of Landauer formula to include inelastic and correlation
effects.
(3) Description of the current in atomic-scale junctions subjected to time-
dependent potentials.

This chapter is rather technical and it can be skipped by those who are
not so interested in the algebra behind the current formulas. Anyway, we
recommend to read the next section about the derivation of the Landauer
formula, since the expression obtained there for the elastic transmission will
be frequently used in subsequent chapters.

8.1 Elastic current: Microscopic derivation of the Landauer

formula

In section 7.4 we discussed two simple examples of atomic-scale contacts. In

both cases we ended up with a Landauer-like formula for the elastic current,

205
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206 Molecular Electronics: An Introduction to Theory and Experiment

Fig. 8.1 Schematic representation of an atomic-scale contact of arbitrary geometry. We

divide this system into three parts: a central region, C, and the two leads, L and R.

the only difference being the expression for the transmission coefficient. In
this section we shall demonstrate that this was not a coincidence and we
shall derive a general expression for the elastic current valid for any type
of atomic and molecular junction.
Let us consider a contact with arbitrary geometry like the one depicted
in Fig. 8.1. Such a contact can be either an atomic contact or a molecular
junction. Since we shall ignore inelastic interaction in this discussion, one
can describe the system in terms of the following generic tight-binding
Hamiltonian

hiα,jβ c†iα,σ cjβ,σ ,

X
H= (8.1)
ij,αβ,σ

where i, j run over the atomic sites and α, β denote the different atomic
orbitals. The number of orbitals in each site can be arbitrary. For the sake
of simplicity, we assume that the local basis is orthogonal. Later in this
section, we shall generalize the results to the case of nonorthogonal basis
sets. Notice also that we are assuming that matrix elements are independent
of the spin, i.e. for the moment we do not consider magnetic situations.
We now distinguish three different parts in this contact: the reservoirs
L and R, and a central region that can have arbitrary size and shape.
In principle, the reservoirs L and R could also have an arbitrary shape
and we assume that an electron in these subsystems has a well-defined
temperature and chemical potential. In other words, these regions play the
role of electron reservoirs, in the spirit of the scattering approach of Chapter
4. The separation of the contact in these three subsystems is somewhat
arbitrary, especially in the linear response regime, and one can play with
that, as we shall discuss below. We also assume that there is no direct
coupling between the reservoirs. With this assumption the Hamiltonian
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Formulas of the electrical current 207

above can be written in the following matrix form

 
HLL tLC 0
H =  tCL HCC tCR  , (8.2)
0 tRC HRR
where the diagonal terms HXX with X = L, C, R are the Hamiltonian of
the three subsystems and the t’s describe the coupling between them.
Our aim is to determine the current through the contact induced by a
constant bias voltage, eV = µL − µR . For this purpose, we first evaluate
the current at the interface between the left lead L and central region C,
which in the tight-binding representation adopts the form (see section 7.4)
ie ³ ´
hiα,jβ hc†iα,σ cjβ,σ i − hjβ,iα hc†jβ,σ ciα,σ i ,
X
I= (8.3)
~
i∈L;j∈C;α,β,σ

where i runs over the atoms of the left electrode which are connected with
the atoms in the central region C, and j runs over the atoms of the central
region coupled to the left electrode (in principle, all of them). The indexes
α and β indicate the different atomic orbitals in every site.
Following the ideas of the last section of the previous chapter, we make
use of nonequilibrium Green’s function techniques to calculate the current.
First of all, we express the expectation values appearing in the current
expression in terms of the Keldysh-Green’s function G+− . This function
gives information about the distribution function of the system and in a
local basis it adopts the following form
′
+−,σσ
Giα,jβ (t, t′ ) = ihc†jβ,σ′ (t′ )ciα,σ (t)i. (8.4)
Using this expression one can write the current as
e X h
+−,σσ +−,σσ
i
I= tiα,jβ Gjβ,iα (t, t) − tjβ,iα Giα,jβ (t, t) . (8.5)
~
i∈L;j∈C;α,β,σ

The current can be expressed in a more compact way in terms of the

hopping matrices tLC and tCL [see Eq. (8.2)] whose elements are given by
(tLC )iα,jβ = hiα,jβ with i ∈ L; j ∈ C (8.6)
†
(tCL ) = (tLC ) .
Analogously, one can define similar matrices for the Green’s functions
G+− . With this new notation, one can express the current as
2e £
I = Tr G+− +−
¤
CL (t, t)tLC − tCL GLC (t, t) , (8.7)
~
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208 Molecular Electronics: An Introduction to Theory and Experiment

where Tr denotes the trace over atoms and orbitals in the central region C.
The prefactor 2 comes from the sum over spins, since for the moment we
do not consider any magnetic situation. For the same reason, we drop the
superindex σ in the Green’s functions.
This transport problem admits a stationary solution and therefore, the
different Green’s functions only depend on the difference of time arguments.
Thus, we can Fourier transform with respect to the difference of the time
arguments and write the current as
2e ∞
Z
dE Tr G+− +−
£ ¤
I= CL (E)tLC − tCL GLC (E) . (8.8)
h −∞
Notice that the current is expressed in terms of the trace of a matrix whose
dimension is the number of orbitals in the central region, which we denote as
NC . At this stage, the problem has been reduced to the determination of the
functions G+− in terms of matrix elements of the Hamiltonian of Eq. (8.1).
We shall calculate these functions considering the coupling terms between
the electrodes and the central region as a perturbation. Then, starting from
the Green’s functions for the three isolated systems, we shall determine the
corresponding functions for the whole system. With this choice, the self-
energies of the problem are the hopping matrices defined in Eq. (8.6) and
the equivalent ones for the interface between the central region and the
right electrode R.
We now follow the ideas of section 7.4.3 and make use of Dyson’s equa-
tion in Keldysh space, see Eq. (7.46), to write the functions G+− as follows1
G+− +− a r +−
LC = gLL tLC GCC + gLL tLC GCC (8.9)
G+−
CL = G+− a
CC tCL gLL + +−
GrCC tCL gLL ,
r,a
where gXXare the (retarded, advanced) Green’s functions of the uncoupled
reservoirs (X = L, R). Introducing this equation in the current expression
and making use of the relation G+− − G−+ = Ga − Gr , we obtain
2e ∞
Z
dE Tr G−+ +− +− −+
£ ¤
I= CC tCL gLL tLC − GCC tCL gLL tLC . (8.10)
h −∞
Then, we determine G+−/−+ by means of the relation
G+−/−+ = (1 + Gr t) g+−/−+ (1 + tGa ) . (8.11)
Taking the element (C,C) in this expression we obtain
+−/−+ +−/−+ +−/−+
GCC = GrCC t̂CL gLL tLC GaCC +GrCR tCR gRR tRC GaCC . (8.12)
1 In order to abbreviate the notation, we do not write the energy argument E explicitly.

Moreover, since there are no inelastic processes involved in this model, the self-energies
Σ+− associated with them vanish.
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Formulas of the electrical current 209

+−/−+
Notice that there is an additional contribution containing gCC that was
left out in the previous expression. The reason for this is that, in analogy
with our discussion of the resonant tunneling model in section 7.4.3, one
can show that such a term does not contribute to the final expression of
the current.
Substitution of the previous equation in the expression of the current
yields
2e ∞
Z
−+ +−
dE Tr GrCC tCR gRR tRC GaCC tCL gLL
£
I= tLC −
h −∞
+− −+
GrCC tCR gRR tRC GaCC tCL gLL
¤
tLC . (8.13)
Let us recall that the unperturbed functions g+− and g−+ satisfy the
following relations2
g+− = (ga − gr ) f = 2i Im (ga ) f
−+ (8.14)
g = (ga − gr ) (f − 1) = 2i Im (ga ) (f − 1),
where f is the Fermi function. Thus, the current can be expressed as
8e ∞
Z
I= dE Tr [GrCC tCR Im {gRR
a
} tRC GaCC tCL Im {gLL
a
} tLC ]
h −∞
× (fL − fR ) . (8.15)
Here, fL/R is the Fermi function of the corresponding electrode, which takes
into account the shift of the chemical potential induced by the voltage.
One can further simplify the expression of the current by defining
Σr,a r,a r,a r,a
L = tCL gLL tLC and ΣR = tCR gRR tRC , (8.16)
These matrices are nothing else but the self-energies of this problem for the
subspace of the central region. These self-energies describe the influence of
the reservoir in the central region and they depend both on the coupling
between the reservoirs and the central region and on the local electronic
structure of the leads. Notice that these matrices have a dimension equal
to the number of orbitals in the central region. Using these definitions, the
current can now be rewritten in the following familiar form
2e ∞
Z
I= dE T (E, V ) (fL − fR ) , (8.17)
h −∞
where T (E, V ) is the energy- and voltage-dependent total transmission
probability of the contact given by
T (E, V ) ≡ 4Tr [ΓL GrCC ΓR GaCC ] . (8.18)
2 Noticethat in Eq. (8.14) we have assumed that that Hamiltonian is real, i.e. there is
time reversal symmetry. One can easily show that this implies that gr (E) = [ga (E)]∗ .
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

210 Molecular Electronics: An Introduction to Theory and Experiment

where we have defined the scattering rate matrices as ΓL,R ≡ Im{ΣaL,R }.3
The voltage dependence of the transmission comes through the scattering
rates (i.e. via the leads), but also through the possible voltage dependence
of the Hamiltonian matrix elements of the central region.
We can further symmetrize this expression by £ using † the cyclic
¤
property
£ † of the trace
¤ and write T (E, V ) = Tr t(E, V )t (E, V ) =
Tr t (E, V )t(E, V ) , where
1/2 1/2
t(E, V ) = 2ΓL GrCC ΓR (8.19)
1/2
is the transmission matrix of the system. The existence of Γ as a real
matrix is warranted by Γ being positive definite (see Exercise 8.1).
Finally, the current adopts the form
2e ∞
Z
dE Tr t† (E, V )t(E, V ) (fL − fR ) ,
£ ¤
I= (8.20)
h −∞
valid for arbitrary bias voltage. In the linear regime this expression reduces
to the standard Landauer formula for the zero-temperature conductance
N
2e2 £ † ¤ 2e2 X
G= Tr t (EF , 0)t(EF , 0) = Ti , (8.21)
h h i=1

where Ti are the eigenvalues of t̂† t (or tt̂† ) at the Fermi level. As one can
see, in principle the number of channel would be NC , which is the dimension
of the matrix t† t. However, as we stated at the beginning of this section, the
separation in three subsystems in somewhat arbitrary and one can evaluate
the current at any point. Thus, it is evident that the actual number of
channels is controlled by the narrowest part of the junction. This fact will
be very important in our discussion of the conduction channels in metallic
single-atom contacts, see section 11.5. Notice also that in this formulation,
the conduction channels , defined as the eigenfunctions of t† t̂, are linear
combinations of the atomic orbitals in the central system.
As a result of the discussion above, we have not only re-derived the
Landauer formula, but more importantly, we have also obtained an explicit
formula for the transmission as a function of the microscopic parameters of
the system. As one can see in Eq. (8.18) or in Eq. (8.19), the determination
of the transmission requires the calculations of both the retarded/advanced
Green’s functions of the central system and the scattering rate matrices.
These functions can be determined from their Dyson’s equation
¤−1
GaCC = (GrCC )† = (E − i0+ )1 − HCC − ΣaL − ΣaR
£
, (8.22)
3 We have assumed without loss of generality that the hopping matrix elements are real.
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Formulas of the electrical current 211

where HCC is the Hamiltonian of the central region and the self-energies
ΣX (X = L, R) are given by Eq. (8.16).
On the other hand, the calculation of the scattering rate matrices,
which are the imaginary part of the self-energies of Eq. (8.16), requires
the knowledge of the Green’s functions of the uncoupled reservoirs, gXX
(with X = L, R). The leads are semi-infinite systems and thus they cannot
possess in practice a very complicated geometry. A typical option is to
describe these leads as ideal surfaces of the corresponding material and the
unperturbed Green’s functions are then computed using special recursive
techniques like the so-called decimation [198].
Let us end this section with a brief technical discussion. The quantity
t(E, V ) appearing in Eq. (8.19) has been called transmission matrix without
a real justification. We should demonstrate that this matrix fulfills the
properties of a transmission matrix. In particular, we should at least prove
that the eigenvalues of tt† are bounded between 0 and 1. Indeed, this
property can be shown using a few algebraic manipulations (see Exercise
8.2).
Another way of showing that t(E, V ) in Eq. (8.19) is indeed the trans-
mission matrix of the contact is via the so-called Fisher-Lee relation [199],
which expresses the elements of the scattering matrix in terms of Green’s
functions. For the readers interested in this route, we recommend the orig-
inal work of Ref. [199] and the discussion on this matter in Chapter 3 of
Ref. [50].

8.1.1 An example: back to the resonant tunneling model

As an application of the general formula derived above and in order to
illustrate its use, let us now re-derive the current formula for the resonant
tunneling model considered in section 7.4.3.
Our starting point is the expression for the transmission of Eq. (8.18).
We need first to compute the retarded/advanced Green’s functions of the
central region. In this case this region consists of a single site with an on-
site energy ǫ0 . Therefore, the Green’s functions of the central region are
scalars with the following form
r,a ¤−1
Gr,a + r,a
£
CC = E ± i0 − ǫ0 − ΣL − ΣR , (8.23)

where the self-energies are the scalars Σr,a 2 r,a

L/R = tL/R gLL/RR . Assuming as in
r,a
section 7.4.3 that the local Green’s functions gLL/RR are purely imaginary
and independent of the energy around EF , the advanced self-energies reduce
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212 Molecular Electronics: An Introduction to Theory and Experiment

to ΣaL/R = iΓL/R , where ΓL/R = t2L/R Im{gLL/RR

a
(EF )} are the scattering
rates at the Fermi energy. Substituting now Eq. (8.23) and the expressions
of the self-energy in Eq. (8.18), we arrive again at the well-known Breit-
Wigner formula
4ΓL ΓR
T (E) = . (8.24)
(E − ǫ0 )2 + (ΓL + ΓR )2
Analogously, we can easily re-derive all the different formulas obtained
in the last chapter like for instance, the current expression of the Exercise
7.5 for a linear tight-binding chain (see Exercise 8.3).

8.1.2 Nonorthogonal basis sets

In the context of molecular electronics the use of nonorthogonal local basis
is quite common. In this sense, we have to discuss how to generalize the
current formula derived above for this type of bases. We shall address this
issue using a simple argument put forward by Emberly and Kirczenow [200]
and we refer the reader to different entry points in the literature for more
rigorous discussions.
In an orthogonal basis set, the overlap between the different basis states
is: hi|ji = Sij = δij , while the corresponding secular equation that provides
the eigenstates of the systems reads: HO − E1 = 0. Here, the subindex
O indicates that we are working with an orthogonal basis set. Finally, the
Green’s functions are simply obtained by inverting the Hamiltonian in the
usual way, i.e. GO = [E1 − HO ]−1 .
For a nonorthogonal basis, the overlap matrix differs from the unity
and the secular equation adopts the form: HN − ES = 0. Here, N denotes
nonorthogonal basis set. The left hand side of the secular equation can be
rewritten as follows
HN − ES = HN − E(S − 1) − E1 ≡ H′N − E1. (8.25)
Notice that the secular equation has now the same form as in the orthog-
onal case, but with an effective energy-dependent Hamiltonian: H′N ≡
HN −E(S−1). In this Hamiltonian, the on-site energies remain unchanged,
as compared with the original one, whereas the hopping matrix elements
become energy dependent: h′ij = hij − ESij . This argument suggests that
the only effect that the nonorthogonal basis introduces is the renormaliza-
tion of the hopping elements and therefore, the current formula is identical
to the one derived above after replacing the orthogonal parameters by the
nonorthogonal ones. Additionally, the retarded/advanced Green’s functions
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Formulas of the electrical current 213

appearing in the expression of the transmission have to be calculated by

means of the following Dyson’s equation
r,a ¤−1
Gr,a + r,a
£
CC = (E ± i0 )SCC − HCC − ΣL − ΣR , (8.26)

where HCC is now the nonorthogonal Hamiltonian of the central region

and SCC is the sector of the overlap matrix corresponding to the central
region. On the other hand, in the expression of the self-energies we have to
replace the hopping matrices tXC by tXC − ESXC , where X = L, R.
There is another way of deriving the result above [201]. The idea is to
transform every quantity from an orthogonal representation to a nonorthog-
onal one via the so-called Löwdin’s transformation. This transformation is
defined by S−1/2 , where S is the overlap matrix and it transforms an opera-
tor MO in the orthogonal basis to the corresponding one in the nonorthog-
onal basis, MN , as follows

MN = S1/2 MO S1/2 . (8.27)

Inserting 1 = S−1/2 S1/2 in the current formula in the orthogonal repre-

sentation, we arrive after some straightforward algebra at the same conclu-
sions as those stated above. For more detailed discussion of the derivation
of this result, we recommend Refs. [202, 203].

8.1.3 Spin-dependent elastic transport

So far we have only considered situations where there was spin symmetry.
We proceed now to generalize the Landauer formula derived above to sit-
uations where the spin symmetry is broken. Those situations include very
prominent examples in molecular electronics such as the transport through
ferromagnetic atomic-sized contacts (see Chapter 12) and molecular junc-
tions with ferromagnetic leads.
For the sake of concreteness, let us first consider the case of a metallic
atomic-sized contact made of a ferromagnetic material (like Fe, Co or Ni).
It is customary to analyze the transport properties of these junctions within
the two-current model put forward by N.F. Mott [204, 205]. Mott realized
that at sufficiently low temperature, where the magnon scattering in a
ferromagnet becomes vanishingly small, electrons of majority and minority
spin, with magnetic moment parallel and antiparallel to the magnetization,
respectively, do not mix in the scattering processes. This means in practice
that the total current can then be expressed as the sum of two independent
contribution coming from the two different spin projections, which implies
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214 Molecular Electronics: An Introduction to Theory and Experiment

that in ferromagnets the current is spin polarized. Therefore, the Landauer

formula of Eq. (8.20) adopts now the form
e ∞
Z
I= dE T (E, V ) (fL − fR ) , (8.28)
h −∞
where T (E, V ) is the total transmission sum of the transmissions of the two
spin bands
X X © ª X
T = Tσ = Tr t†σ tσ = Tnσ , (8.29)
σ=↑,↓ σ n,σ

where tσ is the transmission matrix of the spin sector σ and Tnσ are the
corresponding transmission coefficients. The transmission tσ is given by
Eq. (8.19), where all the quantity are referred to the spin band σ.
The previous current formula describes any (elastic) situation where
there is no mixing of the two spin bands. This is what occurs in most of
the atomic-scale junctions that we have in mind, where the system size is
clearly smaller than the spin-diffusion length. However, this is no longer
true if, for instance, there is a small domain wall of atomic size in the
junction or a strong spin-orbit interaction is present. Let us show how the
formula for the elastic current is modified in those situations.
A system in which the majority and minority spin bands are mixed can
be generically described by the following tight-binding Hamiltonian
†
X ′
H= hσσ
iα,jβ ciασ cjβσ ′ , (8.30)
ijαβσσ ′

where i, j run over the atomic sites, α, β denote the different atomic orbitals,
and σ = ↑, ↓ the spin. Within this model, the current can be computed fol-
lowing the same steps as in the case with spin symmetry and we only sketch
here the main idea and the final result. Briefly, the atomic-scale contact is
divided into three parts, a central region C containing the constriction and
the left/right (L/R) leads. The retarded Green’s functions of the central
part read4
−1
GrCC = [ESCC − HCC − ΣrL − ΣrR ] , (8.31)
where ΣrX = (tCX − ESCX )gXX r
(tCX − ESCX )† are the lead self-energies
(X = L, R). Here, tCX and SCX are the hoppings and overlaps between the
r
C region and the lead X, and gXX is a lead Green’s function. Notice that
the dimension of all the matrices in the previous equation is equal to the
total number of orbitals in the central region multiplied by two. This factor
4 Notice that we take into account the possibility of using non-orthogonal basis sets.
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Formulas of the electrical current 215

two comes from the structure in spin space. As before, the transmission
1/2 1/2
matrix is given by t = 2ΓL GrCC ΓR , but this time the scattering rate
matrices are given by where ΓX = i[ΣrX − (ΣrX )† ]/2. The reason for this is
r a
that, in general, the Hamiltonian is not real and gXX = (gXX )† . Finally,
the current then adopts the standard Landauer form of Eq. (8.28), but now
the trace includes not only a sum over the orbitals in the central part, but
also over spins. Finally, the low-temperature linear conductance can be
written as G = (e2 /h) n Tn , where Tn are the transmission coefficients,
P

i.e. the eigenvalues of t† t at Fermi energy.

8.2 Current through an interacting atomic-scale junction

As we explained in previous chapters, one of the main advantages of the

Green’s functions techniques with respect to the scattering approach is the
possibility to describe the influence of correlation and inelastic effects in
the transport characteristics. The goal of this section is to show how the
Landauer formula derived in the previous section is modified when such
effects are present in an atomic-scale junction. The derivation of the current
formula for an interacting system in the framework of Hamiltonian written
in a local basis was first done by Caroli and coworkers [209]. Later, Meir
and Wingreen re-derived this formula to express the current in a more
appealing way [210]. Although this latter formula is widely used in the
context of mesoscopic physics, its simplest form is not generally valid for
atomic-scale systems (see discussion below). We follow now the formulation
of Caroli and coworkers [209] and then discuss the Meir-Wingreen formula
in section 8.2.2.
Let us consider again the generic junction of Fig. 8.1. For the sake
of simplicity, we assume that the interactions (such as electron-electron or
electron-phonon interactions) are restricted to the central region. The cal-
culation of the current is identical to that of the elastic case up to Eq. (8.10).
At this point we have to determine the functions G+−/−+ of the central re-
gion, which can be done using the general Keldysh relations [see Eqs. (7.47)
and (7.48)]

G+−/−+ = (1 + Gr t) g+−/−+ (1 + tGa ) − Gr Σ+−/−+ Ga , (8.32)

where Σ+−/−+ are the Keldysh components of the self-energy describing

the inelastic effects. Notice that the last term was absent in the elastic
case. We now take the block-element (C,C) in the previous equation and
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216 Molecular Electronics: An Introduction to Theory and Experiment

obtain
+−/−+ +−/−+ +−/−+
GCC = gCC + GrCC t̂CL gLL tLC GaCC + (8.33)
+−/−+ +−/−+ a
GrCR tCR gRR tRC GaCC − GrCC ΣCC GCC .

Here we have used the fact that the interactions are restricted to the central
region, which in practice means that the inelastic self-energies Σ+−/−+
have only a (CC) component. Introducing now these Green’s functions in
Eq. (8.10), one can readily show that the current can be written as the sum
of two contributions: I = Iel + Iinel , where
8e ∞
Z
Iel = dE Tr [GrCC ΓR GaCC ΓL ] (fL − fR ) (8.34)
h −∞
Z ∞
4ie
dE Tr GaCC ΓL GrCC (fL − 1)Σ+− −+
© £ ¤ª
Iinel = CC − fL ΣCC (8.35).
h −∞
Again, the trace in these expressions has to be understood as a sum over all
the orbitals in the central region. The first term, Iel , represents the elastic
current and it has the same form as the Landauer formula derived in the
previous section. The second term, Iinel , which we call inelastic current,
is the new contribution due to the inelastic interactions. Notice that this
term has a rather asymmetric form, which is a consequence of our choice of
computing the current in the left interface. If wanted, one can symmetrize
this expression by combining it with the inelastic current evaluated in right
interface5 and using current conservation to define the inelastic current as
L R
Iinel = (Iinel + Iinel )/2.
From Eq. (8.35) it is not obvious that the inelastic current vanishes at
zero bias. However, this can be shown by using the general relations for a
system in equilibrium

Σ+− (E) = (Σr − Σa ) f (E); Σ−+ (E) = (Σr − Σa ) (f (E) − 1), (8.36)

where f (E) is the Fermi function.

It is important to emphasize that the retarded and advanced Green’s
functions of the central region are computed through a Dyson’s equation
that now also includes the new inelastic self-energies

GaCC = (GrCC )† = [(E − i0+ )1 − HCC − ΣaL − ΣaR − ΣaCC ]−1 , (8.37)
5 Such expression reads
4ie ∞
Z n h io
R +− −+
Iinel =− dE Tr Ga r
CC ΓR GCC (fR − 1)ΣCC − fR ΣCC .
h −∞
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Formulas of the electrical current 217

where ΣaCC is the advanced component of the self-energy describing the

inelastic interactions in the central region and ΣaL,R are given by Eq. (8.16).
The precise form of the inelastic self-energies and in turn of the con-
tribution of the inelastic term to the total current depends on the specific
nature of the inelastic interactions. In order to illustrate the use of this
new current formula, we present in the next subsection an important ex-
ample concerning the role of the electron-phonon interaction in molecular
junctions.

8.2.1 Electron-phonon interaction in the resonant tunnel-

ing model
In most transport experiments in molecular junctions, there is no certainty
that the current is indeed flowing through a molecule. Thus, it is to find
unambiguous signatures of the presence of the molecule, for instance, in
the current-voltage (I-V) characteristics. As we shall discuss extensively
in Chapter 16, presently the most convincing signatures are those related
to the excitation of vibration modes of the molecules used to form the
junctions. For this reason, it has become very important to understand
how the local interaction between the conduction electrons and molecular
vibrations is manifested in the I-V curves. We shall address this issue here
with a toy model that will also serve us to illustrate the use of the inelastic
current formula derived above.
Let us consider the resonant tunneling model that was already discussed
in section 7.4.3. Let us recall that in this model an electronic level with
energy ǫ0 is coupled to two metallic reservoirs via hopping elements tL and
tR , where L and R denote the left and right leads, respectively. Now, we
assume that this resonant level is also coupled to a single local vibrational
mode of energy ~ω. This model is schematically represented in Fig. 8.2.
Our goal is to compute the current-voltage characteristics when a constant
bias voltage, V , is applied. In particular, we shall pay special attention to
the correction of the current due to the electron-vibration interaction.
The Hamiltonian of the system that we have just described has the
following form

H = He + Hvib + He−vib , (8.38)

where He describes the electronic part of this problem as it is given by

Eq. (7.93). The vibrational mode is described as a simple harmonic oscil-
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218 Molecular Electronics: An Introduction to Theory and Experiment

eV
λ hω
L
ε0 R

Metal Molecule Metal

Fig. 8.2 Schematic representation of the resonant tunneling model where the electronic
level is coupled to a single vibrational mode of frequency ω with an electron-phonon
coupling constant λ.

lator of energy ~ω by
µ ¶
1
Hvib = ~ω b† b + , (8.39)
2
where the creation and annihilation operators b† and b satisfy the bosonic
commutation relations, e.g. [b, b† ] = 1. Finally, the interaction between the
vibration mode and the conduction electrons is described by the following
Hamiltonian [174]
He−vib = λc†0 c0 (b† + b), (8.40)
where λ is the electron-vibration coupling constant and c†0 and c0 are the
fermionic operators related to the electronic level.6
In this simple model, the central region consists of a single site and
therefore the Green’s functions, scattering rates and self-energies appearing
in the current formulas of Eqs. (8.34) and (8.35) are just scalars. Such
formulas reduce to the following expressions
Z ∞
8e
Iel = ΓL ΓR dE |Gr |2 (fL − fR ), (8.41)
h −∞
Z ∞
4ie
dE |Gr |2 (fL − 1)Σ+− −+
£ ¤
Iinel = ΓL e−vib − fL Σe−vib . (8.42)
h −∞

Here, the Green’s function Gr (E) refers to the central site or resonant
level. Moreover, as usual, we have assumed that the scattering rates, ΓL,R ,
that describe the strength of the coupling between the resonant level and
the leads are independent of the energy. Now, we have to determine the
6 The spin does not play any role in this problem and we have dropped it in the previous
expression.
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Formulas of the electrical current 219

self-energy associated to the electron-vibration interaction, Σ̌e−vib . The

simplest approximation for this self-energy can be obtained by applying
perturbation theory and keeping only the lowest order correction. Physi-
cally, this means that one only takes single-phonon processes into account.
As it is shown in Exercise 8.5, the first non-vanishing correction to the
self-energy is proportional to λ2 and its different components are given by7
Z ∞
r 2 dE ′ n r ′ −+ o
Σe−vib (E) = iλ D (E )G̃ (E − E ′ ) + D+− (E ′ )G̃r (E − E ′ ) ,
−∞ 2π
dE ′ +− ′ +−
Z
Σ+−
e−vib (E) = −iλ 2
D (E )G̃ (E − E ′ ),
2π
dE ′ −+ ′ −+
Z
−+ 2
Σe−vib (E) = −iλ D (E )G̃ (E − E ′ ). (8.43)
2π
Here, the functions with tilde are the electronic Green’s functions of the
resonant site where the coupling to the leads is taken into account and
the electron-vibration is not included, i.e. these are, loosely speaking, the
unperturbed functions of this problem, which are given by
−1
G̃r (E) = [(E + iη) − ǫ0 + i(ΓL + ΓR )] ,
+− r 2
G̃ (E) = 2i|G̃ (E)| [ΓL fL + ΓR fR ] ,
−+
G̃ (E) = −2i|G̃r (E)|2 [ΓL (1 − fL ) + ΓR (1 − fR )] , (8.44)
where η = 0+ .
On the other hand, the D’s are the phonon Green’s functions of this
problem and their general definitions can be found in Exercise 8.4. Assum-
ing that the vibration mode is in thermal equilibrium at the temperature
of the electrodes, these functions are given by (see Exercise 8.4)
1 1
Dr (E) = − ,
E − ~ω + iη E + ~ω + iη
D+− (E) = −2πi {(nB + 1)δ(E + ~ω) + nB δ(E − ~ω)} ,
D−+ (E) = −2πi {(nB + 1)δ(E − ~ω) + nB δ(E + ~ω)} , (8.45)
where nB = 1/[exp(β~ω) − 1], with β = 1/kB T is the Bose function that
describes the thermal occupation of the vibration mode.
7 There is an additional contribution to Σre−vib (E) which is equal to
Z ∞
dE ′ +− ′
−iλ2 Dr (0) G̃ (E ).
−∞ 2π

This gives a constant contribution that simply renormalizes the position of the resonant
level and we ignore it in what follows.
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220 Molecular Electronics: An Introduction to Theory and Experiment

Now, we expand the current to lowest order in the coupling constant λ.

To do so, in the inelastic term of Eq. (8.42) we just need to introduce the
expressions of Σe−vib and replace the full Gr by G̃r . In the elastic term
of Eq. (8.41) we have to insert the lowest order correction of the Green’s
functions, i.e. Gr ≈ G̃r + G̃r Σre−vib G̃r , and collect all the terms up to second
order in λ. Doing this, the current can be expressed as the sum of three
0
terms: I = Iel + δIel + Iinel , where the different contributions are given by
Z ∞
0 8e
Iel = ΓL ΓR dE |G̃r (E)|2 [fL (E) − fR (E)] , (8.46)
h −∞
Z ∞
16e
δIel = ΓL Γ R dE |G̃r (E)|2 ×
h −∞
n o
Re G̃r (E)Σre−vib (E) [fL (E) − fR (E)] , (8.47)
∞
8eλ2
Z n
Iinel = ΓL ΓR dE (nB + 1)|G̃r (E)G̃r (E − ~ω)|2 ×
h −∞
[fL (E)(1 − fR (E − ~ω)) − fR (E)(1 − fL (E − ~ω))]
+nB |G̃r (E)G̃r (E + ~ω)|2 ×
[fL (E)(1 − fR (E + ~ω)) − fR (E)(1 − fL (E + ~ω))]} . (8.48)
0
The first contribution, Iel , is nothing else but the elastic current in the
absence of electron-vibration interaction that we have studied in section
7.4.3, see Fig. 8.3(a). The third term, Iinel , is the inelastic contribution
coming from the emission and absorption of a vibrational mode. Notice
that the term in Iinel proportional to nB corresponds to the contribution
of processes assisted by the absorption of a mode, see Fig. 8.3(b), whereas
the term proportional to (nB + 1) is the contribution of tunneling pro-
cesses mediated by the stimulated and spontaneous emission of a mode, see
Fig. 8.3(c). At temperatures much lower than ~ω/kB , the second one dom-
inates. Moreover, it is easy to see that at low temperatures the emission
term has a threshold voltage equal to the vibration energy (~ω/e) below
which it vanishes. Above this voltage this term gives always a positive con-
tribution, which means that it gives rise to a step up in the conductance.
The second term, δIel , has a less obvious interpretation. It is an elastic term
that involves the emission and absorption of a virtual vibrational mode, see
Fig. 8.3(d). This term will be referred to as elastic correction.
It is easy to evaluate numerically the different contributions to the cur-
rent for an arbitrary range of parameters. However, in order to gain some
insight, we concentrate here on a limiting case that can be worked out
analytically. Let us assume that the energy dependence in the retarded
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Formulas of the electrical current 221

(a) Elastic process (b) Phonon absorption

eV hω eV
L L
ε0 R ε0 R

Metal Molecule Metal Metal Molecule Metal

(c) Phonon emission (d) Elastic correction

hω eV hω eV
L L
ε0 R ε0 R

Metal Molecule Metal Metal Molecule Metal

Fig. 8.3 Schematic representation of the elastic (a) and inelastic (b-d) tunneling pro-
cesses that can occur in the model in which an electronic level is coupled to a single
vibration mode. Here, we have assumed that the electron-phonon interaction is weak
and the processes (b-d) are responsible for the inelastic correction to the elastic current
up to order λ2 .

electronic Green’s functions can be neglected, i.e. G̃r (E) = G̃r (EF ). This
means in practice that we assume that both the local density of states
and the transmission are energy-independent. This is a good approxima-
tion in two cases: (i) when the coupling to the leads is so strong that
ΓL + ΓR >> ~ω, eV, |EF − ǫ0 | and (ii) when the resonant level is far away
from the Fermi energy, i.e. |EF − ǫ0 | ≫ ΓL,R , eV, ~ω. With this approxi-
mation the different terms can be computed analytically. At temperatures
well below the vibrational energy, the correction to the elastic current is a
competition between the emission term in Iinel and the elastic correction
δIel . Assuming a symmetric junction, ΓL = ΓR = Γ, the three contri-
butions to the zero-temperature differential conductance are given by (see
Exercise 8.6)
G0el
= T,
G0
λ2
½ 2
δGel (V ) T (1 − T )/2; |eV | ≤ ~ω
= 2
G0 Γ T 2 (1 − 2T )/2; |eV | > ~ω
λ2
½
Ginel (V ) 0; |eV | ≤ ~ω
= 2 2 , (8.49)
G0 Γ T /4; |eV | > ~ω
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222 Molecular Electronics: An Introduction to Theory and Experiment

(a) T > 1/2 (b) T < 1/2

dI/dV dI/dV
2
∼(λ/Γ)

_ _ _ _
− hω + hω eV − hω + hω eV

Fig. 8.4 Signature of a vibration mode in the zero-temperature differential conductance

of a resonant level. (a) For transmissions greater than 1/2, the differential conductance
exhibits a step down at eV = ±~ω due to electron-vibration interaction. The height of
the step is mainly determined by the ratio λ2 /Γ2 and it has been exaggerated for clarity.
(b) The signature of the vibration mode in the differential conductance for transmissions
less than 1/2 is a step up at eV = ±~ω.

where T = 4Γ2 |G̃r (EF )|2 is the elastic transmission in the absence of
electron-vibration interaction. Notice that δGel has a discontinuity (step
down) at eV = ±~ω proportional to −T 3 /2, while the emission term con-
tributes to this jump as ∼ +T 2 /4. This means that the sign of the con-
ductance jump depends on the junction transmission and it is given by:
(λ2 /Γ2 )T 2 (1 − 2T )/4. Notice that the magnitude is determined by the ra-
tio of the two relevant coupling constants, λ and Γ, which has been assumed
to be small. On the other hand, the conclusion of this analysis is that the
electron-vibration interaction in this simple model is reflected in the appear-
ance of a jump in the low-temperature conductance at eV = ±~ω. This
jump is seen as a step up in conductance for T < 1/2 and as step down
for T > 1/2. This conclusion is summarized schematically in Fig. 8.4. The
signature of the vibration modes can be seen more clearly in the second
derivative of the current, d2 I/dV 2 , where it appears as a peak or as a dip
depending on the junction transmission.8 The results of this model will be
discussed in much more detail in section 17.1.1.

8.2.2 The Meir-Wingreen formula

As we mentioned in the introduction of this section, Meir and Wingreen
proposed in 1992 [210] an alternative form for the formula of the current
through an interacting region. This formula has been widely used in meso-
scopic physics and, in particular, for studying the transport through all kind
of quantum dots and molecular transistors. For this reason and for the sake
8 The signature is antisymmetric with respect to the voltage polarity in the sense that
if it appears as peak for positive bias, it appears as dip for negative one.
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Formulas of the electrical current 223

of completeness, we include here a short discussion of the derivation of this

formula. Further technical details can be found in Exercise 8.7.
Once more, we consider the system of Fig. 8.1, where the central part
represents an interacting region. The current evaluated at the left interface
is given by Eq. (8.8). Now, to determine the Keldysh-Green’s functions
appearing in that expression we make use of Dyson’s equation in Keldysh
space as follows
G+− ++ +− +− −−
LC = gLL tLC GCC − gLL tLC GCC (8.50)
G+−
CL = G++ +−
CC tCL gLL − G+− −−
CC tCL gLL .

Using the general relations G+− + G−+ = G++ + G−− and G+− − G−+ =
Ga − Gr , it is straightforward to show that the current evaluated at the
left interface, IL , is given by
4ie ∞
Z
dE Tr ΓL G+− r a
© £ ¤ª
IL = CC + (GCC − GCC )fL , (8.51)
h −∞
where the scattering rate ΓL is defined in the usual way.
Analogously, one can obtain the expression of the current, evaluated
this time at the right interface, IR . Writing then the current in a more
symmetric manner as I = (IL + IR )/2, one arrives at
2ie ∞
Z
dE Tr (ΓL − ΓR )G+− r a
© ª
I= CC + (fL ΓL − ΓR fR )(GCC − GCC ) .
h −∞
(8.52)
This is the Meir-Wingreen formula in its most general form. It is completely
equivalent to the expression derived above and in the non-interacting case
it reduces to the Landauer formula (see Exercise 8.7). The “popularity”
of this formula is due to the fact that it takes an appealing form in the
case in which the couplings to the leads differ only by a constant factor,
ΓL (E) = λΓR (E). In this case, the current reads
8e ∞
Z
I= dE Tr {ΓA} (fL − fR ), (8.53)
h −∞
where Γ ≡ ΓL ΓR /(ΓL + ΓR ) and A ≡ i(GrCC − GaCC )/2 is the spectral
function of the central region. The division in the expression of Γ has to
be understood as multiplication by the inverse of the matrix appearing in
the denominator. The nice thing about this formula is that the current is
expressed in terms of the spectral function, A. Unfortunately, the condition
of proportionality of the scattering rates is quite restrictive and most cases
it is not really fulfilled. For applications of this latter formula, see Exercises
8.9 and 8.10.
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224 Molecular Electronics: An Introduction to Theory and Experiment

8.3 Time-dependent transport in nanoscale junctions

Up to now we have only considered stationary situations where the current

was time-independent. In this section we shall illustrate the use of the
Keldysh formalism for computing the transport properties of a system that
is subjected to an externally applied time-dependent drive.
As a model problem, which will be very important for Chapter 20,
we consider here the calculation of the current in an atomic or molec-
ular contact under the presence of an oscillating bias voltage: V (t) =
V + Vac sin(ωt), where V is the dc part of the bias and Vac and ω are the
amplitude and the frequency of this periodic potential, respectively. This
ac field can be simply due to an applied chemical potential difference, but
one can also imagine that it is induced in the junction by the application of
an external radiation, which is a situation is of special interest for us. The
question of how the current through an atomic-scale junction can be modi-
fied by irradiation is a very important subject in molecular electronics [211].
In other contexts, like for instance in the case of superconducting tunnel
junctions, this problem has a long history [212]. From the theory side, the
“photon-assisted” transport has been traditionally addressed following the
seminal work of Tien and Gordon (TG) [213], where this phenomenon was
described by a harmonic voltage at the radiation frequency ω applied to
one of the leads of a junction. Such a simple approach have been quite
successful in gaining a qualitative understanding of radiation-induced cur-
rents in many situations like superconducting systems [212], semiconductor
heterostructures [214], STM [215], and and other mesoscopic systems [211].
Our discussion in this section provides the basis to address similar problems
in the context of atomic and molecular junctions.
Different theoretical approaches have been applied to the problem that
we are about to tackle such as the scattering approach [165] or Floquet
theory [211, 214]. We shall follow here the nonequilibrium Green’s func-
tion formalism used in the previous sections of this chapter (see also
Refs. [216–220]). This approach allows us to describe the photo-transport
in realistic atomic and molecular contacts, in the sense that it can be com-
bined by advanced electronic structure methods.
Let us consider the generic geometry of Fig. 8.1, which again represents
an atomic or molecular contact of arbitrary shape. For simplicity, we as-
sume that the correlation and inelastic effects do not play a mayor role in
this case. In other words, we assume that the transport in the absence of
the ac drive is coherent. We describe the system with the following time-
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Formulas of the electrical current 225

dependent Hamiltonian: H(t) = H0 + H1 (t). Here, H0 is the Hamiltonian

of Eq. (8.1) that contains the full microscopic information about the sys-
tem in the absence of dc and ac voltages.9 The time-dependent part H1 (t)
describes the driving potential and it can be written generically as
Wii (t)c†i ci ,
X
H1 (t) = (8.54)
ij

where Wii (t) = Uidc + Uiac

sin(ωt) describe the shifts in the on-site energies
induced by the dc and ac parts of voltage. Here, the Ui ’s are the amplitudes
of the local potential at site i and equal for all the orbitals in the same atom.
We assume that the potential is spatially constant in the L and R leads and
dc ac
equal to UX (t) = UX + UX sin(ωt), and X = L, R. The applied dc voltage
is V = (UL − UR )/e and the corresponding ac part is Vac = (ULac − URac )/e.
dc dc

We shall calculate the current for an arbitrary potential profile in the central
region (encoded in the functions Ui (t)), the actual shape of which should
in principle be obtained self-consistently [165].
In order to derive the current formula in this situation, we shall follow
the same steps taken in section 8.1 and we shall emphasize here only the
main differences with respect to that calculation. Our starting point is the
expression of the time-dependent current evaluated at the left interface,
which can be written in terms of the Green’s functions as follows
2e £
I(t) = Tr G+− +−
¤
CL (t, t)tLC − tCL GLC (t, t) . (8.55)
~
To determine the Green’s functions we follow the same perturbative ap-
proach as in section 8.1. The essential difference now is that the Green’s
functions depend explicitly on two time arguments (rather than on their dif-
ference), which introduces an extra complication, as we are about to show.
Using the Dyson’s equation [see Eq. (7.46)] we can express the functions
appearing in the current as10
G+− ′
© +− a r +−
ª ′
LC (t, t ) = gLL ◦ tLC ◦ GCC + gLL ◦ tLC ◦ GCC (t, t ) (8.56)
G+− ′ +− a r +− ′
© ª
CL (t, t ) = GCC ◦ tCL ◦ gLL + GCC ◦ tCL ◦ gLL (t, t ),

where the product ◦ is defined by (A ◦ B)(t, t′ ) = dt1 A(t, t1 )B(t1 , t′ ), i.e.

R

it is a convolution over the intermediate time arguments. This means that

any Dyson’s equation is no longer an algebraic equation as before, but rather
an integral equation. Anyway, if we handle carefully this non-commutative
9 We shall assume throughout this discussion that this Hamiltonian is written is a local
orthogonal basis.
10 Here, the time-dependent hopping matrices are defined, for instance, as: t ′
LC (t, t ) =
′
tLC δ(t − t ).
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226 Molecular Electronics: An Introduction to Theory and Experiment

product, the derivation still follows the same steps as in section 8.1. Thus,
we can easily arrive at the following expression for the current
2e £ r
I(t) = Tr GCC ◦ Σ−+ a +−
R ◦ GCC ◦ ΣL −
~
GrCC ◦ Σ+− a −+
¤
R ◦ GCC ◦ ΣL (t, t), (8.57)
which is the analog of Eq. (8.13). Here, we have define the “lead self-
energies”
h i
+−/−+ +−/−+
ΣX (t, t′ ) = tCX ◦ gXX ◦ tXC (t, t′ ), (8.58)

where X = L, R.
The lead Green’s functions have now a more complicated time depen-
dence. Due to the ac voltage they oscillate on time as follows11
′
c
gX (t, t′ ) = e−iφX (t) gX
c
(t − t′ )eiφX (t ) , (8.59)
where c = r, a, +−, −+. Here, ∂φX (t)/∂t = µX (t)/~, where µX (t) is the
chemical potential of the corresponding electrode. Therefore, φX (t) =
dc ac
(UX /~)t + αX cos(ωt), with αX = UX /(~ω).
As usual, it is more convenient to work in energy space and for this
reason we now Fourier transform with respect to the two time arguments
1
Z Z
′ ′
c ′
gX (t, t ) = dE dE ′ e−iEt/~ eiE t /~ gX
c
(E, E ′ ). (8.60)
2π
From Eq. (8.59) it is easy to show that the lead Green’s functions admit a
Fourier expansion of the form
dE −iE(t−t′ )/~ c
X Z
′
c
gX (t, t′ ) = eimωt e gX (E, E + m~ω). (8.61)
m
2π
c
In other words, the functions gX (E, E ′ ) satisfy the following relation
X
c
gX (E, E ′ ) = c
[ĝX ]0,n (E)δ(E − E ′ + n~ω), (8.62)
n
c c
where [ĝX ]0,n (E) ≡ gX (E, E +n~ω). Other Fourier components are related
c c
by [gX ]n,m (E) = [gX ]0,m−n (E + n~ω). These Fourier components can be
seen as the matrix elements of the Green’s functions in energy space. We
11 This time dependence can be shown by solving the Dyson’s equation for the lead

Green’s function, which e.g. for the retarded component reads:

„ «
∂ r
i~ − HXX − WXX (t) gXX (t, t′ ) = ~δ(t − t′ ),
∂t
where WXX (t) is nothing else but a spatially constant term equal to the chemical
potential of the electrode.
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Formulas of the electrical current 227

denote the matrices in this space with a “hat” symbol. The previous rela-
tion is the mathematical expression of the fact that all physical quantities
in this problem oscillate in time with the driving frequency and all its har-
monics.
With the help of the relation
X
eiα cos(ωt) = im Jm (α)eimωt , (8.63)
m

where Jm is the Bessel function of first kind of order m, one can show that
the Fourier components of the lead Green’s functions are given by
c,eq
X
c
[ĝX ]n,m (E) = im−n Jn−l (αX )Jm−l (αX )gX dc
(E − UX + l~ω), (8.64)
l
c,eq
where gX are the equilibrium Green’s functions of the lead X, i.e. the
usual lead Green’s function for us. With these expressions, it is straight-
forward to show that the self-energies, like the ones in Eq. (8.58), and the
corresponding scattering rates are related to the corresponding equilibrium
quantities as follows
X c(l) X (l)
[Σ̂cX ]m,n = [Σ̂X ]m,n , [Γ̂X ]m,n = [Γ̂X ]m,n , (8.65)
l l

where we define the components

(l)
[Γ̂X ]n,m (E) = im−n Jn−l (αX ) Jm−l (αX ) Γeq dc
X (E − UX + l~ω), (8.66)
c(l)
with a similar equation for Σ̂X (E).
The full Green’s functions in the central region have a similar structure
in energy space and their different Fourier components are given by the
following matrix Dyson’s equation
[Ĝr,a
CC ]
−1
= Ê − HCC 1̂ − ŴCC − Σ̂r,a r,a
L − Σ̂R . (8.67)
dc ac
Here [Ê]n,m = (E + n~ω)δm,n 1, [ŴCC ]n,m = WCC
+ δn,m WCC
+ (δn−1,m
δn+1,m )/2. This means that the Fourier components of the Green’s func-
tions can be obtained by inverting the usual matrix, but this time in an
extended space. This is a (∞ × ∞) matrix that has to be truncated and its
actual dimension is determined by the amplitude of the ac voltage.
Now, we can bring all these results into the current expression, see
Eq. (8.57). The first thing to notice is that, as we have already pointed
out, all the quantities in this problem, Green’s functions, self-energies, etc.,
admit a Fourier expansion of the form of Eq. (8.61). It is easy to show
that the convolution (or ◦-product) of two quantities with this property is
a function that also fulfills this property. Therefore, it is obvious that the
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228 Molecular Electronics: An Introduction to Theory and Experiment

current in Eq. (8.57), in which the two time arguments are equal, has the
following time dependence
X
I(t) = Im eimωt , (8.68)
m

i.e. as anticipated, it oscillates with the external frequency and all its har-
monics. We are only interested here in the dc component, I0 , which from
now on we will simply denote as I.
Using the generic Fourier expansion of Eq. (8.61) for all the quantities
appearing in the current expression, see Eq. (8.57), it is easy to show that
the dc current can be written in terms of the different Fourier components
in energy space defined above as
8e ∞ X
Z n o
(n) (n′ ) (n′ ) (n)
I= Tr [Ĝr ]0,k [Γ̂R ]k,l [Ĝa ]l,m [Γ̂L ]m,0 (fL − fR ),
h −∞
k,l,m,n,n
′

(8.69)
(n) dc
where fX (E) = f (E − UX+ n~ω). At this stage it is already obvious
that in the absence of an ac field, this formula reduces to the Landauer
formula derived in section 8.1. We can write the current in numerous
ways by changing summation indices and the integration variable. Thus
for instance, it is not difficult to show that the dc current can be expressed
as follows12
∞ Z ∞
2e X (k)
I(V ; α, ω) = dE [TRL (E, V ; α, ω)fL (E) − (8.70)
h −∞
k=−∞
(k)
TLR (E, V ; α, ω)fR (E)],
dc
where fX (E) = f (E − UX ), the parameter α = αL − αR = eVac /~ω is
the strength of the ac drive and the coefficients appearing inside the energy
integral are given by
(k) (k) (0)
TRL (E) = 4Trω [Ĝr (E)Γ̂R (E)Ĝa (E)Γ̂L (E)], (8.71)
(k) a (k) (0)
TLR (E) = 4Trω [Ĝ (E)Γ̂L (E)Ĝr (E)Γ̂R (E)], (8.72)
where trace Trω includes a summation over the “harmonic” indexes, i.e.
over the Fourier components in energy space, and over the usual site and
(k)
orbital indexes of the central region. Here TRL (E) can be interpreted as
a transmission coefficient that describes processes taking an electron from
left (L) to right (R), under the absorption of a total of k energy quanta
12 For the sake of clarity, we make explicit the dependence of the current on the dc
voltage, V , the frequency, ω, and the strength of the ac drive, α = αL − αR = eVac /~ω.
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Formulas of the electrical current 229

(k)
~ω. The coefficient TLR (E) has a similar interpretation. By the way,
these interpretations are the reason why one usually talks about photo-
assisted processes in this problem, although there is indeed no quantized
electromagnetic field interacting with the conduction electrons in our model.
Let us summarize the discussion above. The current of Eq. (8.70) de-
scribes the dc current in the presence of an oscillating potential and it
adopts a form similar to the standard Landauer formula. The main differ-
ence is that all the quantities have now a matrix structure in an extended
Hilbert space, which includes both the orbital and the energy space. The
appearance of off-diagonal elements in energy space is a natural conse-
quence of the occurrence of the inelastic processes that take place in this
problem. In those inelastic tunneling processes, a certain number of en-
ergy quanta (multiples of ~ω) can be either absorbed or emitted. The
retarded/advanced Green’s functions appearing in the current formula are
determined by solving the matrix equation (8.67), while the scattering rates
are given by Eq. (8.66). All these matrices have, in principle, an infinite
dimension in energy space, but they can be truncated in practice and their
actual dimension is governed by the amplitude of the ac drive, α.
The formalism above has been recently used to discuss both the photon-
assisted transport in atomic [221] and molecular wires [222]. This formalism
is a bit cumbersome and numerically demanding due to the large size of
the matrices involved. However, the current formula above can be greatly
simplified in the case in which we can ignore the energy dependence in the
leads, which is frequently a very good approximation. In this situation the
self-energies Σ̂X become diagonal (see Exercise 8.8)
[Σ̂X ]n,m (E) = ΣX (E)δn,m . (8.73)
If in addition we assume that the ac potential profile is such that it is
constant in the central region (i.e. the drops occur at the interfaces), the
current formula reduces to [165, 211, 214]
∞
2e X h ³ α ´i2
Z
I(V ; α, ω) = Jl dE T (E + l~ω)[fL (E) − fR (E)], (8.74)
h 2
l=−∞

where T (E) is the transmission in the absence of ac drive.13 Moreover, we

have assumed here that the ac potential drops symmetrically at both inter-
faces, i.e. αL,R = ±α/2. The result of Eq. (8.74) is quite remarkable and it
tell us that the current under a periodic time-dependent field depends pri-
marily on the energy dependence of the elastic transmission. This becomes
13 This transmission may include the dc part of the voltage.
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230 Molecular Electronics: An Introduction to Theory and Experiment

even more apparent in the case of the conductance. At low temperatures

and in the linear response regime (vanishing dc bias), the conductance,
which will be referred to as photoconductance, takes the particularly simple
form
∞ h ³ ´i2
X α
G(V = 0; α, ω) = G0 Jl T (EF + l~ω), (8.75)
2
l=−∞

where T (E) is the zero-bias equilibrium transmission. Let us remind that

here l can be interpreted as the number of absorbed or emitted photons,
Jl (x) is a Bessel function of the first kind (of order l), and α = eVac /~ω is
the dimensionless parameter describing the strength of the ac drive. Note
that if the transmission does not depend on energy in the range explored
by the inelastic processes, the conductance reduces to the conductance in
the absence of drive, i.e. G0 T (EF ).
In the limit α ≪ 1 and frequencies small in comparison with the energy
scale in which transmission changes significantly, we can expand T (E) and
the Bessel functions in Eq. (8.75) to leading order in these small quantities,
yielding G(ω) = G0 T (EF ) + G0 (α~ω)2 T ′′ (EF )/16, where T ′′ denotes the
second derivative respect to energy. Defining then the induced conductance
correction ∆G(ω) = G(ω) − G(ω = 0), where G(ω = 0) = G = G0 T (EF ),
the relative correction becomes
∆G(α, ω) (α~ω)2 T ′′ (EF )
= . (8.76)
G 16 T (EF )
We thus see that this quantity gives experimental access to the second
derivative of the transmission function at E = EF . Note that in this ap-
proximation, which can be seen as an adiabatic or “classical” limit [212],
the conductance correction depends only on the driving field through the
ac amplitude Vac = α~ω/e.
Finally, let us mention that Eq. (8.74) may equally well be written in
the form [213, 212]
∞ h ³ ´i2
X α
I(V ; α, ω) = Jl I0 (V + 2l~ω/e), (8.77)
2
l=−∞

where I0 (V ) is the I-V characteristic in the absence of light.

The main assumption leading to these simplified formulas is the fact
that the profile is flat across the central part of the constriction. However,
it has been shown in Refs. [221, 222] that the detailed shape of the profile
does not change significantly the main results, unless the ac amplitude is
very large.
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Formulas of the electrical current 231

8.3.1 Photon-assisted resonant tunneling

In order to illustrate the previous time-dependent formalism, let us now
apply it to the resonant tunneling model (see section 7.4.3). This problem
has been analyzed by Jauho et al. [218]. As we have discussed many times
by now, this simple model gives useful insight into the conduction through
a single-molecule junction. Now, the question is: How is the resonant
transport modified in the presence of radiation? Following the discussion
above, we assume here that an electromagnetic field simply induces an ac
voltage of frequency ω across the junction. If, as usual, we neglect the
energy dependence of the scattering rates, we can analyze this problem in
terms of the simplified formulas presented at the end of the previous section.
The first issue that we want to address is the modification of the non-
linear conductance. For this purpose, we use the expression of Eq. (8.74)
to determine the current-voltage characteristics and in turn the differential
conductance dI(V ; α, ω)/dV , where V is the dc voltage. In this formula
we make use of the expression of Eq. (8.24) for the elastic transmission
through the resonant level in the absence of radiation. We assume that
both the bias voltage and the ac drive drop symmetrically at the interfaces.
This means in practice that the chemical potentials of both electrodes are
shifted by ±eV /2, while the resonant level is not shifted by the bias. An
example of the zero-temperature I-V characteristics for different values of α
is shown in Fig. 8.5(a). In this example, the level position (measured with
respect to the equilibrium chemical potential of the electrodes) is ǫ0 = 5~ω
and ΓL = ΓR = 0.1~ω. The corresponding differential conductance as a
function of the bias voltage is shown in Fig. 8.5(b). Notice that in the
absence of the external ac field, the conductance is simply given by a Breit-
Wigner resonance centered around 2ǫ0 (see curve for α = 0). The factor two
is due to the symmetric voltage profile adopted here. When the radiation is
applied, one can see the appearance of additional steps in the current and
satellite peaks in the conductance with a regular spacing equal to 2~ω. In
the case of the conductance, the peaks on the left hand side of the central
elastic resonance are due to the photon absorption, i.e. due to tunneling
processes in which an incoming electron with energy E = ǫ0 − eV /2 − k~ω
absorbs k photons and crosses the level exactly at resonance. Similarly,
the peaks on the right hand side are due to emission processes in which an
electron loses energy emitting a certain number of photons. The number
of satellite peaks (or side bands) depends on the strength of the ac drive,
α, which is basically a measure of the local field intensity at the junction.
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232 Molecular Electronics: An Introduction to Theory and Experiment

0.6 α = 0.0
I(V,α,ω)/(eω/π)

(a) (b) α = 1.0

G(V,α,ω)/G0
0.4 α = 2.0
0.4 α = 4.0

α = 0.0 0.2
0.2 α = 1.0
α = 2.0
α = 4.0
0 0
6 8 10_ 12 14 6 8 10_ 12 14
eV/hω eV/hω
G(V=0,α,ω)/G(ω=0)

1
200 α = 1.0 α = 0.0
G(V=0,α,ω)/G0
(c) α = 2.0 0.8 (d) α = 1.0
150 α = 4.0 α = 2.0
0.6 α = 4.0
100
0.4
50 0.2
0 0
0 0.5 _ 1 1.5 -3 -2 -1 0 _1 2 3
hω/ε0 (ε0-EF)/hω
Fig. 8.5 Photon-assisted transport in the resonant tunneling model. In this example we
consider a symmetric junction with ΓL = ΓR = 0.1~ω and all the results are obtained at
zero temperature. (a) Current as a function of the dc bias voltage V for ǫ0 − EF = 5~ω
and different values of α = eVac /~ω. (b) The differential conductance corresponding to
the I-V curves of panel (a). (c) Photoconductance normalized by the conductance in the
absence of radiation as a function of the radiation frequency for different values of α.
(d) Photoconductance versus level position measured with respect to the Fermi energy.

An important quantity for us is the photoconductance G(V = 0; α, ω),

i.e. the conductance when the dc voltage is infinitesimally small. In
Fig. 8.5(c) we show an example of this quantity as a function of the ra-
diation frequency. The fact that we want to illustrate here is that when
the frequency matches the distance between the Fermi energy and the level
position, one observes a huge enhancement of the conductance that can
reach up to several orders of magnitude. The additional peaks that one
can see in Fig. 8.5(c) are due to multi-photon processes. Finally, in some
situations the position of the resonant level can be tuned by means of, for
instance, a gate voltage. Therefore, it is interesting to know what is the
expected dependence of the photoconductance on the level position. This
can be seen in Fig. 8.5(d), where one can observe that in this case the in-
elastic tunneling events give rise to satellite peaks that are separated by an
energy equal to ~ω.
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Formulas of the electrical current 233

8.4 Exercises

8.1 Scattering rate matrices: Show that the scattering rate matrices defined
in section 8.1 as ΓX = Im{ΣX } (X = L, R), where ΣX are the self-energies of
Eq. (8.16), are positive definite and therefore their square roots are well-defined.
8.2 Transmission matrix: The goal of this exercise is to show that the matrix
defined in Eq. (8.19) has indeed the basic properties of a transmission matrix. For
this purpose, it must be shown that the eigenvalues of tt† are bounded between
0 and 1. Demonstrate this property following the next steps:
(i) Using the result of the previous exercise, show that tt† is positive definite
and therefore all its eigenvalues are real and positive.
(ii) Use the definition of the scattering rate matrices and Dyson’s equation
for the retarded and advanced Green’s functions to prove the following relation

i
GrCC [ΓL + ΓR ] GaCC = [GrCC − GaCC ] .
2

(iii) Use the previous relation to demonstrate the following relation

1 = rr† + tt† ,

where r is the reflection matrix given by

1/2 1/2
r = 1 − 2iΓL GrCC ΓL .

(iv) Using this last relation, show that the all eigenvalues of tt† are less than
(or equal to) one.
8.3 Formula for the current through an atomic chain: Consider the model
for an atomic chain described in Exercise 7.5. Use the general expression of
Eq. (8.20) to re-derive the formula for the electrical current obtained in that
exercise.
8.4 Phonon Green’s functions: The phonon Green’s functions are defined in
analogy with the electronic ones as
h i h i
Dr (t, t′ ) = −iθ(t − t′ )h A(t), A† (t′ ) i, Da (t, t′ ) = −iθ(t′ − t)h A(t), A† (t′ ) i,

D+− (t, t′ ) = ihA† (t′ )A(t)i, D−+ (t, t′ ) = −ihA(t)A† (t′ )i,
where A = b + b† and the creation and annihilation operators b† and b satisfy
the bosonic commutation relations (see Appendix A). Show that for the case of
a free phonon (or vibration) mode, described by the Hamiltonian of Eq. (8.39),
these functions are by given Eq. (8.45). Hint: Compute first the time evolution
of the bosonic operators by solving the equation of motion of an operator in the
Heisenberg picture.
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234 Molecular Electronics: An Introduction to Theory and Experiment

8.5 Lowest order expansion of the electron-vibration self-energy: The

goal of this exercise is to demonstrate that Eq. (8.43) gives the correct expression
for the lowest order correction to the electronic self-energy in the problem of
section 8.2.1. For this purpose, follow the next steps:
(i) Use the Hamiltonian of Eq. (8.40) as the perturbation in this problem.
With this choice, show that the second order correction in λ of an electronic
Green’s function is equal to
3 Z Z
2 (−i)
G(2)
c (t α , t β ) = λ dt1 dt2 ×
2! c c
h i
hTc c(tα )c† (t1 )c(t1 )[b† (t1 ) + b(t1 )]c† (t2 )c(t2 )[b† (t2 ) + b(t2 )]c† (tβ ) i.

Here, the subindex c indicates that the Green’s functions can be any of the four
components in Keldysh space depending upon where the time arguments, tα and
tβ (α, β = +, −), lie on the Keldysh contour. The integrations above have to be
understood as follows
Z Z ∞ Z ∞
dti = dti,+ − dti,− .
c −∞ −∞

(ii) Apply Wick’s theorem to the previous expression and keep only the con-
tributions of topologically distinct connected diagrams. Show that the only two
relevant self-energy diagrams are the ones shown in Fig. 8.6.

Fig. 8.6 Lowest-order electronic self-energy diagrams associated to the electron-phonon

interaction in the resonant tunneling model. The solid lines represent electronic Green’s
functions, while the dashed ones correspond to phonon Green’s functions.

(iii) Evaluate the contribution of the diagrams of Fig. 8.6 to the different com-
ponents of the self-energy in energy space. Show that the contributions coming
from the diagram on the left hand side lead to the results of Eq. (8.43). Discuss
also the relevance of the contributions coming from the other diagram.
8.6 Signature of a vibrational mode in the differential conductance:
Consider the model used in section 8.2.1 to understand the signature of a vibration
mode in the current through a single resonant level. Assume that the density of
states in that level and the corresponding transmission are energy-independent
and show that the zero-temperature differential conductance is given by Eq. (8.49)
in the case of a symmetric junction.
Hint: The only complicated term in the expression for the current is the elastic
correction, which contains the self-energy Σre−vib . Separating the contributions
of the real and imaginary part of the retarded phonon Green’s function Dr , this
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Formulas of the electrical current 235

self-energy can be written as

∞
dE ′
Z  ff
2~ω
Σre−vib (E)/λ2 = i G̃ −+
(E − E ′
) +
−∞ 2π(E ′ )2 − (~ω)2
1 h −+ i
G̃ (E − ~ω) − G̃−+ (E + ~ω) +
2
(nB + 1)G̃r (E + ~ω) + nB G̃r (E − ~ω).

The first term, which has to be understood as principle value, does not contribute
to the conductance in the case of a symmetric junction, while the others are
responsible for the contribution of the elastic correction to Eq. (8.49).
8.7 The Meir-Wingreen formula:
(i) Follow the steps indicated in section 8.2.2 to show that the current through
an interacting region is given by Eq. (8.52).
(ii) Show that the current given by Eq. (8.52) vanishes in equilibrium.
(iii) Demonstrate that in the noninteracting case the Meir-Wingreen formula
of Eq. (8.52) reduces to the Landauer formula derived in section 8.1.
(iv) Assume that the scattering rates fulfill ΓL (E) = λΓR (E) and prove that
the Meir-Wingreen formula adopts the form given in Eq. (8.53).
8.8 Photo-current formula in the wide-band approximation:
(i) Show that the general formula of Eq. (8.70) for the current in a nanocontact
under an ac field reduces to Eq. (8.74) when (i) the energy dependence of the
density of states in the leads can be neglected (wide-band approximation) and
(ii) the ac potential is assumed to be flat in the central region of the system.
(ii) Starting from Eq. (8.75), show that in the limit of α ≪ 1 and small
frequencies, the conductance correction induced by the ac drive is a measure of the
second derivative of the transmission around the Fermi energy, i.e. demonstrate
Eq. (8.76).
Hint: Use the following properties of Bessel’s functions
∞
X ∞
X
Jn+l (x)Jm+l (x) = δnm , [Jl (x)]2 = 1,
l=−∞ l=−∞

(±x/2)l (±x/2)l+2
Jl (x ≪ 1, l > 0) ≈ − .
l! (l + 1)!

8.9 Linear conductance in the Coulomb blockade regime: As we shall

explain in Chapter 15, the Coulomb blockade is a transport phenomenon that
takes place in weakly coupled quantum dots and molecular junctions. The sig-
natures of Coulomb blockade in the linear conductance (i.e. at vanishingly small
bias voltage) are: (i) the appearance of peaks as a function of the gate voltage
(or chemical potential) known as Coulomb oscillations and (ii) a characteristic
temperature dependence that is described by the derivative of the Fermi function
with respect to energy. The goal of this exercise is to explain these two signa-
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236 Molecular Electronics: An Introduction to Theory and Experiment

tures by combining the Meir-Wingreen formula of Eq. (8.53) and the single-level
Anderson model of Eq. (5.109). For this purpose, carry out the following tasks:
(i) Adapt the Meir-Wingreen formula to the case of a single-level Anderson
model and derive an expression for the linear conductance in terms of the spectral
function in the resonant level.
(ii) Use the approximation of Eq. (5.112) to compute the spectral function in
the weak coupling limit.
(iii) Combine the results of (i) and (ii) to obtain the gate voltage and temper-
ature dependence of the linear conductance and show that this model reproduces
the two signatures described above.
Hint: This problem was addressed by Meir et al. in Ref. [632].
8.10 Kondo effect in molecular transistors: Unitary limit. The Kondo
effect in molecular junctions is manifested in the appearance of a pronounced
resonance in the density of states at the Fermi energy. This many-body effect is
usually described with the help of the Anderson model (see section 6.9). Apply
the Meir-Wingreen formula to this model and show that in the Kondo regime the
low-temperature linear conductance in a symmetric junction (ΓL = ΓR ) is equal
to the conductance quantum (G0 ). This is referred to as the unitary limit. Hint:
Use the Friedel sum rule discussed in section 6.9.1.
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Chapter 9

Electronic structure I: Tight-binding

approach

In the previous chapters we have shown how to compute the transport

properties of an atomic-scale junction once the corresponding Hamilto-
nian is known. Therefore, in order to make our theoretical background
self-contained, at least to a certain extent, we need to discuss how those
Hamiltonians are determined in practice. In other words, we have to de-
scribe adequate methods for the description of the electronic structure of
atomic and molecular junctions. Such methods are based on the stan-
dard approaches for the calculation of the electronic structure of atoms,
molecules and solids that are used in atomic physics, theoretical chemistry
and solid state physics. There is a great variety of electronic structure
methods and, obviously, we cannot review all of them here. We shall fo-
cus our attention on the two methods that have had the largest impact
so far in the field of molecular electronics. First, in this chapter we shall
discuss the tight-binding approach, which is a very intuitive empirical or
semi-empirical method that has been crucial to elucidate the physics of, in
particular, metallic atomic-sized contacts. Then, the next chapter is de-
voted to the density functional theory (DFT), which is the most widely
used approach among the so-called ab initio methods.
The tight-binding approach is reviewed in several textbooks and we
recommend in particular Refs. [223–226] to the physics-oriented readership
and Ref. [227] for a chemistry view on this subject.

9.1 Basics of the tight-binding approach

The main idea of the tight-binding approach was already introduced in Ap-
pendix A and indeed it has been extensively used in the previous chapters
devoted to the Green’s function techniques. Anyway, let us now define

237
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238 Molecular Electronics: An Introduction to Theory and Experiment

more precisely what we mean by tight-binding approach or by a tight-

binding model. The problem that we are interested in is the determination
the electronic structure of a system composed of a collection of atoms that
are located in different positions denoted by Ri . The corresponding Hamil-
tonian, H, of this system can be written in a local basis, i.e. in a basis
formed by single-particle wave functions that are localized around the dif-
ferent atomic positions. This is the spirit of the method known as linear
combination of atomic orbitals (LCAO), which is so popular in theoretical
chemistry. The first approximation in the tight-binding approach is to as-
sume that the Hamiltonian adopts the form of Eq. (A.67), which in first
quantization language reads (using Dirac’s notation)1
X
H= Hiα,βj |iαihjβ|, (9.1)
ij,αβ

where |iαi denotes the state that corresponds to the localized orbital α
that is centered around Ri , i.e. hr|iαi = φiα (r) = φα (r − Ri ). This generic
form for the Hamiltonian implies that either the many-body interactions
such as the electron-electron interaction are neglected or they are taken
into account in a mean field manner by an appropriate choice of the matrix
elements. In the former case, the matrix elements are rigorously defined as
~2 2
Z · ¸
∗
Hiα,jβ = dr φα (r − Ri ) − ∇ + V (r) φβ (r − Rj ), (9.2)
2m
where V (r) is the potential that describes the Coulomb interaction between
the electrons and ions. Finally, in the tight-binding approach, as it is used
in this book, the matrix elements are not determined from first principles,
i.e. from a direct evaluation of the integral in Eq. (9.2), but they are used
merely as parameters that may be derived approximately or may be fitted
to experiment or to other theories. Thus, by tight-binding model we mean
here a model in which the system is described in terms of a single-particle
Hamiltonian written in a local basis, the elements of which are determined
in a empirical or semi-empirical way. The different tight-binding models
differ in the way in which these parameters are obtained.
There are two situations where the wave function associated to a tight-
binding model can be determined in a straightforward manner. The first
one corresponds to the case of a small finite system such as a molecule and
1 This Hamiltonian in our usual second quantization language reads
Hiα,βj c†iα cjβ .
X
H=
ij,αβ
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Electronic structure I: Tight-binding approach 239

the second one corresponds to the case of an infinite periodic system. In

the first case, the Hamiltonian can be diagonalized by writing first the wave
function as a combination of the localized orbitals:
X
Φ(r) = ciα,jβ φjβ (r). (9.3)
jβ

This leads immediately to the following set of equations for the coefficients
(see Exercise 9.1)
X
[Hiα,jβ − ESiα,jβ ] ciα,jβ = 0, (9.4)
jβ

where E is the energy and

Z
Siα,jβ = dr φ∗α (r − Ri )φβ (r − Rj ), (9.5)

is the overlap between the states |iαi and |jβi. Here, we have taken into
account the possibility that the localized orbitals centered in different atoms
can be non-orthogonal. These equations have non-trivial solutions if
det (H − ES) = 0, (9.6)
where the symbol “det” denotes the determinant of the matrix appearing
inside the brackets. The roots of this secular equation yield the eigenen-
ergies or energy levels of the finite problem and the eigenfunctions are the
corresponding waves functions (or molecular orbitals) of this system. The
dimension of the matrices in Eq. (9.6) is simply the total number of local-
ized orbitals in the problem and therefore, the solution of the generalized
eigenvalue problem of Eq. (9.6) requires typically to resort to numerics.
In the case of an infinite periodic system, typical of solid state physics,
one can diagonalize the Hamiltonian making use of Bloch’s theorem (see
for instance Ref. [223]). The idea goes as follows. Consider a periodically
replicated unit cell, where the lattice vectors are denoted as Rm , with a set
of atoms i located at positions bi in each unit cell. Associated with each
atom is a set of atomic-like orbitals φiα , where α denotes both the orbital
and angular quantum number of the atomic state. The Hamiltonian can
be easily diagonalized in reciprocal space as follows. We first construct the
following wavefunctions (Bloch sums)
1 X
Φkiα (r) = √ exp(ik · Rn )φiα (r − Rn − bi ), (9.7)
N n
where k is the Bloch wave vector, which is restricted to the Brillouin zone,
and N is the number of unit cells in the sum. The solution to Schrödinger
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240 Molecular Electronics: An Introduction to Theory and Experiment

equation for wave vector k then requires the diagonalization of the Hamil-
tonian matrix using the basis functions of Eq. (9.7). Since the Hamilto-
nian has the periodicity of the lattice, this basis will block-diagonalize the
Hamiltonian, with each block having a single value of k. Within one of
these blocks, the matrix elements can be written in the form
X Z
Hiα,jβ (k) = exp(ik · Rn ) φ∗iα (r − Rn − bi )Hφjβ (r − bj )d3 r, (9.8)
n
where we have used the translation symmetry of the lattice to remove one of
the sums over the lattice vector R (see Exercise 9.2). In the same way, one
can also define the overlap matrix in reciprocal space where the different
elements adopt the form
X Z
Siα,jβ (k) = exp(ik · Rn ) φ∗iα (r − Rn − bi )φjβ (r − bj )d3 r. (9.9)
n
The corresponding secular equation reads this time
det (H(k) − ES(k)) = 0. (9.10)
The solution of this generalized eigenvalue problem yields the different en-
ergy bands, ǫµ (k) of the solid and the corresponding eigenvectors Qµ (k).
Notice that the number of bands, i.e. the number of solutions of Eq. (9.10),
equals the number of atoms in the unit cell times the number of orbitals per
atom. Thus, in some simple cases the solution can be found analytically
and, in general, this problem can be easily solved numerically.
An important quantity for many purposes is the density of states (DOS)
per unit energy E. The local DOS projected onto a given atom, orbital and
spin (summarized by the index ν) is defined in terms of the energy bands
ǫµ (k) as follows
1 X
ρν (E) = |Qν,µ (k)|2 δ(ǫµ (k) − E) (9.11)
Nk
k,µ
Ωcell X
Z
= dk |Qν,µ (k)|2 δ(ǫµ (k) − E),
(2π)d µ BZ
where BZ denotes the Brillouin zone, Ωcell is the volume of the unit cell
and d is the dimensionality of the system.
In the case of infinite non-periodic systems, like the atomic-scale junc-
tions that we are interested in, the determination of the wavefunction is
literally impossible. However, the use of the Green’s function techniques
described in Chapter 5 allows to extract most of the relevant information
about the electronic structure from a tight-binding Hamiltonian.
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Electronic structure I: Tight-binding approach 241

9.2 The extended Hückel method

The history of quantum chemistry is plagued with examples of approxima-

tions in the framework of the LCAO method, which fall into our definition
of tight-binding approach. One of the oldest and most familiar of such
approaches in quantum chemistry is the “extended Hückel approximation”
[228]. Let us explain briefly the idea behind this approach. It was developed
by Roald Hoffmann in 1963 [228] to describe the electronic structure of a
variety of organic molecules. It is based on the Hückel method [229–232]
but, while the original Hückel method only considers π-orbitals, the ex-
tended method also includes the σ-orbitals. The idea goes as follows. We
seek matrix elements of the Hamiltonian between atomic orbitals on adja-
cent atoms, hi|H|ji. If |ji were an eigenstate of the Hamiltonian, we could
replace H|ji by ǫj |ji, where ǫj , the on-site energy of the atom j, is the
eigenvalue. Then, if the overlap hi|ji is written Sij , the matrix element be-
comes ǫj Sij . This, however, treats the two orbitals differently, so we might
use the average instead of ǫj . Finding that this does not give good values,
we introduce a scale factor K, to be adjusted to fit the properties of heavy
molecules (a value of K = 1.75 is usually taken); this leads to the extended
Hückel formula2

hi|H|ji = KSij (ǫi + ǫj )/2. (9.12)

In the extended Hückel method, only valence electrons are considered;

the core electron energies and wave functions are supposed to be more or
less constant between atoms of the same type. The method uses a series of
parameterized energies calculated from atomic ionization potentials or the-
oretical methods to fill the diagonal of the Hamiltonian matrix. After filling
the non-diagonal elements (with the formula above) and diagonalizing the
resulting Hamiltonian matrix, the energies (eigenvalues) and wavefunctions
(eigenvectors) of the valence orbitals are found.
The extended Hückel approximation and a wide range of methods that
may be considered as descendents of it have enjoyed considerable success
in theoretical chemistry. This method can be used for determining the
molecular orbitals, but it is not very successful in determining the structural
geometry of an organic molecule. It can however determine the relative
energy of different geometrical configurations. It is common in quantum
chemistry to use the extended Hückel molecular orbitals as a first guess in
2 This formula is indeed due to M. Wolfsberg and L.J. Helmholtz [233].
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242 Molecular Electronics: An Introduction to Theory and Experiment

the determination of the molecular orbitals by ab initio quantum chemistry

methods.

9.3 Matrix elements in solid state approaches

In the context of solid state physics most of the semi-empirical tight-binding

applications are largely based on the seminal work of Slater and Koster (SK)
[234], in which they proposed a modified LCAO method to interpolate the
results of first-principles electronic structure calculations. At that time
(1954), it was computationally impossible to directly evaluate the large
number of integrals occurring in the LCAO method. However, since this
approach shows all the correct symmetry properties of the energy bands
as well as providing solutions of the single-particle Schrödinger equation at
arbitrary points in the Brillouin zone, they suggested that these integrals
could be considered as adjustable constants to be determined from the
results of other, more efficient, calculations. In order to understand the
basis of the simplified LCAO/tight-binding method proposed by Slater and
Koster, we need first to discuss in certain detail the nature of the matrix
elements that appear in the tight-binding approach. Thus, the explanation
of the SK-method will be postponed until the next section.
The tight-binding approach benefits from the consideration of the sym-
metries of the basis orbitals and the crystal or molecule. On each site of the
physical system, the atomic-like functions can be written as radial functions
multiplied by spherical harmonics,

φnlm (r) = φnl (r)Ylm (r̂), (9.13)

where r = |r|, r̂ = r/r and n indicates different functions with the same
angular momentum. We shall work frequently with real basis functions √ that
+ ∗
can be defined using the√ real angular functions S lm = (Ylm + Y lm )/ 2 and
− ∗
Slm = (Ylm − Ylm )/(i 2). The examples of real s (l = 0), p (l = 1) and
d (l = 2) orbitals are given in Fig. 9.1. The analytical expressions of the
angular dependence of these real orbitals can be found in many textbooks,
see e.g. Chapter 1 of Ref. [224] or Chapter 3 of Ref. [227].
The key problem in a tight-binding model is the determination of the
matrix elements (or integrals) that appear both in Eq. (9.8) and Eq. (9.9).
Those matrix elements can be divided into one-, two-, and three-center
terms. The simplest is the overlap matrix in Eq. (9.9), which involves only
one center if the two orbitals are on the same site and two centers otherwise.
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Electronic structure I: Tight-binding approach 243

m=0 m =+
−1

m=0 m =+
−1 m =+
−2

Fig. 9.1 Boundary surfaces for real s-, p-, and d-orbitals. The index m indicates the
quantum number corresponding to the z-component of the orbital angular momentum.

The Hamiltonian matrix elements appearing in Eq. (9.8) consist of kinetic

and potential terms

~2 2 X
H=− ∇ + Vk (r − Rn − bk ), (9.14)
2m
nk

where the first term is the usual kinetic energy and the second is the po-
tential decomposed into a sum of spherical terms centered on each site k in
the unit cell. The kinetic part of the Hamiltonian matrix element always
involves one or two centers. However, the potential terms may depend upon
the positions of other atoms; they can be divided into the following.

• One-center, where both orbitals and the potential are centered on the
same site. These terms have the same symmetry as an atom in free
space.
• Two-center, where the orbitals are centered on different sites and the
potential is on one of the two. These terms have the same symmetry
as other two-center terms.
• Three-center, where the orbitals and the potential are all centered on
different sites. These terms can also be classified into various symme-
tries based upon the fact that three sites define a triangle.
• A special class of two-center terms with both orbitals on the same site
and the potential centered on a different site. These terms add to the
one-center terms above, but depend upon the crystal symmetry.
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244 Molecular Electronics: An Introduction to Theory and Experiment

ssσ spσ sdσ

ppσ
ppπ

pdσ
pdπ

ddπ ddδ
ddσ

Fig. 9.2 The 10 irreducible SK-parameters for the s, p and d orbitals, which are classi-
fied by the angular momentum about the axis with the notation σ (m = 0), π (m = 1)
and δ (m = 2). The orbitals shown are the real combinations of the angular momen-
tum eigenstates. Positive and negative lobes are denoted by solid and dashed lines,
respectively.

9.3.1 Two-center matrix elements

Two-center matrix elements play a special role in most practical tight-
binding approaches and are considered here in more detail. The analysis
applies to all overlap terms and to any Hamiltonian matrix element that
involves only orbitals and potential on two sites. For these integral the
problem is the same as for a diatomic molecule in free space with cylin-
drical symmetry. The orbitals can be classified in terms of the azimuthal
angular momentum about the line between the centers, i.e. the value of m
with the axis chosen along the line, and the only non-zero matrix elements
are between orbitals with the same m. If Klm,l′ m′ denotes an overlap or
two-center Hamiltonian matrix element for states lm and l′ m′ , then in the
standard form with orbitals quantized about the axis between the pair of
atoms, the matrix elements are diagonal in mm′ and can be written as
Klm,l′ m′ = Kll′ m δm,m′ . The quantities Kll′ m are independent matrix ele-
ments that are irreducible, i.e. they cannot be further reduced by symmetry.
By convention the states are labeled with l or l′ denoted by s, p, d, ..., and
m = 0, ±1, ±2, ..., denoted by σ, π, δ, ..., leading to the notation Kssσ ,
Kspσ , Kppπ , etc.
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Electronic structure I: Tight-binding approach 245

z
y
spx sp σ
x

R R

ppσ

px pz R

R ppπ

Fig. 9.3 Schematic representation of two examples of two-center matrix elements of s

and p orbitals for atoms separated by a displacement vector R. Matrix elements are
related to σ and π integrals by the transformation to a combination of orbitals that are
aligned along R and perpendicular to R. The top figure illustrates the transformation
to write a real matrix element Ks,px in terms of Kspσ : the s orbital is unchanged and
the px orbital is written as a sum of the σ orbital, which is shown, and the π orbitals,
which are not shown because there is no spπ matrix element. The lower figure illustrates
the transformation needed to write Kpx ,pz in terms of Kppσ and Kppπ . The coefficients
of the transformation for all s and p matrix elements are given in Table 9.1.

In Fig. 9.2 we show the orbitals for the non-zero σ, π, and δ matrix
elements for s, p, and d orbitals. The orbitals shown are actually the real
±
basis functions Slm defined as combinations of the ±m angular momentum
eigenstates. These are oriented along the axes defined by the line between
the neighbors and two perpendicular axes. All states except the s state
have positive and negative lobes. Note that states with odd l are odd under
inversion. Their sign must be fixed by convention (typically one chooses
the positive lobe along the positive axis). The direction of the displacement
vector is defined to lie between the site denoted by the first index and that
denoted by the second index. For example, in Fig. 9.2, the Kspσ matrix
element in the top center has the negative lobe of the p function oriented
toward the s function. Interchange of the indices leads to Kpsσ = −Kspσ
′
and, more generally, to Kll′ m = (−1)l+l Kl′ lm .
An actual set of basis functions is constructed with the quantization
axis fixed in space, so that the functions must be transformed to utilize the
standard irreducible form of the matrix elements. Examples of two-center
matrix elements of s and pi = px , py , pz orbitals for atoms separated by
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246 Molecular Electronics: An Introduction to Theory and Experiment

Table 9.1 Table of two-center matrix elements for either the overlap or the Hamil-
tonian, with real orbitals s and px , py , pz . The vector R between sites, as shown
in Fig. 9.3, is defined to have direction components R̂ ≡ x, y, z. The matrix el-
ements are then expressed in terms of these coordinates and the four irreducible
matrix elements: Kssσ , Kspσ , Kppσ and Kppπ . Other matrix elements can be
found by permuting elements.

Element Expression

Ks,s Kssσ
Ks,px xKspσ
Kpx ,px x2 Kppσ + (1 − x2 )Kppπ
Kpx ,py xy(Kppσ − Kppπ )
Kpx ,pz xz(Kppσ − Kppπ )

the displacement vector R are shown in Fig. 9.3. Each of the orbitals on
the left-hand side can be expressed as a linear combination of orbitals that
have the standard form oriented along the rotated axes, as shown on the
right. An s orbital is invariant and a p orbital is transformed to a linear
combination of p orbitals. The only non-zero matrix elements are the σ
and π matrix elements, as shown. The top row of the figure illustrates the
transformation of the px orbital needed to write the matrix element Ks,px
in terms of Kspσ and the bottom row illustrates the relation of Kpx ,pz to
Kppσ and Kppπ . Specific relations for all s and p matrix elements are given
in Table 9.1. Expressions for d orbitals are given in Refs. [234, 224, 225].

9.4 Slater-Koster two-center approximation

Now we are in position to describe the Slater and Koster approach [234].
These authors proposed that the Hamiltonian matrix elements can be ap-
proximated with the two-center form and fitted to theoretical calculations
(or empirical data) as a simplified way of describing and extending cal-
culations of electronic bands. Within this approach, all matrix elements
have the same symmetry as for two atoms in free space (see Fig 9.3 and
Table 9.1). This is a great simplification that leads to an extremely useful
approach to understanding electrons in materials.
Slater and Koster gave extensive tables for matrix elements, including
the s and p matrix elements given in Table 9.1. In addition, they presented
expressions for the d states and analytical formulas for bands in several
crystal structures. Examples of the latter are presented in the next sec-
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Electronic structure I: Tight-binding approach 247

tion to illustrate the useful information that can be derived. However, the
primary use of the SK approach in electronic structure has become the de-
scription of complicated systems, including the bands, total energies, and
forces for relaxation of structures and molecular dynamics. These applica-
tions have very different requirements that often lead to different choices of
SK parameters.
For the bands, the parameters are usually designed to fit selected eigen-
values for a particular crystal structure and lattice constant. For example,
the extensive tables derived by Papaconstantopoulos [235] are very useful
for interpolation of results of more expensive methods. It has been pointed
out by Stiles [236] that for a fixed ionic configuration, effects of multi-center
integrals can be included in two-center terms that can be generated by an
automatic procedure. This makes it possible to describe any band struc-
ture accurately with a sufficient number of matrix elements in SK form.
However, the two-center matrix elements are not transferable to different
structures.
On the other hand, any calculation of total energies, forces, etc., requires
that the parameters be known as a function of the position of the atoms.
Thus, the choices are usually compromises that attempt to fit a large range
of data. Such models are fit to structural data and, in general, are only
qualitatively correct for the bands. Since the total energy depends only
upon the occupied states, the conduction bands may be poorly described
in these models. Of particular note, Harrison [224, 225] has introduced a
table that provides parameters for any element or compound. The forms
are chosen for simplicity, generality, and ability to describe many properties
in a way that it is instructive and useful. The basis is assumed to be
orthonormal, i.e. Smm′ = δmm′ . The diagonal Hamiltonian matrix elements
are given in a table for each atom. Any Hamiltonian matrix element for
orbitals on neighboring atoms separated by a distance R is given by a factor
′
times 1/R2 for s and p orbitals and 1/Rl+l for l > l′ .
Many other SK parameterizations have been proposed, each tailored
to particular elements and compounds. Some additional examples can be
found in Chapter 14 of Ref. [226].

9.5 Some illustrative examples

Let us illustrate the tight-binding approach with the analysis of some simple
situations.
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248 Molecular Electronics: An Introduction to Theory and Experiment

9.5.1 Example 1: A benzene molecule

Often in molecular electronics one wants to establish a relation between the
transport properties of a molecular junction and the corresponding the elec-
tronic structure of an isolated molecule. Let us illustrate how this electronic
structure can be described by means of simple tight-binding models. For
this purpose, we consider here the case of a benzene molecule, which was
introduced in section 3.2. Benzene is an emblematic example of a molecule
in which the relevant electronic structure is determined by a conjugated
π-system. This means that the electrons in the highest occupied orbitals
reside in π orbitals, which in this case are formed by the 2pz orbitals of
the six carbon atoms (here z is the direction perpendicular to the plane of
the molecule). The word conjugated refers to the fact that this π-system
extends over several neighboring atoms, which is the way in which the
binding energy is increased in these molecules. In molecular orbital theory
in chemistry, benzene is often described within the simplified Hückel ap-
proximation [229, 232, 227]. This approximation is based on the following
basic assumptions: (i) only π-orbitals are considered (the σ-orbitals are
much more strongly bound and they can be ignored), i.e. only one orbital
per carbon atom is taken into account, (ii) the overlap integrals between
different orbitals are set to zero: Sij = δij , (iii) all the diagonal matrix
elements of the Hamiltonian are ascribed the same value: Hii = ǫ0 , and
(iv) the off-diagonal elements are set equal to zero except for those between
neighboring atoms, all of which are set equal to −t, where t is positive.
This model for benzene is summarized schematically in Fig. 9.4(a).
Following our discussion on finite systems in section 9.1, see in particular
Eq. (9.6), the energy levels in this model are the roots of the following
secular equation
¯ ¯
¯ ǫ0 − E −t 0 0 0 −t ¯¯
¯
¯ −t ǫ − E −t 0 0 0 ¯¯
¯ 0
¯ 0 −t ǫ0 − E −t 0 0 ¯
¯ ¯
¯ = 0. (9.15)
¯ 0 0 −t ǫ0 − E −t 0 ¯
¯
−t ǫ0 − E −t ¯¯
¯ ¯
¯ 0 0 0
¯
¯ −t 0 0 0 −t ǫ0 − E ¯
In this determinant we have followed the order sketched in Fig. 9.4(a) and
the π-orbitals in the different C atoms are denoted by |ii, with i = 1, ..., 6.
This equation can be solved analytically (see Exercise 9.3) and the different
eigenenergies are given by
E1 = ǫ0 − 2t; E2 = E3 = ǫ0 − t; E4 = E5 = ǫ0 + t; E4 = ǫ0 + 2t, (9.16)
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Electronic structure I: Tight-binding approach 249

4 ε0+ 2t
(a) (b) (6)

−t −t
5
3
(4)
ε0+ t ε0+ t (5)

−t −t Energy
(2)
ε0− t ε0− t (3)

2 6
−t −t ε0− 2t
(1)
1

Fig. 9.4 (a) Schematic representation of the Hückel model for the benzene molecule,
as described in the text. (b) Energy level diagram of benzene as obtained from this
approximation. The levels are labeled from 1 to 6 following Eq. (9.16). We also show
charge-density plots of the molecular orbitals obtained from a density-functional-theory
calculation to show that indeed the Hückel approximation reproduces the character of
the orbitals, see Eq. (9.17). The two colors indicate different signs of the wavefunctions.
The ground state is obtained by doubly occupying the three lowest energy levels.

and the corresponding molecular orbitals (eigenfunctions) read

1
φ1 = √ (|1i + |2i + |3i + |4i + |5i + |6i)
6
1
φ2 = √ (2|1i + |2i − |3i − 2|4i − |5i + |6i)
12
1
φ3 = (|2i + |3i − |5i − |6i)
2
1
φ4 = √ (2|1i − |2i − |3i + 2|4i − |5i − |6i)
12
1
φ5 = (|2i − |3i + |5i − |6i)
2
1
φ6 = √ (|1i − |2i + |3i − |4i + |5i − |6i) (9.17)
6
These orbitals indeed describe correctly the symmetry and extension of the
molecular orbitals that one obtains with more sophisticated methods, as
one can see in Fig. 9.4(b), where we show the orbitals as obtained from a
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250 Molecular Electronics: An Introduction to Theory and Experiment

density-functional-theory calculation. The ground state of benzene is that

in which the six π electrons are occupying the three lowest energy levels,
as shown in Fig. 9.4.(b) and it has a total energy of (6ǫ0 − 8t). Notice that
the gap between the highest occupied molecular orbital (HOMO) and the
lowest unoccupied molecular orbital (LUMO) is equal to 2t. This result can
be used to obtain the value of the hopping parameter t by comparing with
spectroscopical data or ab initio calculations. A fit to a density-functional-
theory calculation gives a value of around 2.6 eV.
It is worth stressing that the molecule gains stability or binding energy
by delocalizing the electrons over the entire molecule (conjugation). This
fact can be quantified in the following manner. If the molecule were de-
scribed as having three unconjugated π-bonds, its total π-electron energy
would have been 6(ǫ0 − t). By means of the conjugation, the molecule has
gained an energy equal to −2t (this gain is sometimes called delocalization
energy).
On the other hand, notice that the form of the orbitals is determined
solely by the symmetry of the molecule. Notice also that the six electrons
just complete the molecular orbitals with net bonding effect, leaving unfilled
the orbitals with net antibonding character. Another feature of the energy
of levels of benzene is that the array of levels is symmetrical: to every
bonding level there corresponds an antibonding level. This symmetry is a
characteristic feature of alternant hydrocarbons and can be traced to the
topology of the molecules.
This has been a simple example of the insightful molecular orbital the-
ory, which is widely used in theoretical chemistry. For more examples, see
for instance Chapter 8 of Ref. [227].

9.5.2 Example 2: Energy bands in line, square and cubic

Bravais lattices
In molecular electronics it is important to know the bulk electronic struc-
ture of typical metals that are used in molecular junctions. For this reason,
we consider now a bulk solid with a single atom per unit cell. The simplest
possible example of bands is that of a lattice in which we have a single
relevant orbital per site with s-symmetry. As a further simplification, we
consider the case of orthogonal basis states and non-zero Hamiltonian ma-
trix elements hi|H|ji = t only if i and j are nearest neighbors. The on-site
energy can be chosen to be zero, hi|H|ii = 0. There are three cases (line,
square and cubic lattices) that can be treated together. For the cubic lat-
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Electronic structure I: Tight-binding approach 251

0.3 0.2
1 (a) 1D (b) 2D (c) 3D
DOS (1/t)

0.2
0.1
0.5
0.1

0 0 0
-2 -1 0 1 2 -4 -2 0 2 4 -6 -4 -2 0 2 4 6
E/t E/t E/t
Fig. 9.5 Density of states per spin (DOS) vs. energy for an s-band in (a) a one-
dimensional line, (b) a two-dimensional square, and (c) a three dimensional simple cubic
lattice with nearest neighbor interactions t.

tice with spacing a the general expressions (9.8) and (9.10) reduce to (see
Exercise 9.4)

ǫ(k) = H(k) = 2t [cos(kx a) + cos(ky a) + cos(kz a)] . (9.18)

The bands for the square lattice in the xy-plane are given by this expression,
omitting the kz term; for a line (or chain) in the x-direction, only the kx
term applies. From this expression one can easily deduce several interesting
consequences. First, the bands are symmetric about ǫ(k) = 0 in the sense
that every state at +ǫ has a corresponding state at −ǫ. This can be seen in
Fig. 9.5, where we show the density of states (DOS) for one, two and three
dimensions. The shapes can be found analytically in this case (see Exercise
9.6). Notice that the bandwidth is determined by the hopping element t.
In the case of a metal, this parameter has a value of around 1 eV.
In the square lattice, the energy ǫ(k) = 0 at a face zone k = (π/a, 0).
This is a saddle point since the slope vanishes and the bands curve upward
and downward in different directions. This leads to a density of states with
a logarithmic divergence at ǫ = 0. Furthermore, for a half-filled band (one
electron per cell), the Fermi surface is at ǫ(k) = 0. This leads to the result
that the Fermi surface is a square rotated by π/4 with half the volume of
the Brillouin zone, and the density of states diverges at ǫ = EF as shown
in Fig. 9.5(b). If there are second-neighbor interactions, the symmetry of
the bands in ±ǫ is broken and the Fermi surface is no longer square.
Let us assume now that the states are no longer orthogonal, but the
overlap between nearest neighbors is equal to s. Then the solution for the
bands, Eq. (9.18), is generalized to (Exercise 9.7)
H(k) 2t [cos(kx a) + cos(ky a) + cos(kz a)]
ǫ(k) = = . (9.19)
S(k) 1 + 2s [cos(kx a) + cos(ky a) + cos(kz a)]
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252 Molecular Electronics: An Introduction to Theory and Experiment

In this case, the symmetry about ǫ = 0 is broken, so that the conclusions

on bands and the Fermi surface no longer apply. In fact s has an effect like
longer range Hamiltonian matrix elements, indeed showing strictly infinite
range but rapid exponential decay.

9.5.3 Example 3: Energy bands of graphene

As an example of a lattice with more than one atom per unit cell, we con-
sider now the case of graphene. Graphene is a two-dimensional system
formed by a single sheet of carbon atoms. Although the graphene band
structure was already discussed theoretically more than 50 years ago [237],
only recently it has been shown to exist in reality [238] and its physical
properties are attracting a great attention [239–241]. Graphene has the
planar honeycomb structure shown in Fig. 9.6(a). The corresponding Bril-
louin zone is a hexagon. Full calculations show that the band of graphitic
systems at the Fermi energy are π bands, composed of electronic states
that are odd in reflection in the plane. For graphene the π bands are well
represented as linear combinations of pz orbitals of the C atoms, where z is
perpendicular to the plane. Since graphene has two atoms per cell, the pz
states form two bands. If there is a nearest neighbor Hamiltonian matrix
element t, the bands are given by (Exercise 9.8)
¯ ¯
¯ −ǫ(k) H12 (k) ¯
|H(k) − ǫ(k)| = ¯¯ ∗ ¯ = 0, (9.20)
H12 (k) −ǫ(k) ¯
where (with the lattice oriented as in Fig. 9.6(a))
h √ √ i
H12 (k) = t eiky a/ 3 + 2e−iky a/2 3 cos(kx a/2) , (9.21)

and a is the lattice constant. This is readily solved to yield the bands
h √ i1/2
ǫ(k) = ±t 1 + 4 cos( 3ky a/2) cos(kx a/2) + 4 cos2 (kx a/2) . (9.22)

The most remarkable feature of the graphene bands is that they touch at
the corners of the hexagonal Brillouin zone, e.g. the points denoted K± =
(kx = ±4π/3a, ky = 0). Note also that the bands are symmetric in ±ǫ.
Since there is one π electron per atom, the band is half-filled and the bands
touch with finite slope at the Fermi energy, i.e. a Fermi surface consisting of
points. Indeed, one can show (see Exercise 9.8) that the dispersion relation
around the points K± (Dirac points) is √ linear, i.e. ǫ(q) = ~v|q|, where
q = k − K± with a velocity given by v = ( 3a/2~)t. This linear dispersion
relation resembles that of Dirac’s massless fermions and it is the origin of
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Electronic structure I: Tight-binding approach 253

(a) (b) 0.4

0.3
a1

DOS
0.2
a2
0.1

0
-3 -2 -1 0 1 2 3
E/t
Fig. 9.6 (a) Honeycomb lattice for a graphene sheet: the lattice is triangular and there
√
are two atoms per unit cell. Two primitive vectors are ~a1 = a(1, 0) and ~a2 = a/2(1, − 3),
where a is the lattice constant. (b) Local density of states (per spin) projected onto an
atom of the unit cell as a function of the energy normalized by the hopping parameter t.
Notice that the DOS vanishes at E = 0 and there are van Hove singularities at E = ±t.

the extraordinary properties of this material [239–241]. Finally, if we have

a look at the density of states, see Fig. 9.6(b), we can see that (undoped)
graphene is a zero-bandgap semiconductor.

9.6 The NRL tight-binding method

There is a basic difficulty in generating tight-binding models that can de-

scribe very different structures. In models that have only two-center matrix
elements, the values of the matrix elements must take into account effects
of three-center terms. These effects change drastically between structures.
There are two primary approaches toward making tight-binding models that
are transferable between different structures. One is to define environment-
dependent matrix elements, the values of which depend upon the presence of
other neighbors. The other approach involves non-orthogonal tight-binding,
which is more transferable than orthogonal forms.
The goal of this section is to describe in certain detail a sophisticated
tight-binding parameterization in the spirit of the SK approach that meets
the two requirements discussed in the previous paragraph. Moreover, this
parameterization, which has been developed by Cohen, Mehl, and Papa-
constantopoulos [242, 243], also allows us the computation of total energies
and related quantities. This method has been widely used, in particular, for
the analysis of the transport properties of metallic atomic-sized contacts,
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254 Molecular Electronics: An Introduction to Theory and Experiment

as we will show in later chapters. This parameterization is referred to as

NRL3 and it is nicely described in several review articles [244, 245].
Up to this point we have mainly discussed tight-binding parameteriza-
tions of the band structure alone.4 Total-energy information is typically not
given by these calculations, although a band energy can be readily deter-
mined from the sum of the eigenvalues over the occupied states. However,
in single-particle band theory this sum is only a partial contribution to the
total energy. In the Kohn-Sham single particle density functional theory
(DFT) Ansatz, which will be explained in the next chapter, the total energy
is given by
d3 k X
Z
E= ǫn (k) + F [n(r)], (9.23)
(2π)3 n
where the integral is over the first Brillouin zone, the sum is over occupied
states, and F [n(r)] is a functional of the density which includes the repulsion
of the ionic cores, correlation effects, and part of the Coulomb interaction.
Note that the value of the integral depends upon the choice of zero for the
Kohn-Sham potential vKS (r) which generates the eigenvalue spectrum:
−∇2 ψn (r) + vKS (r)ψn (r) = ǫn ψn (r). (9.24)
This choice is arbitrary. In the method developed at NRL [242, 243], the
potential vKS in the previous equation is shifted by an amount
V0 = F [n(r)]/Ne , (9.25)
where Ne is the number of electrons in the unit cell. Then, the total energy
of the system is
d3 k X d3 k X
Z Z
E= 3
ǫn (k) + F [n(r)] = ǫn (k) + Ne V0
(2π) n (2π)3 n
d3 k X
Z
= [ǫn (k) + V0 ], (9.26)
(2π)3 n
If we now define a shifted eigenvalue: ǫ′ (k) = ǫn (k) + V0 , then to get the
total energy we just sum the shifted eigenvalues of the occupied states:
d3 k X ′
Z
E= ǫ (k). (9.27)
(2π)3 n
3 NRL stands for Naval Research Laboratory, which is located in Washington D.C. For

more practical information about this parameterization, visit the web page: http://cst-
www.nrl.navy.mil/bind/.
4 The only exception was the method put forward by Harrison that was briefly mentioned

in section 9.4.
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Electronic structure I: Tight-binding approach 255

Note that V0 depends upon the structure of the crystal, as well as the
original method for determining the energy zero. Notice also that the ǫ′ (k)
are in some sense “universal”. That is, if any two band structure methods
are sufficiently well converged, they will give the same total energy, and the
eigenvalues derived from the two methods will differ by only a constant.
Then the definition of V0 for each method will be such that the shifted
eigenvalues ǫ′ (k) are identical.
In the NRL method, the authors construct a first-principles5 database of
eigenvalues ǫ(k) and total energies E for several crystal structures at several
volumes. Then, they find V0 for each system, and shift the eigenvalues.
Next, they attempt to find a set of parameters which will generate non-
orthogonal, two-center SK Hamiltonians which will reproduce the energies
and eigenvalues in the database.
Let us now describe how the TB parameters for elemental systems are
constructed. One assumes that the on-site terms are diagonal and sensitive
to the environment. For single-element systems one assigns atom i in the
crystal an embedded-atom-like “density”
X
ρi = exp(−λ2 Rij )F (Rij ), (9.28)
j

where the sum is over all the atoms j within a range Rc of atom i; λ is the
first fitting parameter, squared to ensure that the contributions are greater
from the nearest neighbors; and F (R) is a cut-off function,
F (R) = θ(Rc − R)/ {1 + exp [(R − Rc )/l + 5]} , (9.29)
where θ(z) is the step function. Typically one takes Rc between 10.5 and
16.5 Bohr and l between 0.25 and 0.5 Bohr.
One then defines the angular-momentum-dependent on-site terms by
2/3 4/3
hil = al + bl ρi + cl ρi + dl ρ2i , (9.30)
where l = s, p, or d. These (a, b, c, d)l form the next 12 fitting parameters.
In the spirit of the two-center approximation, one assumes that the
hopping integrals depend only upon the angular momentum of the orbitals
and the distance between the atoms. As we showed in section 9.3.1, all
the two-center (spd) hopping integral can then be constructed from ten
independent parameters, the SK parameters, Hll′ m , where
(ll′ m) = ssσ, spσ, ppσ, ppπ, sdσ, pdσ, pdπ, ddσ, ddπ, and ddδ. (9.31)
5 Thefirst-principle methods used by the authors are typically the augmented plane
wave method (APW) or the linearized augmented plane wave method (LAPW).
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256 Molecular Electronics: An Introduction to Theory and Experiment

Then, it is assumed the following polynomial × exponential form for these

parameters:

Hll′ m (R) = (ell′ m + fll′ m R + gll′ m R2 ) exp(−h2ll′ m R)F (R), (9.32)

where R is the separation between these atoms and F (R) is the cut-off
function defined above. The parameters (ell′ m , fll′ m , gll′ m , hll′ m ) constitute
the next 40 fitting parameters.
Since this is a non-orthogonal calculation, one must also define a set of
SK overlap functions. These represent the overlap between two orbitals sep-
arated by a distance R. They have the same angular momentum behavior
as the hopping parameters:

Sll′ m (R) = (pll′ m + qll′ m R + rll′ m R2 ) exp(−s2ll′ m R)F (R), (9.33)

The parameters (pll′ m , qll′ m , rll′ m , sll′ m ) make up the final 40 fitting pa-
rameters for a monoatomic system, giving in total 93 fitting parameters
that are chosen to reproduce the contents of the first-principles database,
as noted above.6
So in summary, this parameterization uses an analytical set of two-
center integrals, nonorthogonal parameters and on-site parameters that de-
pend on the local environment. The method reproduces not only the band
structure, but also the total energy of the system. It has been demonstrated
that this method reproduces very well structural energy differences, elas-
tic constants, phonon frequencies, vacancy formation energies, and surface
energies for both transition metal and noble metals.
As an application of this parameterization, we have computed the bulk
density of states of six different metals that play an important role in molec-
ular electronics.7 The results can be seen in Fig. 9.7. Notice that in the
cases of Ag and Au (noble metals), the Fermi energy lies in the region
where the DOS is dominated by the s band. In the case of Al and Pb,
the s and p bands dominate the DOS around the Fermi energy. The main
difference between these two metals is that Pb has 4 valence electrons and
therefore, the Fermi energy lies well inside the p band. Finally, Nb and Pt
are examples of transition metals, where the d band dominates the DOS at
the Fermi energy and for this reason, the d orbitals play a fundamental in
the transport properties of these metals.
6 These parameters for many different elementary solids can be found in the following
web page: http://cst-www.nrl.navy.mil/bind/.
7 In particular, we shall analyze in Chapter 11 the conductance of single-atom contacts

of these six materials.

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Electronic structure I: Tight-binding approach 257

3 3
Ag Au
DOS (1/eV)

5s 6s
2 5p 2 6p
4d 5d
1 1

0 0
-10 -5 0 5 10 15 -10 -5 0 5 10 15

Al 3s Pb 6s
DOS (1/eV)

0.4 0.8
3p 6p
3d
0.2 0.4

0 0
-10 -5 0 5 10 15 -10 -5 0 5 10 15
3 3
Nb 5s Pt 6s
DOS (1/eV)

2 5p 2 6p
4d 5d

1 1

0 0
-10 -5 0 5 10 15 -10 -5 0 5 10 15
E-EF (eV) E-EF (eV)
Fig. 9.7 Bulk DOS as a function of energy for Ag, Au, Al, Pb, Nb, and Pt computed
using the NRL-tight-binding parameterization. The DOS is projected onto the s, p, and
d orbitals that give rise to the bands around the Fermi energy (EF ).

9.7 The tight-binding approach in molecular electronics

In this final section we shall explain how the tight-binding approach is used
in practice to describe the transport properties of atomic-scale junctions
and we shall also review very briefly the impact of this approach in the
field of molecular electronics to date.
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258 Molecular Electronics: An Introduction to Theory and Experiment

9.7.1 Some comments on the practical implementation of

the tight-binding approach
In the previous chapters we have learned how the Green’s function tech-
niques can be combined with the knowledge of the Hamiltonian matrix
elements in a local basis to describe both equilibrium and transport prop-
erties of atomic-scale junctions. Thus, the application of the tight-binding
approach to the description of the physics of atomic or molecular junctions
is now rather straightforward. Anyway, let us mention some of technical
issues and difficulties that one encounters in the practical application of
this method.
One of the most common theoretical problems in molecular electronics
is the calculation of the elastic current of an atomic-scale junction. Such
calculation proceeds in a series of steps that we now proceed to describe.
Step 1: Geometry of the contact. As a first step one has to define the
geometry of the junction. This geometry includes a central part of arbi-
trary shape and two ideal leads or electrodes that, for practical reasons,
must have a regular structure (see Fig. 8.1). Ideally, one should deter-
mine the junction geometry by doing, for instance, molecular dynamics
simulations or geometry optimization. In principle, this is possible with
sophisticated tight-binding approaches like the NRL-method described in
section 9.6 which allows us to compute the forces between the atoms or the
total energy of the system. In practice, the determination of the geometry
with tight-binding models has been restricted to the case of atomic con-
tacts and for molecular junctions one needs to resort to more sophisticated
methods like density function theory (see next chapter).
Step 2: Hamiltonian matrix elements. Once the geometry is defined,
one proceeds to the determination of the matrix elements of the Hamilto-
nian, as explained in the previous sections. For instance, in the approaches
based on the two-center approximation (see section 9.4), one can construct
those matrix elements between two neighboring atoms by projecting the
irreducible SK-parameters according to their relative position.
Step 3: Calculation of the Green’s functions. The retarded and ad-
vanced Green’s functions of the central part of the junction contain all the
relevant information about both the equilibrium and transport properties
of the system. These functions are computed via their Dyson’s equation,
see Eq. (8.22). This requires to previously calculate the self-energies and
in turn the Green’s functions of the leads. This is indeed the most com-
plicated step in the whole calculation. The leads are semi-infinite systems
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Electronic structure I: Tight-binding approach 259

and the lack of periodicity complicates the calculation of their Green’s func-
tions. There are different solutions for this problem. For instance, one can
describe the electrodes with simple structures like Bethe lattices [246]. A
more satisfactory solution to avoid artificial interface resistances is to de-
scribe the leads as ideal surfaces and compute the Green’s functions with
recursive methods like the decimation technique of Ref. [198].8
Step 4: Computation of the current. The final step is the calculation
of the current from the knowledge of the Green’s functions, which is done
using Eqs. (8.18-8.20).
In general, the “recipe” described above has to be carried out numeri-
cally, but the computer codes can be developed by a single person in a few
weeks. Moreover, the tight-binding approach is extremely efficient, com-
putationally speaking, and the calculations of the transport properties of
realistic systems can be done in standard PC’s. Of course, the level of accu-
racy of these calculations depends on the quality tight-binding parameters,
which in turn depends on the system under study.

9.7.2 Tight-binding simulations of atomic-scale transport

junctions
The tight-binding approach has been used to describe a great variety of
problems related to the electronic transport in atomic-scale junctions. Our
goal in this subsection is to mention very briefly some these applications and
we refer the reader to Todorov’s review [248] for a more detailed discussion
and for a more complete list of references.
One of the first applications of the tight-binding approach was the anal-
ysis of the operation of the STM and the interpretation of images taken with
this instrument [194, 249–252]. This approach has been very important to
elucidate the role of the tip-substrate distance and it allows to identify the
characteristic signature of many different adsorbates.
In the context of molecular junctions, at the end of the 1980’s Sautet
and Joachim pioneered the use of the tight-binding approach (within the
extended Hückel approximation) to compute the current and conductance
of single-molecule junctions [253, 254]. Later, Ratner and coworkers used
simple tight-binding models to address a number of issues such as the depen-
dence of the conductance on the length of the molecular wires [255–259].9
8 An extension of the decimation technique to the case of non-orthogonal basis sets can

be found in the appendix C of Ref. [247].

9 In Chapter 13 we shall make use of simple tight-binding models to discuss basic issues
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260 Molecular Electronics: An Introduction to Theory and Experiment

Also in the middle of the 1990’s, Datta and coworkers employed the tight-
binding approach to describe the current-voltage characteristics of different
organic molecules and to establish a detailed comparison with the experi-
ments [260, 261].
In the context of metallic atomic-sized contacts,10 the tight-binding for-
malism was first used, in combination with molecular dynamics simula-
tions, to elucidate the origin of the conductance jumps observed during
the formation of these nanowires [262]. Tight-binding models were then
used to establish the relation between the conduction channels of single-
atom contacts and the detailed chemistry of the metal atoms [263, 264].
The tight-binding approach has also been extended to calculate changes
to interatomic forces under electrical current flow in atomic-scale conduc-
tors [265] and this formalism has been used to model electromigration and
current-induced fracture of atomic wires [266, 267].

9.8 Exercises

9.1 Secular equation for a finite system: Use the Schrödinger equation to
show that the coefficients of the expansion of Eq. (9.3) satisfy the set of equations
given by Eq. (9.6).
9.2 Bloch’s theorem: Using the translational invariance in a Bravais lattice,
show that matrix elements of the Hamiltonian with basis functions Φkiα and
Φk′ jβ are non-zero only for k = k′ , and derive the expression of Eq. (9.8).
9.3 Energy spectrum of benzene: Solve analytically the secular equation
(9.15) and show that within the Hückel approximation the energy levels and the
corresponding molecular orbitals are given by Eq. (9.16) and Eq. (9.17), respec-
tively.
9.4 Molecular orbital structure of butadiene: The 1,3-butadiene molecule
(C4 H6 ) shown in Fig. 9.8 is a simple conjugated diene, i.e. it is a hydrocarbon
which contains two double bonds. Determine its energy levels and molecular
orbitals using the Hückel approximation.
9.5 Energy bands of s-bands in line, square and cubic Bravais lattices:
Show that for an s-band in a line, square lattice, and simple cubic lattice with
only nearest neighbor Hamiltonian matrix elements, the energy bands are given
by Eq. (9.18).
9.6 Density of states of s-bands in line, square and cubic Bravais lat-

concerning the coherent transport through molecular junctions.

10 The physics of these metallic nanowires is described in Part 3, where in particular we

shall make extensive use of the tight-binding approach.

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Electronic structure I: Tight-binding approach 261

Fig. 9.8 Composition and structure of the 1,3-butadiene molecule.

tices: Reproduce the results of Fig. 9.5 for the density of states of s-band in a line,
square lattice, and simple cubic lattice with only nearest neighbor Hamiltonian
matrix elements.
9.7 Energy bands of s-bands in line, square and cubic Bravais lattices
in a non-orthogonal model: Show that the expression for bands with non-
orthogonal basis orbitals, Eq. (9.19), is correct. Why are the bands in this case
no longer symmetric about ǫ = 0?
9.8 Electronic structure of graphene: Consider the model for graphene de-
tailed in section 9.5.3. Carry out the following tasks: (i) Determine the Brillouin
zone of the honeycomb lattice, (ii) show that the energy bands are given by
Eq. (9.22), (iii) demonstrate that the dispersion relation around the Dirac’s point
is linear, and (iv) compute the local density of states and show that it is given
by the result of Fig. 9.6(b).
9.9 The NRL tight-binding method: An interesting project for graduate
students and advanced undergraduate students is to write a computer code (in
whatever language) to calculate the energy bands and bulk density of states [see
Fig. 9.7] of elementary solids within the NRL tight-binding method described in
section 9.6.
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262 Molecular Electronics: An Introduction to Theory and Experiment

January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Chapter 10

Electronic structure II: Density

functional theory

This second chapter about electronic structure calculations provides a ba-

sic introduction to the density functional theory (DFT). This theory is
presently the most successful (and also the most promising) approach for
computing the electronic structure of matter. Its applicability ranges from
atoms, molecules and solids to nuclei and quantum and classical fluids.
Thus for instance, in chemistry DFT is widely used to predict a great vari-
ety of molecular properties: molecular structures, vibrational frequencies,
atomization energies, ionization energies, electric and magnetic properties,
reaction paths, etc. Originally, DFT was designed to provide the electron
density and total energy of the ground state of (non-magnetic) electronic
systems. However, meanwhile the theory has been generalized to deal with
many different situations: spin polarized systems, multicomponent systems
such as nuclei and electron hole droplets, free energy at finite temperatures,
superconductors with electronic pairing mechanisms, relativistic electrons,
time-dependent phenomena and excited states, bosons, molecular dynam-
ics, etc.
More importantly for the scope of this book, DFT is at the moment
the theoretical approach with the largest impact in molecular electronics.
In this sense, we believe that DFT should be part of the general culture of
the researchers working in this field. For this reason, we have included here
a concise introduction to the standard formulation of DFT, which can be
read almost independently of the rest of the book. Our goals are: (i) To
explain what kind of information this theory can provide, (ii) to describe
how it is used in molecular electronics, and (iii) to discuss its advantages and
limitations. For those readers who want to get deeper into the subtleties and
performance of this theory, the following entry points into the literature are
recommended. First of all, one has of course the original papers [268, 269]

263
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264 Molecular Electronics: An Introduction to Theory and Experiment

and Kohn’s Nobel lecture [270]. Among the DFT reviews, we recommend
the one of Ref. [271]. Finally, let us say that this chapter is based on the
monographs of Refs. [272, 226] and specially on that of Ref. [273]. Those
readers familiar with DFT who only want to know how it is applied in
molecular electronic are advised to jump directly to the last section of this
chapter.

10.1 Elementary quantum mechanics

In order to pave the way for the understanding of the basic formulation of
density function theory, we start reminding some basic issues in quantum
mechanics.1

10.1.1 The Schrödinger equation

The ultimate goal of most theoretical approaches in solid state physics and
quantum chemistry is the solution of the time-independent, non-relativistic
Schrödinger equation2
~ 1 , ..., R
HΨi (~x1 , ..., ~xN , R ~ M ) = Ei Ψi (~x1 , ..., ~xN , R
~ 1 , ..., R
~ M ), (10.1)
where H is the Hamiltonian for a system consisting of M nuclei and N
electrons. In the absence of external fields, H has the following form:
N M N XM N N M M
1X 2 1X 1 X ZA X X 1 X X ZA ZB
H=− ∇i − ∇2A − + + .
2 i=1 2 MA i=1
riA i=1 j>i rij RAB
A=1 A=1 A=1 B>A
(10.2)
Here, A and B run over the M nuclei while i and j denote the N electrons in
the system. The first two terms describe the kinetic energy of the electrons
and nuclei. The other three terms represent the attractive electrostatic
interaction between the nuclei and the electrons and repulsive potential
due to the electron-electron and nucleus-nucleus interactions. Let us stress
that, following the common practice in most textbooks, we shall use atomic
units throughout this chapter.3
1 Throughout this chapter we shall be using the more standard first quantization for-

mulation of quantum mechanics.

2 In this chapter the operators will be written in boldface, while the vector character of

a variable will be indicated by an arrow on top of it.

3 In this system of units the masses are measured in units of the electron mass, the

charges in units of the electron charge, ~ is the unit of action, the energy is measured in
Hartrees (27.211 eV) and the length unit is the Bohr (0.52910 Å).
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Electronic structure II: Density functional theory 265

The Schrödinger equation [Eq. (10.1)] can be simplified using the Born-
Oppenheimer approximation. Due to their masses the nuclei move much
slower than the electrons, which implies that we can consider the electrons
as moving in the field of fixed nuclei, i.e. the nuclear kinetic energy is
zero and their potential energy is merely a constant. Thus, the electronic
Hamiltonian reduces to
N N M N N
1 X 2 X X ZA X X 1
Helec = − ∇i − + = T + VN e + Vee . (10.3)
2 i=1 i=1
riA i=1 j>i rij
A=1

The solution of the Schrödinger equation with Helec is the electronic

wave function Ψelec and the electronic energy Eelec . The total energy Etot
is then the sum of Eelec and the constant nuclear repulsion term Enuc , i.e.
Helec Ψelec = Eelec Ψelec , (10.4)
and
M
M X
X ZA ZB
Etot = Eelec + Enuc where Enuc = . (10.5)
RAB
A=1 B>A

In principle, our main problem now is to solve Eq. (10.4), which is simply
impossible to accomplish in general.4

10.1.2 The variational principle for the ground state

A general strategy for the search of the lowest-energy solution of
Schrödinger equation [Eq (10.4)] is provided by the variational principle.
This idea goes as follows. When a system is in the state Ψ, the expectation
value of the energy is given by
hΨ|H|Ψi
Z
E[Ψ] = where hΨ|H|Ψi = Ψ∗ HΨ d~x. (10.6)
hΨ|Ψi
The variational principle states that the energy computed from a guessed
Ψ is an upper bound to the true ground-state energy E0 . Full minimization
of the functional E[Ψ] with respect to all allowed N -electrons wave functions
will give the true ground state Ψ0 and energy E[Ψ0 ] = E0 ; that is
E0 = min E[Ψ] = min hΨ|T + VN e + Vee |Ψi, (10.7)
Ψ→N Ψ→N

where Ψ → N indicates that Ψ is an allowed N -electron wave function.

4 Fromnow one we shall only consider the electronic problem and subscript “elec” will
be dropped.
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266 Molecular Electronics: An Introduction to Theory and Experiment

For a system of N electrons and given nuclear potential Vext , the vari-
ational principle defines a procedure to determine the ground-state wave
function Ψ0 , the ground-state energy E0 [N, Vext ], and other properties of
interest. In other words, the ground state energy is a functional of the
number of electrons N and the nuclear potential Vext
E0 = E[N, Vext ]. (10.8)

10.1.3 The Hartree-Fock approximation

Although the variational principle offers a strategy for finding the ground
state wave function, it is simply impossible to solve Eq. (10.4) by searching
through all acceptable many-body wave functions. We need to define a
suitable subset, which offers a reasonable approximation to the exact wave
function without being unmanageable in practice. The Hartree-Fock ap-
proximation provides the simplest, yet physically sound, solution to this
problem. Let us briefly explain the basic idea behind this approach.
Suppose that the ground state wave function, Ψ0 , is approximated as
an antisymmetrized product of N orthonormal spin orbitals ψi (~x), each
a product of a spatial orbital φk (~r) and a spin function σ(s), the Slater
determinant
¯ ¯
¯ ψ1 (~x1 ) ψ2 (~x1 ) ... ψN (~x1 ) ¯
¯ ¯
1 ¯¯ ψ1 (~x2 ) ψ2 (~x2 ) ... ψN (~x2 ) ¯¯
Ψ0 ≈ ΨHF = √ ¯ .. .. .. ¯. (10.9)
N ! ¯¯ . . . ¯
¯
¯ ψ (~x ) ψ (~x ) ... ψ (~x ) ¯
1 N 2 N N N

The Hartree-Fock approximation is the method whereby the orthogonal

orbitals ψi are found that minimize the energy for this determinantal form
of Ψ0
EHF = min E [ΨHF ] . (10.10)
(ΨHF →N )

The expectation value of the Hamiltonian operator with ΨHF is given

by (Exercise 10.1)
N N
X 1 X
EHF = hΨHF |H|ΨHF i = Hi + (Jij − Kij ) , (10.11)
i=1
2 i,j=1

where
· ¸
1
Z
Hi ≡ ψi∗ (~x) − ∇2 + Vext (~x) ψi (~x) d~x (10.12)
2
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Electronic structure II: Density functional theory 267

defines the contribution due to the kinetic energy and the electron-nucleus
attraction and
1 ∗
Z Z
Jij = ψi (~x1 )ψi∗ (~x1 ) ψ (~x2 )ψj (~x2 )d~x1 d~x2 , (10.13)
r12 j
1
Z Z
Kij = ψi∗ (~x1 )ψj (~x1 ) ψi (~x2 )ψj∗ (~x2 )d~x1 d~x2 . (10.14)
r12
The integrals are all real, and Jij ≥ Kij ≥ 0. The Jij are called Coulomb
integrals, the Kij are called exchange integrals. We have the property Jii =
Kii .
The variational freedom in the expression of the energy [Eq. (10.11)] is in
the choice of the orbitals. TheR minimization of the energy functional with
the normalization conditions ψi∗ (~x)ψj (~x)d~x = δij leads to the Hartree-
Fock differential equations (see Exercise 10.2)
f ψi = ǫi ψi , i = 1, 2, ..., N. (10.15)
These N equations have the appearance of eigenvalue equations, where
ǫi are the eigenvalues of the operator f . The Fock operator f is an effective
one-electron operator defined as
M
1 X ZA
f = − ∇2i − + VHF (i). (10.16)
2 riA
A
The first two terms are the kinetic energy and the potential energy due
to the electron-nucleus attraction. VHF (i) is the Hartree-Fock potential,
the average repulsive potential experienced by the i-th electron due to the
remaining N -1 electrons, and it is given by
N
X
VHF (~x1 ) = (Jj (~x1 ) − Kj (~x1 )) , (10.17)
j
1
Z
Jj (~x1 ) = |ψj (~x2 )|2 d~x2 . (10.18)
r12
The Coulomb operator J represents the potential that an electron at
position ~x1 experiences due to the average charge distribution of another
electron in spin orbital ψj .
The second term in Eq. (10.17) is the exchange contribution to the HF
potential. It has no classical analog and it describes the modification of the
energy that can be ascribed to the effects of spin correlation. It is defined
through its effect when operating on a spin orbital
1
Z
Kj (~x1 ) ψi (~x1 ) = ψj∗ (~x2 ) ψi (~x2 ) d~x2 ψj (~x1 ). (10.19)
r12
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268 Molecular Electronics: An Introduction to Theory and Experiment

Two important remarks to conclude this section. First, the HF potential

is non-local and it depends on the spin orbitals. Thus, the HF equations
must be solved self-consistently. Second, the Koopman’s theorem [274]
provides a physical interpretation of the orbital energies: it states that
the orbital energy ǫi is an approximation of minus the ionization energy
(IE) associated with the removal of an electron from the orbital ψi , i.e.
i
ǫi ≈ EN − EN −1 = −IE(i).

10.2 Early density functional theories

In this section we shall introduce the electron density, which is the funda-
mental quantity in DFT, and we shall briefly review some early attempts
to develop a density functional theory.
The electron density is defined as the integral over the spin coordinates
of all electrons and over all but one of the spatial variables (~x ≡ ~r, s)
Z Z
ρ(~r) = N ... |Ψ(~x1 , ~x2 , ..., ~xN )|2 ds1 d~x2 ...d~xN . (10.20)

The electron density ρ(~r) determines the probability of finding any of the N
electrons within volume element d~r. Clearly, ρ(~r) is a non-negative function
of only the three spatial variables which vanishes at infinity and integrates
to the total number of electrons, i.e.
Z
ρ(~r → ∞) = 0; ρ(~r)d~r = N. (10.21)

Moreover, unlike the wave function, the electron density is an observable

and it can be measured experimentally, e.g. by X-ray diffraction.
At this stage, one may wonder whether the central role of the com-
plicated N-electron wave function, which depends on 3N spatial variables,
could be played by a simpler function such as the electron density. As early
as in the 1920’s, several authors conjectured that indeed the total energy
could be a functional of the electronic density alone. Probably, the most
famous example of such an early density functional theory is the so-called
Thomas-Fermi model, which was put forward in 1927. Based on the uni-
form electron gas, Thomas and Fermi proposed independently the following
functional for the kinetic energy
3
Z
TT F [ρ(~r)] = (3π 2 )2/3 ρ5/3 (~r)d~r. (10.22)
10
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Electronic structure II: Density functional theory 269

This functional was combined with the classical expression for the electron-
nuclei potential and the electron-electron potential to write down the fol-
lowing functional for the energy of an atom
3
Z
2 2/3
ET F [ρ(~r)] = (3π ) ρ5/3 (~r)d~r
10
ρ(~r) 1 ρ(~r1 )ρ(~r2 )
Z Z Z
−Z d~r + d~r1 d~r2 . (10.23)
r 2 r12
Notice that the energy is given completely in terms of the electron density.
In order to determine the correct density to be included in Eq. (10.23),
they employed a variational principle. They assumed that the ground state
of the system is connected
R to the ρ(~r) for which the energy is minimized
under the constraint of ρ(~r)d~r = N . The obvious question at this stage is:
does this variational principle make sense? The Hohenberg-Kohn theorems
discussed in the next section will prove that this approach can be rigorously
justified.

10.3 The Hohenberg-Kohn theorems

Density functional theory as we know it today is founded in the so-called

Hohenberg-Kohn theorems that were put forward in 1964 [268]. In this
section we present these theorems and discuss some of their basic implica-
tions. The proofs of these theorems will not be detailed here and they can
be found in any of the references given at the beginning of this chapter.
The first Hohenberg-Kohn theorem states that the electron density
uniquely determines the Hamiltonian operator and thus all the properties
of the system. To be precise, this theorem states that the external potential
Vext (~r) is (to within a constant) a unique functional of ρ(~r); since, in turn
Vext (~r) fixes H we see that the full many particle ground state is a unique
functional of ρ(~r).
Thus, ρ(~r) determines N and Vext (~r) and hence all the properties of
the ground state, for example the kinetic energy T [ρ], the potential energy
V [ρ], and the total energy E[ρ]. Now, we can write the total energy as
Z
E[ρ] = EN e [ρ] + T [ρ] + Eee [ρ] = ρ(~r)VN e (~r)d~r + FHK [ρ], (10.24)

FHK [ρ] = T [ρ] + Eee [ρ]. (10.25)

Here, we have separated the contributions that depend on the actual
system, i.e. the potential energy due to the electron-nuclei attraction,
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270 Molecular Electronics: An Introduction to Theory and Experiment

R
EN e [ρ] = ρ(~r)VN e (~r)d~r, from those which are universal, FHK [ρ]. This
functional FHK [ρ] is the holy grail of density functional theory. If it was
known, we would be able to solve the Schrödinger equation exactly and for
any system. This functional contains the functional for the kinetic energy
T [ρ] and that for the electron-electron interaction, Eee [ρ]. The explicit
form of both these functionals is unknown. However, from the latter we
can extract at least the classical part J[ρ],
1 ρ(~r1 )ρ(~r2 )
Z Z
Eee [ρ] = d~r1 d~r2 + Encl = J[ρ] + Encl [ρ]. (10.26)
2 r12
Encl is the non-classical contribution to the electron-electron interaction:
self-interaction correction, exchange and Coulomb correlation. The explicit
form of the functionals T [ρ] and Encl [ρ] is the major challenge of DFT.
Let us now address the following question: how can we be sure that a
certain density is the ground-state density that we are looking for? The
second Hohenberg-Kohn theorem answers this question. This theorem
states that FHK [ρ], the functional that delivers the ground state energy of
the system, delivers the lowest energy if and only if the input density is the
true ground state density. This is nothing but the variational principle
E0 ≤ E[ρ̃] = T [ρ̃] + EN e [ρ̃] + Eee [ρ̃]. (10.27)
In other words, this means that for any trial density ρ̃(~
R r), which satisfies
the necessary boundary conditions such as ρ̃(~r) ≥ 0, ρ̃(~r)d~r = N , and
which is associated with some external potential Ṽext , the energy obtained
from the functional of Eq. (10.24) represents an upper bound to the true
ground state energy E0 . E0 results if and only if the exact ground state
density is inserted in Eq. (10.27).
Let us summarize what we have learned so far and some basic conse-
quences of the previous theorems:

• All the properties of a system defined by an external potential Vext are

determined by the ground state density. In particular, the ground state
energy associated with a density ρ is available through the functional
Z
ρ(~r)Vext d~r + FHK [ρ]. (10.28)
• This functional attains its minimum value with respect to all allowed
densities if and only if the input density is the true ground state density,
i.e. for ρ̃(~r) ≡ ρ(~r).
• The applicability of the variational principle is limited to the ground
state. Hence, we cannot easily transfer this strategy to the problem of
excited states.
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Electronic structure II: Density functional theory 271

• The explicit form of the functional FHK [ρ] is unknown and this remains
as the major challenge of DFT.

10.4 The Kohn-Sham approach

We have seen that the ground state energy of a system can be written as
µ Z ¶
E0 = min FHK [ρ] + ρ(~r)VN e d~r , (10.29)
ρ→N

where the universal functional FHK [ρ] contains the contributions of the ki-
netic energy, the classical Coulomb interaction and the non-classical portion
FHK [ρ] = T [ρ] + J[ρ] + Encl [ρ]. (10.30)
Of these, only J[ρ] is known. The main problem is to find the expressions
for T [ρ] and Encl [ρ]. The Thomas-Fermi model of section 10.2 provides an
example of density functional theory. However, its performance is really
bad due to the poor approximation of the kinetic energy. To solve this
problem Kohn and Sham proposed in 1965 [269] the following approach.
They suggested to calculate the exact kinetic energy of a non-interacting
reference system with the same density as the real, interacting one
N N X
1X X
TS = − hψi |∇2 |ψi i, ρS (~r) = |ψi (~r, s)|2 = ρ(~r), (10.31)
2 i i s

where the ψi are the orbitals of the non-interacting system. Of course,

TS is not equal to the true kinetic energy of the system. Kohn and Sham
accounted for that by introducing the following separation of the functional
FHK [ρ]
FHK [ρ] = TS [ρ] + J[ρ] + EXC [ρ], (10.32)
where EXC , the so-called exchange-correlation energy is defined through
Eq. (10.32) as
EXC [ρ] ≡ (T [ρ] − TS [ρ]) + (Eee [ρ] − J[ρ]) . (10.33)
The exchange and correlation energy EXC is the functional that contains
everything that is unknown.
Now the question is: How can we uniquely determine the orbitals in
our non-interacting reference system? In other words, how can we define
a potential VS such that it provides us with a Slater determinant which
is characterized by the same density as our real system? To solve this
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272 Molecular Electronics: An Introduction to Theory and Experiment

problem, we write down the expression for the energy of the interacting
system in terms of the separation described in Eq. (10.32)
E[ρ] = TS [ρ] + J[ρ] + EXC [ρ] + EN e [ρ], (10.34)
where
1 ρ(~r1 )ρ(~r2 )
Z Z Z
E[ρ] = TS [ρ] + d~r1 d~r2 + EXC [ρ] + VN e ρ(~r)d~r
2 r12
N N N Z Z
1X 1 XX 1
=− hψi |∇2 |ψi i + |ψi (~r1 )|2 |ψj (~r2 )|2 d~r1 d~r2
2 i 2 i j r12
N Z X
M
X ZA
+EXC [ρ] − |ψi (~r1 )|2 d~r1 . (10.35)
i
r1A
A
The only term for which no explicit form can be given is EXC . We now
apply the variational principle and ask: What condition must the orbitals
{ψi } fulfill in order to minimize this energy expression under the usual
constraint hψi |ψj i = δij ? The resulting equations are the Kohn-Sham
equations:
µ ¶
1
− ∇2 + Vef f (~r1 ) ψi = ǫi ψi , (10.36)
2
where the effective potential Vef f (~r1 ) is given by
M
ρ(~r2 ) ZA
Z X
Vef f (~r1 ) = d~r2 + VXC (~r1 ) − . (10.37)
r12 r1A
A

Thus, once we know the various contributions in Eq. (10.37), we can

insert the potential Vef f into the one-particle equations, which in turn de-
termine the orbitals and hence the ground state density and the ground
state energy employing Eq. (10.35). Notice that Vef f depends on the den-
sity, and therefore the Kohn-Sham equations have to be solved iteratively.
One term in the above equations needs some additional comments. The
exchange-correlation potential, VXC is defined as the functional derivative
of EXC with respect to ρ, i.e. VXC = δEXC /δρ. It is very important to
realize that if the exact forms of EXC and VXC were known, the Kohn-Sham
strategy would lead to the exact energy.
A question of special relevance for the use of DFT in molecular elec-
tronics is: Do the Kohn-Sham orbitals and eigenvalues mean anything? It
is often said that the Kohn-Sham orbitals and eigenvalues have no physical
meaning. In particular, the eigenvalues are not the energies to add or sub-
tract electrons from the interacting many-body system. There is only one
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Electronic structure II: Density functional theory 273

exception [275]: The highest eigenvalue in a finite system, which is minus

the ionization energy.5 Anyway, several authors have lately pointed to the
interpretative power of the Kohn-Sham orbitals in traditional qualitative
molecular orbital schemes (see section 5.3.3 in Ref. [273]) and in solid state
physics it is customary to use these orbitals as an approximation for the
true spectrum of an electronic system, see Ref. [271]. After all, these or-
bitals are not only associated with a one-electron potential which includes
all non-classical effects, they are also consistent with the exact ground state
density.
Let us close this section by saying that the Kohn-Sham (KS) approach
provides a practical strategy to find both the electron density and the total
energy of the ground state of any electronic system. However, there are
still two important issues that we have to address. First, we need to find
reasonable approximations for the exchange-correlation functional and sec-
ond, we have to discuss how to solve in practice the Kohn-Sham equations.
These two issues are the subject of the next sections.

10.5 The exchange-correlation functionals

The genius of the Kohn-Sham approach described in the previous section

is two-fold. First, the Ansatz leads to tractable single-particle equations
that hold the hope of solving interacting many-body problems. Second, by
explicitly separating the independent-particle kinetic energy and the long-
range Hartree terms, the remaining exchange-correlation functional EXC [ρ]
can be reasonable approximated as a local or nearly local functional of the
density. Even though the exact functional EXC [ρ] must be very complex,
great progress has been made with remarkably simple approximations. This
sections is devoted to a brief description of some of those approximations.
For more details about this topic, see for instance Chapter 8 of Ref. [226]
and Chapter 6 of Ref. [273].

10.5.1 LDA approximation

The local density approximation (LDA) is the basis of all approximate
exchange-correlation functionals. At the center of this model is the idea
of a uniform electron gas. This is a system in which electrons move on
5 The asymptotic long-range density of a bound system is governed by the occupied

state with the highest eigenvalue; since the density is assumed to be exact, so must the
eigenvalue be exact. No other eigenvalue is guaranteed to be correct.
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274 Molecular Electronics: An Introduction to Theory and Experiment

a positive background charge distribution such that the total ensemble is

neutral.
The central idea of LDA is the assumption that we can write EXC in
the following form
Z
LDA
EXC [ρ] = ρ(~r)ǫXC (ρ(~r)) d~r. (10.38)

Here, ǫXC (ρ(~r)) is the exchange-correlation energy per particle of a uniform

electron gas of density ρ(~r). This energy per particle is weighted with the
probability ρ(~r) that there is an electron at this position. The quantity
ǫXC (ρ(~r)) can be further split into exchange and correlation contributions,
ǫXC (ρ(~r)) = ǫX (ρ(~r)) + ǫC (ρ(~r)). (10.39)
The exchange part, ǫX , which represents the exchange energy of an
electron in a uniform electron gas of a particular density, was originally
derived by Bloch and Dirac in the late 1920’s and it is given by
µ ¶1/3
3 3ρ(~r)
ǫX = − . (10.40)
4 π
No such explicit expression is known for the correlation part, ǫC . How-
ever, highly accurate numerical quantum Monte-Carlo simulations of the
homogeneous electron gas are available from the work of Ceperly and Alder
[276]. On the basis of these results various authors have presented analyti-
cal expressions of ǫC based on sophisticated interpolation schemes.
Up to this point the local density approximation was introduced as a
functional depending solely on ρ(~r). If we extend the LDA to an unre-
stricted case, i.e. to a case without spin symmetry, we arrive at the local
spin-density approximation, or LSDA, where the two spin densities, ρ↑ (~r)
and ρ↓ (~r), with ρ(~r) = ρ↑ (~r) + ρ↓ (~r), are employed as the central input. In
this approximation, instead of Eq. (10.38) one now writes
Z
LSDA
EXC [ρ↑ , ρ↓ ] = ρ(~r)ǫXC (ρ↑ (~r), ρ↓ (~r)) d~r. (10.41)

As for the simple spin-symmetric situation, there are related expressions

for the exchange and correlation energies per particle of the uniform electron
gas characterized by ρ↑ (~r) 6= ρ↓ (~r), the so-called spin polarized case. In the
following we do not differentiate between the local and the local spin-density
approximation and use the abbreviation LDA for both.
We conclude this subsection with some brief comments about the per-
formance of LDA, especially in the context of molecular physics:
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Electronic structure II: Density functional theory 275

• The accuracy of the LDA for the exchange energy is typically within
10%, while the normally much smaller correlation energy is generally
overestimated by up to a factor 2. The two errors typically cancel
partially.
• Experience has shown that the LDA gives ionization energies of atoms,
dissociation energies of molecules and cohesive energies with a fair ac-
curacy of typically 10-20%. However, the LDA gives bond lengths of
molecules and solids typically with an astonishing accuracy of ∼ 2%.
• The moderate accuracy that LDA delivers is insufficient for most appli-
cations in chemistry. For this reason, for many years, where LDA was
the only approximation for the exchange-correlation functional, DFT
was mostly used by solid-state physicists and it hardly had any impact
in quantum chemistry.

10.5.2 The generalized gradient approximation

The first step beyond the local approximation is a functional of the mag-
nitude of the gradient of the density ∇ρ(~r) as well as the value of ρ(~r) at
each point. Such a gradient expansion approximation (GEA) was already
suggested in the original paper of Kohn and Sham. The low-order expan-
sion of the exchange and correlation energies is known. However, the GEA
does not lead to consistent improvement over the LDA. It violates exact
sum rules and other relevant conditions and, indeed, often leads to worse
results. The basic problem is that gradients in real systems can be so large
that the expansion breaks down.
The term generalized gradient approximation (GGA) denotes a variety
of ways proposed for functionals that modify the behavior at large gradients
in such way as to preserve the desired properties. These functionals are the
workhorses of current DFT and they can be generically written as
Z
GGA
EXC [ρ↑ , ρ↓ ] = f (ρ↑ , ρ↓ , ∇ρ↑ , ∇ρ↓ ) d~r. (10.42)

GGA
In practice, EXC is usually split into its exchange and correlation con-
GGA GGA GGA
tributions, EXC = EX + EC , and approximations for the two terms
GGA
are sought separately. With respect to the exchange part EX , it can be
written as
X Z
GGA LDA
EX = EX − F (sσ )ρ4/3
σ (~r) d~r. (10.43)
σ=↑,↓
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276 Molecular Electronics: An Introduction to Theory and Experiment

The argument of the function F is the reduced density gradient for spin σ

|∇ρσ (~r)|
sσ = 4/3
. (10.44)
ρσ

Numerous forms for the function F above have been given. We just
mention here three of the most widely used ones that were proposed by
Becke in 1986 (B86) [277], Perdew also in 1986 (P) [278], and Perdew, Burke
and Ernzerhof in 1996 (PBE) [279]. In all these cases, F is a complicated
rational function of the reduced density gradient that we shall not write
here explicitly.
The corresponding gradient-corrected correlation functionals have even
more complicated analytical forms and cannot be understood by simple
physically motivated reasoning. Among the most widely used choices is
the correlation counterpart of the 1986 Perdew exchange functional [278],
usually referred to as P or P86. This functional employs an empirical
parameter, which was fitted to the correlation energy of the neon atom. A
few years later Perdew and Wang [280] refined their correlation functional,
leading to the parameter free PW91. Another, nowadays even more popular
correlation functional is due to Lee, Yang, and Parr (LYP) [281]. This
functional was derived from an expression for the correlation energy of
the helium atom. The LYP functional contains one empirical parameter
and it differs from other GGA functionals in that it contains some local
components.
In principle, each exchange functional could be combined with any of the
correlation functionals, but only a few combinations are currently in use.
The exchange part is usually chosen to be Becke’s functional which is either
combined with Perdew’s 1986 correlation functional or the LYP one. These
combinations are termed BP86 and BLYP, respectively. Sometimes also
the PW91 correlation functional is employed, corresponding to BPW91. It
is worth stressing that these combinations lead to results that are of very
similar quality.
As a general statement about the performance of GGA-based function-
als, let us say that they have reduced the LDA errors of, in particular,
atomization energies of standard set of small molecules by a factor 3-5.
This improved accuracy has made DFT one of the most widely used tools
in quantum chemistry.
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Electronic structure II: Density functional theory 277

10.5.3 Hybrid functionals

Usually the exchange contributions are significantly larger than the cor-
responding correlation effects. Therefore, an accurate expression for the
exchange functional is a prerequisite for obtaining meaningful results from
density functional theory. In this sense, it is important to remind that the
exchange energy of a Slater determinant can be computed exactly (see dis-
cussion of the Hartree-Fock (HF) approximation in section 10.1.3). This
fact has motivated the construction of functionals called hybrid because
they are a combination of orbital-dependent Hartree-Fock and an explicit
density functional. These are the most accurate functionals available as far
as the energetics is concerned and are the method of choice in the quantum
chemistry community.
The hybrid functionals differ in the way in which the exchange HF en-
ergy is mixed with the exchange-correlation energy of a density functional.
Becke [282] has argued that the total exchange-correlation energy can be
approximated by
1 ¡ HF DFA
¢
EXC = EX + EXC , (10.45)
2
where DFA denotes an LDA or GGA functional. Later Becke presented
parameterized forms that are accurate for many molecules, such as “B3P91”
[282, 283], a three-parameter functional that mixes Hartree-Fock exchange,
the exchange functional of Becke (B88), and correlation from Perdew and
Wang (PW91).
Currently the most popular hybrid functional is the so-called B3LYP
[284] that uses the LYP correlation functional. In this case the definition
of the exchange-correlation energy is
LDA
¡ HF DFA
¢ Becke
EXC = EXC + a0 EX − EX + ax EX + ac EC , (10.46)
where the three coefficients ai are empirically adjusted to fit atomic and
molecular data.

10.6 The basic machinery of DFT

In this section we shall address how the Kohn-Sham single-particle equa-

tions are solved in practice. There are three main types of methods that
are applied to this problem with their own advantages and disadvantages.
The first one are the plane wave and grid methods that provide general ap-
proaches for the solution of differential equations, including the Schrödinger
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278 Molecular Electronics: An Introduction to Theory and Experiment

and Poisson equations. A second family is formed by the atomic sphere

methods that are the most general methods for precise solution of the
Kohn-Sham equations. The basic idea is to divide the electronic structure
problem, providing efficient representation of atomic-like features that are
rapidly varying near each nucleus and smoothly varying functions between
atoms. Finally, the third type of methods is based on localized atomic-
(like) orbitals (LCAO) that provide a basis that captures the essence of
the atomic-like features of solids and molecules. They provide a satisfying,
localized description of electronic structure widely used in chemistry. Since
this latter method is, in principle, better adapted to the type of systems
studied in molecular electronics, we shall devote the rest of this section to
describe how it is actually used to solve the Kohn-Sham equations. The
other two types of methods are extensively discussed in Ref. [226].

10.6.1 The LCAO Ansatz in the Kohn-Sham equations

Recall the central ingredient of the Kohn-Sham (KS) approach to density
functional theory, i.e. the one-electron KS equations,
  
N Z 2 M
1
− ∇2 + 
X |ψ (~
j 2r )| X Z A 
d~r2 + VXC (~r1 ) − ψi = ǫi ψi .
2 j
r12 r1A
A
(10.47)
The term in square brackets defines the Kohn-Sham one-electron oper-
ator and Eq. (10.47) can be written more compactly as
f KS ψi = ǫi ψi . (10.48)
Most of the applications in chemistry of the Kohn-Sham density func-
tional theory make use of the LCAO expansion of the Kohn-Sham orbitals.
Indeed, the way to proceed is almost identical to the case of the tight-
binding approach that we discussed in the previous chapter. Let us assume
that we are dealing with a finite system and introduce a set of L predefined
basis functions {ηµ } and linearly expand the K-S orbitals as
L
X
ψi = cµi ηµ . (10.49)
µ=1

We now insert Eq. (10.49) into Eq. (10.48) and obtain

L
X L
X
f KS (~r1 ) cνi ην (~r1 ) = ǫi cνi ην (~r1 ). (10.50)
ν=1 ν=1
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Electronic structure II: Density functional theory 279

If we now multiply this equation from the left with an arbitrary basis
function ηµ and integrate over space we get L equations6
XL Z L
X Z
cνi ηµ (~r1 )f KS (~r1 )ην (~r1 )d~r1 = ǫi cνi ηµ (~r1 )ην (~r1 )d~r1 , (10.51)
ν=1 ν=1
where i runs from 0 to L.
The integrals on the left hand side of this equation define the Kohn-
Sham matrix, FKS , with the corresponding elements defined as
Z
KS
Fµν = ηµ (~r1 )f KS (~r1 )ην (~r1 )d~r1 , (10.52)

and on the right hand side we can identify the overlap matrix, S, the ele-
ments of which are given by
Z
Sµν = ηµ (~r1 )ην (~r1 )d~r1 . (10.53)

Both matrices are L × L dimensional. The previous equation can be rewrit-

ten compactly as a matrix equation
FKS C = SCǫ. (10.54)
Hence, through the LCAO expansion we have translated the non-linear
optimization problem into a linear one, which can be expressed in the lan-
guage of standard algebra.
By expanding f KS into its components, the individual elements of the
KS matrix become
M
à !
1 2 X ZA ρ(~r2 )
Z Z
KS
Fµν = ηµ (~r1 ) − ∇ − + d~r2 + VXC (~r1 ) ην (~r1 )d~r1 .
2 r1A r12
A
(10.55)
The first two terms describe the kinetic energy and the electron-nuclear
interaction, and they are usually combined since they are one-electron in-
tegrals7
M
à !
1 2 X ZA
Z
hµν = ηµ (~r1 ) − ∇ − ην (~r1 )d~r1 . (10.56)
2 r1A
A
For the third term we need the charge density ρ which takes the following
form in the LCAO scheme
XL X L X
N X L
ρ(~r) = |ψi (~r)|2 = cµi cνi ηµ (~r)ην (~r). (10.57)
i i µ ν
6 Weassume without loss of generality that the basis functions are real.
7 Theseare the hooping matrix elements introduced in the frame of the tight-binding
approach.
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280 Molecular Electronics: An Introduction to Theory and Experiment

The expansion coefficients are usually collected in the so-called density

matrix P with elements
XN
Pµν = cµi cνi . (10.58)
i

Thus, the Coulomb contribution in Eq. (10.55) can be expressed as

L XL
1
X Z Z
Jµν = Pλσ ηµ (~r1 )ην (~r1 ) ηλ (~r2 )ησ (~r2 )d~r1 d~r2 . (10.59)
σ
r12
λ

Up to this point, exactly the same formulas also apply in the Hartree-
Fock case. The difference is only in the exchange-correlation part. In the
Kohn-Sham scheme this is represented by the integral
Z
XC
Vµν = ηµ (~r1 )VXC (~r1 )ην (~r1 )d~r1 , (10.60)

whereas the Hartree-Fock exchange integral is given by

L XL
1
X Z Z
Kµν = Pλσ ηµ (~x1 )ηλ (~x1 ) ην (~x2 )ησ (~x2 )d~x1 d~x2 . (10.61)
σ
r12
λ

The L2 /2 one-electron integrals contained in hµν can be easily com-

puted. The computational bottle-neck is the calculation of the ∼ L4 two-
electron integrals in the Coulomb and exchange-correlation terms. For
a discussion about efficient ways of computing these latter integrals, see
Ref. [273].

10.6.2 Basis sets

In order to complete our discussion of the LCAO approach in DFT, we shall
now describe the main types of localized basis functions that are used. A
first type of orbitals are the Gaussian-type-orbitals (GTOs), which
have been inherited from wave-functions-based methods like Hartree-Fock.
The GTO basis functions have the following general form
η GTO = N xl y m z n exp −αr2 .
£ ¤
(10.62)
Here, N is a normalization factor which ensures that hηµ |ηµ i = 1, but
note that the ηµ are not orthogonal. The orbital exponent α determines
how compact or how diffusive the resulting function is. L = l + m + n
is used to classify the GTO as s-functions (L = 0), p-functions (L = 1),
etc. The advantage of this type of basis functions lies in the existence of
very efficient algorithms for calculating analytically the huge number of
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Electronic structure II: Density functional theory 281

multi-center integrals appearing in the Coulomb and exchange-correlation

terms.
On the other hand, from a physical point of view, Slater-type-orbitals
(STO) seem to be the natural choice for basis functions. They are expo-
nential functions that mimic the exact eigenfunctions of the hydrogen atom.
A typical STO is expressed as
η STO = N rn−1 exp [−βr] Ylm (Θ, φ). (10.63)
Here, n corresponds to the principal quantum number, the orbital exponent
is termed β and Ylm are the usual spherical harmonics. Unfortunately,
many-center integrals are very difficult to compute with STO basis, and
they do not play a major role in the DFT community.
The so-called contracted Gaussian functions (CGF) try to combine
the advantages of the two previous type of orbitals. In this case, several
primitive Gaussian functions are combined in a fixed linear combination:
A
X
ητCGF = daτ ηaGTO . (10.64)
a

The original motivation for contracting was that the contraction coefficients
daτ can be chosen in a way that the CGF resembles as much as possible a
single STO function. In density functional theory, CGF basis sets enjoy a
strong popularity.
A fourth type of basis functions are the numerical basis functions.
In this case, the orbitals are represented numerically on atomic centered
grids. These functions can be generated, for instance, by numerically solv-
ing the atomic KS equations with a given approximation for the exchange-
correlation functional. Obviously, in this approach the different integrals
are computed numerically.
Irrespective of the type of functions used, the basis sets can be classified
in the following simple way that already gives a hint about their quality.
The simplest (and smallest) basis functions are those that use a single basis
function for each atomic orbital up to and including the valence orbitals.
These basis sets are called, for obvious reasons, minimal basis sets. A
typical representative is the STO-3G basis set, in which three primitive
GTO functions are combined into one CGF. For carbon, this basis set
consists of five functions, one describing the 1s atomic orbital, another one
for the 2s orbital and three more for the 2p shell. One should expect no
more than only qualitative results from minimal sets and nowadays they
are hardly used anymore.
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282 Molecular Electronics: An Introduction to Theory and Experiment

In the next level of sophistication are the double-zeta basis sets. Here,
the set of functions is doubled, i.e. there are two functions for each orbital.
If only the valence orbitals are doubled, and each core atomic orbitals is
still described by a single function, the resulting basis set is called split-
valence basis set. Typical examples are the 3-21G or 6-31G Gaussian basis
sets. In most applications, such basis sets are augmented by polarization
functions, i.e. functions of higher angular momentum than those occupied in
the atom. Polarized double-zeta or split valence basis sets are the mainstay
of routine quantum chemical applications since usually they offer a balance
compromise between accuracy and efficiency. Finally, it is obvious how
these schemes can be extended by increasing the number of functions in
the various categories. This results in triple- or quadruple-zeta basis sets
which are augmented by several sets of polarization functions.
If the molecules or solids of interest contain elements heavier than, say
krypton, one usually employs effective core potentials, also called pseudopo-
tentials, to model the core electrons. For a detailed discussion of the theory
of pseudopotentials see Ref. [226].

10.7 DFT performance

Our discussion about density functional theory would not be complete with-
out answering, at least partially, the most obvious question at this stage,
namely: how much should one trust DFT? In other words, what is the
accuracy of DFT at present, i.e. with the existent approximations for the
exchange-correlation functional? A detailed answer to this question is out
of the scope of this book and we just pretend to give here a flavor about
DFT’s performance in the case of the systems of interest in molecular elec-
tronics.
Let us remind again that the standard DFT, as presented here, gives
only results for the ground state energy and density of a system and related
properties. In the context of chemistry (or molecular physics), this means in
practice that one can expect from DFT information about the structure of
molecules, vibrational frequencies, atomization energies, dipole moments,
reactions paths and other similar properties. In what follows, we shall illus-
trate DFT’s performance with a very brief discussion of its predictions for
some basic molecular properties. We follow here Ref. [273], where an excel-
lent discussion of “goodness” of DFT in the context of quantum chemistry
can be found.
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Electronic structure II: Density functional theory 283

Table 10.1 Calculated and experimental bond lengths for different bonding
situations [Å]. The LDA calculations were done with the 6-31G(d) basis set
and the GGA ones with the 6-311++G(d,p) basis set.

Bond LDA BLYP BP86 BPW91 Experiment

H-H RH−H 0.765 0.748 0.752 0.749 0.741

H3 C-CH3 RC−C 1.510 1.542 1.535 1.533 1.526
RC−H 1.101 1.100 1.102 1.100 1.088
H2 C=CH2 RC−C 1.331 1.339 1.337 1.336 1.339
RC−H 1.098 1.092 1.094 1.092 1.085
HC≡CH RC−C 1.203 1.209 1.210 1.209 1.203
RC−H 1.073 1.068 1.072 1.070 1.061

Molecular structures: DFT calculations provide the electronic part

of the energy of a molecule. If this information is combined with the clas-
sical nuclear energy in Eq. (10.5), the total energy of the molecule can be
minimized with respect to the position of the nuclei to find the most stable
structure. This is one of the main applications of DFT, which gives the
bond lengths of a large set of molecules with a precision of 1-2%. Gradient-
corrected and hybrid functionals have improved the LDA results, which
for this property are already surprisingly good. The degree of accuracy
is illustrated in Table 10.1, where we show the calculated bond distances
for several basic covalently bound molecular structures with different func-
tionals and their comparison with experimental results (data taken from
Ref. [273]).
Vibrational frequencies: The frequencies of molecular vibrational
modes can be calculated by evaluation of second derivatives of the total
energy with respect to Cartesian coordinates. DFT predicts the vibrational
frequencies of a broad range of molecules within 5-10% accuracy.
Scott and Radom [285] have investigated the performance of a variety
of gradient-corrected and hybrid functionals for predicting vibrational fre-
quencies of a large set of 122 test molecules. By fitting computed data to a
basis of 1066 experimental vibrations, they obtained scaling factors relating
the computed frequencies to experimental values. Some of the results are
reproduced in Table 10.7.
Atomization energies: The most common way of testing the perfor-
mance of new functionals is the comparison with the experimental atom-
ization energies (the energies needed to break up a molecule into its con-
stituent atoms) of well-studied sets of small molecules. These comparisons
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284 Molecular Electronics: An Introduction to Theory and Experiment

Table 10.2 Vibrational frequencies of a set of 122 molecules: functional, fre-

quency scaling factor (f ), root mean square (RMS) error after scaling in cm−1
and percentage of frequencies that fall outside the experimental values by more
than 10%.

Functional f RMS 10%

BLYP 0.9945 45 10
BP86 0.9914 41 6
B3LYP 0.9614 34 6
B3P86 0.9558 38 4
B3PW91 0.9573 34 4

have established the following hierarchy of functionals:

LDA < GGA < hybrid functionals .

The hybrid functionals are progressively approaching the desired accu-

racy in the atomization energies, and in many cases they deliver results
comparable with highly sophisticated post-HF methods.
Ionization and affinity energies: The energies needed to remove
(IE) or to add an electron (EA) can be determined with hybrid functionals
with an average error of around 0.2 eV for a large variety of molecules.
The discussion above shows the impressive accuracy that DFT is achiev-
ing in many situations. However, it is worth stressing that DFT (with the
present approximations) is still failing in situations where the density is
not a slowly varying function. An important example of the failure of DFT
is the description of systems where the binding is dominated by van der
Waals interactions, which is something essential for supramolecular chem-
istry. Another example is the description of electronic tails evanescing into
the vacuum near the surfaces of bounded electronic systems, which is a key
problem in the context of the STM.

10.8 DFT in molecular electronics

In this final section we shall discuss how DFT is used in practice in the field
of molecular electronics. For this purpose, we shall first discuss how DFT
can be combined with the nonequilibrium Green’s function (NEGF) tech-
niques presented in previous chapters to describe the electronic transport
in atomic-scale junctions. Then, we shall end this section with some com-
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Electronic structure II: Density functional theory 285

Fig. 10.1 Schematic representation of a molecular junction. We distinguish regions:

the left (L) and right (R) semi-infinite electrodes and the central region or “extended
molecule” that contains the molecule and part of the leads.

ments about the advantages and limitations of such a combination. For a

more detailed discussion of the use of DFT to compute the transport prop-
erties of nanostructures, we recommend the excellent review of Pecchia and
Di Carlo [286].

10.8.1 Combining DFT with NEGF techniques

In section 10.6 we have shown how DFT is applied to the description of
the electronic structure of finite systems like molecules or atomic clusters,
in particular within the LCAO approach. For periodic systems like infinite
solids, one proceeds in a similar manner, but in this case the Kohn-Sham
equations are solved in reciprocal space. In both types of systems the
dimension of the problem, i.e. the number of Kohn-Sham equations, is
finite. In the first case this dimension is mainly determined by the number
of atoms in the system, whereas in the latter it is governed by the size of
the unit cell. In molecular electronics we are interested in the description of
the electronic structure and transport properties of atomic-scale junctions,
like the one depicted in Fig. 10.1. These junctions are neither finite nor
periodic, which makes more complicated the application of DFT. Moreover,
we are also interested in situations in which these systems are driven out
of equilibrium, for instance by the application of a bias voltage. Such
situations are out of the scope of the standard ground state DFT. The
goal of this subsection is to show how DFT can be combined with the
Green’s function techniques of Chapters 5-8 to describe the equilibrium
and transport properties of nanoscale junctions.
When applying DFT to systems like the one in Fig. 10.1, one is con-
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286 Molecular Electronics: An Introduction to Theory and Experiment

fronted with the following two questions: (i) how to compute the charge
density? and (ii) how to make finite the dimension of the problem? Both
questions can be answered with the help of Green’s function methods as
follows. First, we divide the junction into three parts: the left (L) and right
(R) electrodes and a central part or “extended molecule” that contains the
narrowest part of the junction (the molecule in Fig. 10.1) and part of the
electrodes.8 Second, within the LCAO approach, the charge density is com-
puted in terms of the density matrix, see Eqs. (10.57) and (10.58), which
in turn can be computed in terms of Green’s functions in the following
way. Let us assume that the system is in equilibrium. The retarded and
advanced Green’s functions Gr,a µν referred to the local basis functions µ and
ν can be written via their spectral representation [see Eq. (5.14)] as follows
X cµi c∗iν
Gr,a
µν (E) = . (10.65)
i
E ± iη − Ei
Here, the c’s are the coefficients of the expansion of the system eigenfunc-
tions (or molecular orbitals) in terms of local orbitals and Ei are the cor-
responding eigenenergies. Notice that i cµi c∗iν is nothing but the element
P

Pµν of the density matrix, see Eq. (10.58).9 Therefore, the density matrix
in the central part of the junction can be calculated from the retarded or
advanced Green’s function matrix of this part of the system as10
1 ∞
Z
P=∓ dE Im {Gr,a (E)} f (E), (10.66)
π −∞
where f (E) is the Fermi function that ensures that only the occupied states
contribute to the electron density. Now, these Green’s functions can be
computed via their Dyson’s equation [see Eq. (8.26)]
−1
Gr,a = [(E ± iη)S − H − Σr,a r,a
L − ΣR ] , (10.67)
where S is the overlap matrix, H is the one-electron Kohn-Sham Hamilto-
nian of the central part and Σr,aL/R are the left and right self-energies (see
section 8.1). The calculation of these self-energies requires the computation
of the Hamiltonian and Green’s functions of the electrodes. This issue will
be discussed in detail below.
This discussion shows that DFT can be applied to describe nanoscale
junctions by using Eq. (10.66) for determining the density matrix, rather
8 The reason for dividing the system in this way will become clear below.
9 In Eq. (10.58), we assumed that the c’s were real, but in principle, these coefficients
can be complex numbers and then, i cµi c∗iν is the most general definition of Pµν .
P
10 In what follows, we shall not write explicitly the subindexes CC to refer to the central

part of the junction, as we did, for instance, in section 8.1.

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Electronic structure II: Density functional theory 287

Im(z)
C

R
Re(z)

Fig. 10.2 The integral of a retarded Green’s function Gr (z), considered as a function
of a complex variable z, is the same along the contour C and along the real axis R.
However, Gr (z) is much smoother away from the real axis and for this reason, it is
advantageous to integrate the Green’s functions in Eq. (10.66) along a contour like C.
The lower limit of this contour has to be below the lowest lying states of the system,
while the upper limit should be the chemical potential of the system.

than solving the Kohn-Sham equations. The evaluation of the density ma-
trix requires the calculation of the Green’s functions via Eq. (10.67) from
the knowledge of the Kohn-Sham Hamiltonian of the central part. Since this
Hamiltonian depends on the charge density (or density matrix), Eqs. (10.66)
and (10.67) are coupled and they have to be solved in a self-consistent man-
ner. Finally, when these equations are solved, one can compute the different
equilibrium properties of a junction such as charge density, total energy, lo-
cal density of states, etc.
A technical comment is pertinent at this point. Usually Green’s func-
tions vary rapidly as a function of energy, which complicates the integration
appearing in Eq. (10.66). One can get around this problem by making use
of the fact that the Green’s functions are analytical functions and they
can be extended into the complex plane. This means in practice that the
integral in Eq. (10.66) can be done by integrating along a contour in the
complex plane, see Fig. 10.2, where these functions are very smooth. Thus,
one needs a much smaller number of points to carry out the numerical
integration.
The previous discussion also suggests a straightforward way of general-
izing this approach to nonequilibrium situations. In this case, the density
matrix can be expressed in terms of the Keldysh-Green’s functions as
Z ∞
1
P= dE G+− (E), (10.68)
2πi −∞
where G+− can be computed in terms of the retarded and advanced func-
tions as [see Eq. (8.12)]
G+− (E) = 2iGr [ΓL fL + ΓR fR ] Ga . (10.69)
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288 Molecular Electronics: An Introduction to Theory and Experiment

Here, the scattering rates ΓL/R are the imaginary part of the self-energies
ΣaL/R (see section 8.1) and fL/R are the Fermi functions of the left and
right electrodes that include the energy shift caused by the applied bias
voltage. In this latter equation, the retarded and advanced functions can
be calculated from a Dyson’s equation like Eq. (10.67) taking into account
the presence of the bias voltage. Again, Eqs. (10.69) and (10.69) are coupled
and they have to be solved self-consistently. Once this is done, the different
transport properties can be computed as described in Chapter 8. It is
worth mentioning that again the integration in Eq. (10.68) can be done
more efficiently in the complex plane, although this time the integration
close to the Fermi energy requires to modify the contour shown in Fig. 10.2
(see Ref. [287] for details).
The key step to make our generic problem finite was the division of
the system into three parts, see Fig. 10.1. In this division one assumes
that the electrodes are not perturbed by the central part and therefore,
their Hamiltonians and charge densities can be obtained from a separate
(bulk-like) calculation, which only needs to be done once. This assumption
is based on the idea that deep inside a solid the Kohn-Sham potential
approaches the bulk potential. This approximation is often referred to as
the screening approximation and it provides natural boundary conditions
for the potential of the open system. In any calculation, it should be checked
that the potential of the central part actually matches that of the bulk
calculation. Such a check defines in practice the size of the central part and
this size depends on the nature of the electrodes.
The practical implementations of the DFT-NEGF combination differ
mainly in the way in which the electrodes Green’s functions are determined
and how the potential and Hamiltonian of the central part are forced to
match the corresponding ones in the leads. Roughly speaking, one can
grouped all the existent approaches into the following two families:
1.- Methods based on quantum chemistry software. In order to take ad-
vantage of the powerful and well-tested existent quantum chemistry codes,
several groups have implemented the DFT-NEGF approach as follows. The
diagonalization in these codes of the Kohn-Sham Hamiltonian of the finite
central system is replaced by Eqs. (10.68) and (10.69), which are solved
self-consistently. The self-energies required to compute the retarded and
advanced Green’s functions appearing in Eq (10.69) are obtained from a
separate calculation, from which one extracts the bulk Hamiltonian as well
as the coupling matrix elements between the central part and the leads.
This separate calculation can be done at different levels of sophistication.
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Electronic structure II: Density functional theory 289

Isolated system Infinite system

DFT DFT
ρ, P H ρ, P H

Equilibrium Stat. Mechanics NEGF

H
ρ, P H ρ, P ΣL , ΣR

Fig. 10.3 Schematic description of the self-consistent loop in DFT for the determina-
tion of the electronic structure of a finite system (left panel) and an infinite non-periodic
system (right panel) [293]. For an isolated system, the density matrix is constructed by
occupying the states of the Kohn-Sham Hamiltonian H with N electrons. For an infi-
nite system, the density matrix is computed from the nonequilibrium Green’s functions,
see Eq. (10.68), which requires the determination of the self-energies from a separate
calculation.

Thus for instance, some authors describe the leads in terms of simple pa-
rameterized tight-binding methods [288–292] and others extract the bulk
Hamiltonian from DFT calculations of finite clusters [293, 294, 247]. The
bulk parameters are then used to construct surface Green’s functions using
recursive methods like those described in Refs. [198, 295, 296]. Following
Damle et al. [293], we summarize in Fig. 10.3 this approach and emphasize
the main differences with the standard method used for finite equilibrium
systems.
In this approach it is implicitly assumed that the central Hamiltonian
is a functional of the charge density only in the central system. This is
indeed the case for the contributions coming from the kinetic energy and the
electron-nuclei interaction, see Eq. (10.56). It is also true for the exchange-
correlation potential, see Eq. (10.59), but it is not really the case for the
classical Coulomb contribution, see Eq. (10.60). In this latter term, there
are non-local contributions coming from the leads, which are not easy to
describe correctly in the approach discussed in the previous paragraph. The
lack of these non-local contributions causes sometimes severe problems in
the convergence procedure in this method. However, it seems that, as shown
in Ref. [247], if the central system is sufficiently large, those additional
contributions do not play a major role in the physical quantities of interest,
and the self-consistent loop is not crucial in equilibrium systems.
2.- Methods based on solid state software. In the implementations of
the DFT-NEGF method based on the computer codes specially designed
for the description of (infinite) solid states systems, the Hartree potential
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290 Molecular Electronics: An Introduction to Theory and Experiment

VH , or classical Coulomb term of Eq. (10.59), is obtained in the central

region as a solution of the Poisson equation (in Hartree atomic units)
∇2 VH = −4πρ(~r). (10.70)
This equation needs to be solved with appropriate boundary conditions
given by the contact potentials, which are obtained from a separate bulk
calculation. This Poisson equation can be solved with different strategies.
For instance, in the TRANSIESTA code [287], which is an extension of
the SIESTA code for equilibrium systems, this equation is solved via a
fast Fourier transformation algorithm by constructing a periodic supercell.
In other implementations, this equation is solved in real space with 3D
multigrid algorithms [297, 298].
In this approach, the Hamiltonian of the leads, necessary for the con-
struction of the self-energies, is determined with recursive methods similar
to those employed in the method described above. This second approach

Calculation of the
bulk Hamiltonians of
the electrodes

Guess input density

Solve the Poisson equation

with the proper boundary
conditions for the external
potential. This gives the
Hartree potential

Self−consistent Hartree potential + xc potential

loop

Compute the Hamiltonian

of the device and load
the lead Hamiltonians

Solve for the NEGF

of the system and
compute the density
matrix from Eq. (10.68)

Fig. 10.4 Flowchart of the self-consistent loop for the solution of the nonequilibrium
transport problem based on the solution of the Poisson equation [286].
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Electronic structure II: Density functional theory 291

Table 10.3 List of implementations of the combination of DFT and Green’s func-
tion techniques for the description of equilibrium and nonequilibrium properties of
nanoscale junctions. We provide the name of the code, if any, some characteristics,
the reference equilibrium DFT code in which it is based on, and a reference where
details about it can be found. Methods 1 and 2 refer to the methods described in
the text, BC means boundary conditions and TB corresponds to tight-binding.

Name Key features Basis code Ref.

McDCAL Method 2, real space, SIESTA [297]

non-linear transport
TRANSIESTA Method 2, periodic BC, SIESTA [287]
non-linear transport
— complex band structure method, FIREBALL [299]
linear transport
— Periodic BC + Wannier functions, Dacapo [300]
linear transport
SMEAGOL Method 2, periodic BC, non-linear SIESTA [301]
and spin-dependent transport
ALACANT Method 1, TB for the leads GAUSSIAN [290]
— Method 1, DFT for the leads, GAUSSIAN [294]
non-linear transport
— Method 1, TB for the leads, GAUSSIAN [292]
non-linear transport
Cluster-based Method 1, DFT for the leads, TURBOMOLE [247]
method linear transport

is summarized in the flowchart of Fig. 10.4.

We conclude this discussion by listing a few implementations of the
combination of DFT and Green’s function techniques, see Table 10.3. This
list is by no means complete and it is included here just to give some entry
points into the literature where one can find the technical details that we
have skipped in our discussion of the use of DFT for transport problems.

10.8.2 Pluses and minuses of DFT-NEGF-based methods

Let us end this section with some brief comments about the advantages and
drawbacks of the DFT-NEGF approach for the description of the proper-
ties of nanoscale junctions. As we have seen in previous sections, DFT
was designed to describe the ground state energy and related equilibrium
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292 Molecular Electronics: An Introduction to Theory and Experiment

properties of a system. In this sense, from DFT we can expect to obtain an

excellent description of, for instance, contact geometries, breaking forces,
vibration modes and electron-phonon coupling constants. This constitutes
a very valuable information for the understanding of issues like the struc-
ture and formation mechanisms of molecular junctions as well as for the
description of the vibration-assisted inelastic transport (see Chapter 17) or
the phonon contribution to heat conduction in these systems.
On the other hand, the use of DFT for transport problems has clear
limitations. As it is used in these problems, DFT just provides a mean field
approach which is unable to describe strong electronic correlations like the
ones that give rise to phenomena such as Coulomb blockade or the Kondo
effect (see Chapter 15). Furthermore, the use of Kohn-Sham orbitals as the
molecular orbitals of a system is just an approximation that in some cases
leads to large errors in the position of the relevant energy levels responsible
for the transport. In the case of molecular junctions, the transport often
proceeds through the tails of the molecular orbitals closest to the Fermi
energy. Thus, small errors in the position of those levels can lead to big
errors in the transport properties. It is also worth stressing that the use
of DFT in nonequilibrium situations, as it was described above, is just a
reasonable Ansatz, but it is not really justified at the same level as the
corresponding ground state theory. For all these reasons, the DFT-NEGF
combination should be seen as a first step towards a quantitative theory of
transport in nanoscale junctions.
In summary, DFT provides crucial information for the description of the
transport properties of atomic and molecular junctions, but it is important
to be aware of its limitations. In Part 4, we shall discuss in detail the per-
formance of this theory when applied to different aspects of the electronic
and thermal transport of molecular junctions.

10.9 Exercises

10.1 Energy in the Hartree-Fock approximation: Show that the expecta-

tion value of the electronic Hamiltonian of Eq. (10.3) in the Hartree-Fock approx-
imation [see Eq. (10.9)] is given by Eq. (10.11).
10.2 Hartree-Fock equations: Use the variational principle to derive the
Hartree-Fock equations [see Eqs. (10.15) and (10.16)].
10.3 Kohn-Sham equations: Use the variational principle to derive the Kohn-
Sham equations [see Eq. (10.36)].
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PART 3

Metallic atomic-size contacts

293
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294
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Chapter 11

The conductance of a single atom

In order to understand the electrical and thermal conduction through

molecular junctions, which is the main goal of this monograph, it is neces-
sary to first understand the corresponding properties of the metallic atomic
contacts that are used as electrodes in these nanoscale circuits. The con-
duction through atomic-scale wires constitutes a field of its own that started
at the beginning of the 1990’s and it has reached maturity in the last years.
Metallic wires of atomic dimensions have become a marvelous playground
where many basic concepts of quantum transport have been tested [15].
The physics of these nanocontacts and the progress made in this field up
to 2003 have been reviewed in a magnificent article by Agraı̈t, Levy Yeyati
and van Ruitenbeek [15].1 For this reason, we shall not make any attempt
to provide a historical revision of this field or to give a complete list of
references. Instead, we shall present here a short elementary introduction
to some basic aspects that will be useful in subsequent chapters where the
physics of molecular transport junctions is described.
With this idea in mind, we initiate here a series of two chapters devoted
to the electrical conduction through metallic atomic-size contacts. In this
first chapter, we shall focus our attention on the conduction through non-
magnetic contacts, with special emphasis in the simplest structures, namely
single-atom junctions and monoatomic chains. Our main goal here is to es-
tablish the relation between the transport characteristics of these nanowires
and the quantum properties of those atoms used as building blocks. The
next chapter will be devoted to the spin-dependent transport through mag-
netic atomic-size contacts. This is a topic in which a lot of progress has
been made in recent years and these advances are not covered in the review

1A brief introduction to the transport properties of metallic atomic contacts can be

found in [302].

295
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296 Molecular Electronics: An Introduction to Theory and Experiment

of Ref. [15].

11.1 Landauer approach to conductance: brief reminder

Before discussing the experimental results for the conductance of atomic

contacts, it is convenient to say a few words about how this transport prop-
erty is described theoretically. The metallic point contacts and nanowires
that we are considering here have characteristic dimensions that are much
smaller than the typical elastic and inelastic scattering lengths of metals. In
particular, the electron mean free path for elastic scattering on defects and
impurities near the contact is usually much larger than the contact size.2
The main source of (elastic) scattering in these nanocontacts are the walls
forming the boundary of the system. Thus, the transport through atomic
contacts is phase-coherent and it can be described within the framework of
the scattering or Landauer approach, which was extensively described in
Chapter 4.
Within this approach, the low-temperature linear conductance G is
given by Landauer formula
G = G0 T (EF ) = G0 Tr t† t (EF ),
© ª
(11.1)
where T (E) = Tr{t† t}(E) is the energy-dependent total transmission of
the structure, EF is the Fermi energy and G0 = 2e2 /h = 77.5 nS = (12.9
kΩ)−1 is the conductance quantum.3 Here, t is the transmission matrix
of the contact whose elements tmn give the probability amplitude for an
electron wave in mode n on the left of the contact to be transmitted into
mode m on the right of the contact. Since the trace is an invariant, one
can choose to write Eq. (11.1) in the basis that diagonalizes the matrix t† t
and then the conductance expression adopts the following simplified form
Nc
X
G = G0 Tn (EF ), (11.2)
n=1

where Tn (with 0 ≤ Tn ≤ 1) are the transmission coefficients defined as

the eigenvalues of t† t. A simple estimate of the number channels Nc in 3D
metallic contact is given by Nc ≈ (πR/λF )2 , where R is the radius of the
contact radius and λF the Fermi wavelength of the conduction electrons.
For one-atom thick contacts this number is between 1 and 3 for most metals.
2 Moreover, the spin-diffusion and the phase-breaking lengths are also typically much

larger than the contact dimensions.

3 The factor 2 is due to the spin degeneracy in non-magnetic situations.
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The conductance of a single atom 297

8
7
Conductance (2e /h) 6
Gold
2

5
4
3
2
1
0
0 50 100 150 200 250 300
Piezo-voltage (V)

Fig. 11.1 Three typical recordings of the conductance G measured in atomic-size con-
tacts for gold at helium temperatures, using the MCBJ technique. The electrodes are
pulled apart by increasing the piezo-voltage. The corresponding displacement is about
0.1 nm per 25 V. After each recording the electrodes are pushed firmly together, and
each trace has new structure. Reprinted with permission from Ref. [74].

As we shall see later in this chapter, the actual number channels that give
a significant contribution to the conductance depends on the geometry of
the narrowest part of the contacts and on the number of valence orbitals of
the atoms of the corresponding metal.

11.2 Conductance of atomic-scale contacts

The first question that we want to address is: what is the conductance
of a metallic atomic contact? As we discussed in Chapter 2, a metallic
contact of atomic size can be fabricated with various techniques, but the
most widely used ones are the scanning tunneling microscope (STM) and
the mechanically controllable break junction (MCBJ). In Fig. 11.1 one can
see some typical examples of the conductance measured during breaking of
a gold contact at low temperatures, using a MCBJ device.4 Notice that the
conductance decreases by sudden jumps, separated by “plateaus”, which
have a negative slope, the higher conductance the steeper. Some of the
plateaus are remarkably close to multiples of the conductance quantum,
G0 ; in particular the last plateau before loosing contact is nearly flat and
very close to 1 G0 .5 This behavior resembles the conductance quantization
4 In these atomic contacts the current-voltage characteristics are typically linear at low

voltages (below, let us say, 100 mV) and for this reason we shall mainly talk about the
linear conductance as the central transport property.
5 As it will become clear later in this chapter, the last conductance plateaus most likely

correspond to contacts with one atom in cross section and, in particular, long plateaus,
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298 Molecular Electronics: An Introduction to Theory and Experiment

3 4
conductance (2e /h)

(a) Al (b) Pb
2

3
2
2
1
1

0 0
0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16
displacement (Å) displacement (Å)
Fig. 11.2 Evolution of conductance vs tip-sample relative displacement for several rep-
resentative nanocontacts of Al and Pb in STM experiments at low temperatures (4.2 K
for Al and 1.5 K for Pb). The black and grey curves correspond to elongation (open-
ing of the contact) and contraction (closing of the contact), respectively. Adapted with
permission from [264]. Copyright 1998 by the American Physical Society.

that occurs in point contacts defined in 2D electron gases (2DEG), see sec-
tion 4.6.1 and in particular Fig. 4.11. Indeed, different authors interpreted
the step-like evolution of the conductance as an evidence of conductance
quantization in atomic contacts. However, closer inspection of Fig. 11.1
shows that many plateaus cannot be identified with integer multiples of
the quantum unit, and the structure of the steps is different for each new
recording. Also, the height of the steps is of the order of the quantum unit,
but they can vary by more than a factor of 2, where both smaller and larger
steps are found.
The conductance traces not only change from realization to realiza-
tion, but they are also clearly distinct for different metals. In Fig. 11.2
we show several examples of conductance curves for aluminum and lead
wires obtained in the last stages of the breaking of contacts formed with a
STM at low temperatures. In the case of aluminum, one finds that many
plateaus have an anomalous slope: the conductance increases when pulling
the contact, in contrast to the results for gold. For aluminum, the last
plateau before breaking is still close to the quantum conductance, but one
frequently observes the conductance diving below this value, and then re-
covering to nearly 1 G0 , before contact is lost. Lead, on the other hand,
has a last conductance value, which is clearly above 1 G0 and the slope is
positive, i.e. the conductance is reduced upon stretching.
It is worth mentioning that, as one can see in the examples of Fig. 11.2,
like the one in the left curve in Fig. 11.1, are a signature of the formation of a monoatomic
chain.
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The conductance of a single atom 299

20

conductance (2e²/h) 15

10

5
(a)
0

2
(b)
0
∆F
force (nN )

-2

-4

-6

-8

-1 0
0 .0 0 .5 1 .0 1 .5
tip displacem ent (nm )

Fig. 11.3 Simultaneous measurement of force and conductance on atom scale point
contacts for Au. The sample is mounted on a cantilever beam and the force between tip
and sample is measured by the deflection of the beam using an AFM. The measurements
are done in air at room temperature. Reprinted with permission from [58]. Copyright
1996 by the American Physical Society.

the conductance traces recorded when opening the contacts differ from
those recorded during the closing of the contacts. The reason for this lies
in the different atomic arrangements which can be achieved when stretching
as opposed to the ones when pushing the electrodes together. Furthermore,
as we shall explain later, the shape of the conductance traces also depends
on the technique used for fabricating the nanowires.
The previous examples raise several basic questions. The main one is
related to the origin of the conductance steps. In the case of point contacts
defined in 2DEGs, these steps are due to continuous change in the number
of conduction channels as the width varies. In that case the abruptness of
the jumps depends in particular on the shape of the confinement potential.
However, in the case of atomic contacts the cross section cannot be changed
continuously. Early molecular dynamic simulations [303, 304, 262] already
suggested that these jumps could be due to sudden atomic rearrangements.
The idea goes as follows. Upon stretching of the contact, the stress accumu-
lates elastic energy in the atomic bonds over the length of a plateau. This
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300 Molecular Electronics: An Introduction to Theory and Experiment

energy is suddenly released in a transition to a new atomic configuration,

which will typically have a smaller contact size.
The direct proof of the relation between atomic rearrangements and con-
ductance steps was provided in an experiment by Rubio, Agraı̈t and Vieira
[58], where the conductance for atomic-size gold contacts was measured si-
multaneously with the force on the contacts, see Fig. 11.3. Notice that the
stress accumulation on the plateaus and the coincidence of the stress relief
events with the jumps in the conductance can be clearly distinguished.

11.3 Conductance histograms

As shown in the previous section, the conductance of an atomic con-

tact changes from realization to realization, but there are features that
are certainly reproducible, like the last plateau in gold contacts. In or-
der investigate objectively the intrinsic conductance of atomic junctions,
several authors introduced a method [305, 306], which consists in record-
ing histograms of conductance values encountered in a large number of
runs.6 The most studied metal has been gold, which has been investi-
gated with various techniques and under very different conditions, see e.g.
Refs. [60, 70, 307–315].7 In Fig. 11.4 we show conductance histograms of
gold contacts that were measured at different temperatures and different
bias voltages using the MCBJ technique [315].8 Notice that the histograms
are largely dominated by the presence of a peak located very close to 1 G0 ,
while some additional peaks close 2 and 3 G0 are also visible. Similar his-
tograms to that of gold are found for the other two noble metals and an
example for silver can be seen in the upper left panel of Fig. 11.6.
It is worth stressing that, although it does not seem to be very obvi-
ous in the case of gold, the conductance histograms are in general rather
sensitive to experimental conditions such as temperature, voltage, breaking
speed, environmental conditions, etc., see e.g. Ref. [315]. To illustrate this
point, let us briefly discuss here the influence of the experimental technique.
As mentioned in the previous section, the shape of the conductance traces
6 This method was adopted later by Xu and Tao to study the conductance of single-

molecule junctions [549].

7 This metal plays a very important role in molecular electronics since it is by far the

most common material used for the electrodes in molecular junctions.

8 The Ph.D. thesis of A.I. Yanson [315] contains the most systematic study of the con-

ductance histograms of various metals published to date (see in particular Chapter 4 of

this work). Another systematic analysis can be found in Ref. [313].
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

The conductance of a single atom 301

(a) (b)

Fig. 11.4 (a) Conductance histograms of gold at 4.2, 77 and 295 K using the notched-
wire MCBJ technique. The inset shows the first peaks on the expanded scale. (b)
Conductance histograms of gold built from 2000 traces recorded at 1.25 V bias and
12 K (gray). The low temperature histogram (4.2 K) from the left panel is shown for
comparison (black). Note that the vertical axis is in logarithmic scale. Reprinted with
permission from [315].

depends on the sample fabrication method and this variation is reflected in

the histograms. This fact can be easily understood with the help of a me-
chanical model of the atomic contact and its leads [316]. The nanowire can
be modeled as a series of the atomic contact between the left and the right
lead. The leads are modeled as one effective spring with a spring constant
Ks . Obviously, when the spring constant of the leads is smaller than the
effective spring constant of the atomic contact, only a small fraction of the
total applied stretching force is concentrated at the atomic contact. When
pulling the electrodes apart, the leads are elongated and the atomic contact
remains almost unchanged. The conductance trace would thus display hor-
izontal plateaus. The plateaus however do in fact not correspond to values
which are favored electronically but by the minimum force. When however,
the spring representing the atomic contact is softer than the lead springs (in
Fig. 11.5 this is modeled as atomic contact without spring, thus an infinite
spring constant of the lead), the majority of the strain is concentrated at
the contact. It has to respond to the changing separation of the leads result-
ing in plateaus with rich substructure and rather broad conductance peaks
in the histogram. When comparing the various techniques with respect to
this property, we can deduce the following rule of thumb: Techniques with
short free-standing electrodes such as the lithographic MCBJ technique are
supposed to give rise to plateaus with fine structure and wide histogram
peaks, while STM techniques and notched-wire MCBJs are expected to
show straight plateaus and narrower peaks for the same metal.
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302 Molecular Electronics: An Introduction to Theory and Experiment

(a) (b)

Fig. 11.5 (a) Molecular dynamics simulation of the elongation of a Au nanocontact.

Inset: Initial atomic configuration. Top panel: Tensile force on the contact during
elongation: Elastic straining of the metallic bonds interrupted by mechanical instabilities
and processes. Middle panel: Conductance and minimum cross section area (rs 51.6 Å is
the Wigner-Seitz radius for Au). Lower panel: Transmissions coefficients of conductance
channels. For smaller contacts, stages appear where the transmission is carried by a
few almost fully transmitting channels, giving rise to a conductance plateau slightly
downshifted from an integer value (G = {1, 3, 6}G0 ). (b) The same as in the left panel
but for a contact in series with a spring, in order to account for finite stiffness of the
experimental setup. The spring constant is here taken to be Ks = 25 N/m corresponding
to a typical value for contacts fabricated with an STM. Reprinted with permission from
[316]. Copyright 1997 by the American Physical Society.

For alkali metals (Na, K, etc.) one finds histograms at low temperatures
with peaks near 1, 3, 5 and 6 times G0 [306]. An example for sodium is
shown in the upper right panel of Fig. 11.6. The fact that peaks near 2
and 4 G0 are absent points at an interpretation in terms of a smooth, near-
perfect cylindrical symmetry of the sodium contacts. The alkali metals can
be described to a good approximation as free electron systems. Within this
framework, it can be shown that in smooth cylindrical contacts with con-
tinuously adjustable contact diameter [143, 317], the conductance increases
from zero to 1 G0 as soon as the diameter is large enough, so that the first
conductance mode is occupied. When increasing the diameter further, the
conductance increases by two units because the second and third modes are
degenerate. In a similar way, one can explain the absence of a peak at 4 G0
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

The conductance of a single atom 303

Number of counts [arb. units]

Ag Na

Al Pb

Nb Pt

0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7
2 2
Conductance (2e /h) Conductance (2e /h)
Fig. 11.6 Conductance histograms of several metals obtained using the MCBJ tech-
nique. All the histograms were recorded at 4.2 K, except for Nb, which was obtained at
16 K. The conductance was measured at 20 mV for Ag and Nb, 10 mV for Na and Al
and 100 mV for Pb and Pt. Adapted with permission from [315].

and the presence of peaks at 5 and 6 G0 .

The analysis of atomic contacts of monovalent metals (alkali and noble
metals) suggests that there is certain tendency to observe quantized values
of the conductance, at least in the very last stages of the formation of these
atomic contacts. However, this tendency is by no means universal. Indeed,
most multivalent metals only show a rather broad first peak, which reflects
the conductance of a single-atom contact (see discussion below). This peak
can generally not be identified with an integer value of the conductance.
This is illustrated in Fig. 11.6 for Pb, Nb and Pt, which exhibit peaks at
roughly 1.7, 2.3 and 1.6 G0 , respectively. On the other hand, there are a
few examples of multivalent metals, which show pronounced peaks in the
histograms, like for instance Al [318] (see Fig. 11.6), Zn [315, 319, 320]
and Mg [321]. As we shall discuss in the next section, the histogram for Al
throws doubt upon a straightforward interpretation of the histogram peaks
in terms of conductance quantization.
To conclude this section and following Ref. [315], we can summarize the
findings concerning the conductance histograms of atomic contacts in the
following way:
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304 Molecular Electronics: An Introduction to Theory and Experiment

• With the exception of alkali metals, the highest peak is always lying at
the lowest conductance value.
• The position of this peak for all the elements falls in the range between
0.7 and 2.3 G0 . There is no structure related to metallic conductance
in the histograms below the position of the first peak.
• For free electron-like alkali metals the first peak is extremely sharp and
is located almost exactly at 1 G0 . This statement also extends to the
almost free electron-like noble metals.
• For divalent metals (zinc, magnesium) and trivalent ones (aluminum)
the first peak is rather sharp and located slightly below 1 G0 . Other
multivalent metals, and in particular transition metals, exhibit a broad
first peak located well above 1 G0 and in some case like niobium it lies
even above 2 G0 .

11.4 Determining the conduction channels

As the Landauer formula indicates [see Eqs. (11.1) and (11.2)], the con-
ductance measurements gives us only access to the total transmission
P
T = n Tn at the Fermi energy. Obviously, the experimental determina-
tion of the individual transmission coefficients, Tn , could provide a valuable
insight into the origin of the differences between atomic contacts of different
metals. From a mathematical point of view, it is clear that the extraction
of the set {Tn } requires the analysis of transport properties that depend on
the transmission coefficients on a non-linear manner. As we saw in section
4.7, the shot noise is an example of such a quantity. Indeed, the experimen-
tal study of shot noise has provided very important information about the
conduction channels of both atomic contacts and single-molecule junctions.
This is discussed in detail in Chapter 19.
In this section we shall focus our attention on the first method that was
used to extract the individual transmission coefficients of an atomic con-
tact and which continues to be the most precise one. This method was put
forward by Scheer et al. [77] and it is based on the analysis of the subgap
structure in superconducting contacts. Let us explain this idea in certain
detail. Many simple metals, like Al, Pb, Nb, etc., are superconducting be-
low a critical temperature of the order of a few K. In the superconducting
state these metals exhibit a gap in density of states, ∆, which is typically
between 0.1 and 1 meV. This gap strongly influences the transport prop-
erties of superconducting contacts (including atomic junctions) leading to
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The conductance of a single atom 305

1e 2e 3e
(a) (b) (c)
e h e
e
h
2∆ e 2∆ 2∆ h
e e
1
0 1
0 1
0 1
0 11
00 1
0
1
0 1
0 1
0 1
0 11
00 1
0

e e e

> 2∆/1
eV − − 2∆/2
eV > − 2∆/3
eV >

Fig. 11.7 Schematic representation of the multiple Andreev reflection (MAR) that take
place in a contact between two superconductors with gap ∆. We have sketched the
density of states of both electrodes, which exhibits a singularity at the gap edges. In order
to simplify these graphical representations, we have not shifted the DOS of the leads with
bias voltage, but equivalently we have taken into account the fact the quasiparticles gain
an energy eV every time they cross the junction. (a) This panel describes the process in
which a single electron tunnels through the system overcoming the gap due to a voltage
eV ≥ 2∆. (b) Andreev reflection process in which an electron is reflected as a hole
transferring a Cooper pair to the other electrode. This process has a threshold voltage
equal to ∆/e and its probability is proportional to T 2 . (c) MAR of order 3 in which a
quasiparticle is reflected twice before it finds an available state in the right electrodes.
In this process three electron charges are transferred across the junction, the threshold
voltage is 2∆/3e and its probability is proportional to T 3 . Higher-order processes with
contributions proportional to T n can also occur when the bias voltage is larger than
2∆/ne (with n integer).

highly non-linear current-voltage (I-V) characteristics. In order to under-

stand why this is so, let us consider a junction with a single conduction
mode of transmission T . In the limit T ≪ 1, we have essentially a tunnel
junction and the I-V characteristic for a superconducting tunnel junction
is known to directly reflect the gap [702]. As illustrated in Fig. 11.7(a), no
current flows until the applied bias exceeds 2∆/e, after which the current
jumps to approximately the normal-state resistance line. For eV > 2∆
single quasiparticles can be transferred from the occupied states at EF − ∆
on the low voltage side of the junction to empty states at EF + ∆ at the
other side. For eV < 2∆ this process is forbidden, since there are no states
available in the gap.
If the transmission of the junction is not too low, one can still have cur-
rent for voltages smaller than 2∆/e due to higher-order tunnel processes.
Figure 11.7(b) illustrates a process, known as Andreev reflection, in which
an electron is reflected as hole leading to the transfer of a Cooper pair to
the other side of the junction.9 The Andreev process is allowed for eV > ∆
9 This process can also be viewed as the simultaneous tunneling of two quasiparticles.
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306 Molecular Electronics: An Introduction to Theory and Experiment

5 7
(a) 6 (b) T = 0.2
4 0.95 T = 0.4
1.0 5 T = 0.8
eI/GN∆

G/GN
3 4
0.8
2 3
0.6
0.4 2
1 0.2 1
0.01
0 0
0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3
eV/∆ eV/∆
Fig. 11.8 (a) Zero-temperature I-V characteristics of a single channel superconducting
quantum point contact for different values of the normal transmission coefficient (indi-
cated in the graph). Notice that the current has been normalized with the normal state
conductance GN = G0 T to see all the curves in the same scale. (b) The corresponding
differential conductance G = dI/dV for three different values of the transmission. As
a guide for the eyes, the vertical dotted lines indicate the position eV = 2∆/n with
n = 1, . . . , 6. From Ref. [326].

and its onset causes a step in the current at V = ∆/e. The height of
the current step is smaller than the step at 2∆/e by a factor T , since the
probability for two particles to tunnel is T 2 . Depending on the junction
transparency, similar processes of order n involving the transfer of n parti-
cles can occur. These processes give rise to current onsets at eV = 2∆/n
with a step height proportional to T n . An example for n = 3 is illustrated in
Fig. 11.7(c). These processes are referred to as multiple Andreev reflections
(MARs) [323].10 The microscopic theory of MARs for a single-channel point
contact was developed in the late 1980’s and in the 1990’s by several groups
independently [324–328]. In Fig. 11.8 we show the zero-temperature I-V
curves and the corresponding differential conductance for different values of
the normal transmission coefficient. Notice the appearance of a pronounced
structure in the I-Vs close to voltages V = 2∆/ne (with n integer) as a re-
sult of the onset of the different MAR processes. This structure, which is
known as subharmonic gap structure, is more clearly seen in the differential
conductance a series of maxima, see Fig. 11.8(b).
The subharmonic gap structure had been measured in the context of
atomic contacts by several authors [69, 329, 330], but Scheer et al. [77]
10 These multiple processes were first described by Schrieffer and Wilkins [322] in the
limit of low transparent junctions. These authors coined the name multiple particle
tunneling (MPT) for these tunnel events. It is now understood that the concepts of
MAR and MPT are indeed equivalent.
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The conductance of a single atom 307

Fig. 11.9 Current–voltage characteristics for four atom-sized contacts of aluminum us-
ing a lithographically fabricated MCBJs at 30 mK (symbols). The right inset shows
the typical variation of the conductance, or total transmission T = G/G0 , as a func-
tion of the displacement of the electrodes, while pulling. The data in the main panel
have been recorded by stopping the elongation at the last stages of the contact (a–c) or
just after the jump to the tunneling regime (d) and then measuring the current while
slowly sweeping the bias voltage. The current and voltage are plotted in reduced units,
eI/G∆ and eV /∆, where G is the normal state conductance for each contact and ∆ is
the measured superconducting gap, ∆/e = (182.5 ± 2.0)µV. The solid lines have been
obtained by adding several theoretical curves for a single channel contact and optimiz-
ing the set of transmission values. The curves are obtained with: (a) three channels,
P
T1 =0.997, T2 =0.46, T3 =0.29 with a total transmission Tn =1.747, (b) two channels,
P
T1 =0.74, T2 =0.11, with a total transmission Tn =0.85, (c) three channels, T1 =0.46,
P
T2 =0.35, T3 =0.07 with a total transmission Tn =0.88. (d) In the tunneling range a
P
single channel is sufficient, here Tn = T1 =0.025. Reprinted with permission from [77].
Copyright 1997 by the American Physical Society.

were the first to realize that the highly non-linear dependence of the su-
perconducting I-Vs on the transmission coefficient offers the possibility to
extract the transmission coefficients of a few-atom thick contacts. The
principle is illustrated in Fig. 11.9. Using lithographic MCBJs, Al atomic
contacts were formed at very low temperatures (30 mK). During the break-
ing of the Al wires, I-V at low bias (. 1 mV) were recorded along the
conductance plateaus. Examples of these I-Vs can be seen in the main
panel of Fig. 11.9 for different realizations of the contacts. Notice in par-
ticular that curves (b) and (c) correspond to similar values of the normal
state conductance (i.e. for voltages much larger than the Al gap). This
indicates that while these two junctions are almost indistinguishable in
the normal state, they exhibit clearly distinct superconducting I-Vs, which
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308 Molecular Electronics: An Introduction to Theory and Experiment

means that their set of transmission coefficients are very different. The
I-V curves were fitted very accurately with the single-channel I-V curves of
Fig. 11.8(a) using as adjustable parameters both the number of conduction
channels and the transmission coefficients. The authors of Ref. [77] showed
that for the smallest contacts, the set of transmission probabilities can be
unambiguously determined.
The most important finding in these experiments was that in the last
“plateau” in the conductance, just before the breaking of the contact, typ-
ically three channels with different T ’s are required for a good description.
This is surprising since the conductance for such contacts is typically below
1 G0 (see Fig. 11.2(a) and the Al histogram in Fig. 11.6), and it would in
principle require only a single conductance channel. Contacts at the verge
of breaking are expected to consist of a single atom, and this atom would
then admit three conductance channels, but each of the three would only be
partially open, adding up to a conductance close to 1 G0 . This very much
contradicts a simple picture of quantized conductance in atomic-size con-
tacts, and poses the question as to what determines the number of channels
through a single atom.

11.5 The chemical nature of the conduction channels of

one-atom contacts

In order to answer the question posed at the end of the previous section,
Cuevas, Levy Yeyati and Martin-Rodero [263] put forward a minimal model
to compute the conductance of atomic contacts within the framework of
Landauer approach. This model is based on a combination of a simple tight-
binding (TB) model and nonequilibrium Green’s functions techniques, in
the spirit of what we have discussed in Chapters 7-9, and it contains the fol-
lowing three basic ingredients. First, a proper description of the electronic
structure of atomic contacts, and in turn of their transport properties, re-
quires the inclusion in the TB model of at least the atomic orbitals that
give the major contribution to the bulk density of states at the Fermi en-
ergy. As one can see in Fig. 9.7, this means in practice to include the s
orbitals for alkali and noble metals, the s and p orbitals for metals like Al
and Pb and the s and d orbitals in the case of transition metals. Second,
since often we do not have direct information about the geometry of the
atomic contacts, it is important to study the influence of the precise atomic
arrangements. Finally, metals often exhibit local charge neutrality due to
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The conductance of a single atom 309

their small screening length. In this respect, it is important to impose such

neutrality in the TB model via a self-consistent determination of the on-site
energies.
With this minimal model, the authors of Ref. [263] focused on the anal-
ysis of the conductance of one-atom thick contacts like the one shown in
Fig. 11.10(a). In such geometries the current proceeds mainly through the
central atom. Thus, it is convenient to compute the transmission matrix
at the central atom, where its dimension is just the number of orbitals in-
cluded in the basis set.11 This simple idea already tell us that the number
of channels is determined by the number of valence orbitals of the central
atom. This means in practice that the number of conduction channels for
monovalent metals (alkali and noble ones) is limited to one, this number is
at maximum four in the case of sp-like metals like Al or Pb and it can be
up to 6 for transition metals due to the contribution of the s and d bands.
It is worth stressing that this rule of thumb should be taken as an upper
limit since some of the channels may be closed for symmetry reasons. The
case of Al nicely illustrates this fact. Al in its atomic form has an electronic
configuration [Ne]3s2 3p1 , and a total of four orbitals would be available for
current transport: one s orbital and three p orbitals (px , py and pz ). The
calculations of Ref. [263] showed that for a single-atom Al contact there are
three channels that give a significant contribution adding up to a total con-
ductance of the order of 1 G0 . There is a dominant channel that originates
from a combination of the s and pz orbitals of the central atom (where the
z coordinate is taken in the current direction), and two smaller identical
contributions coming from the px and py orbitals.12 The degeneracy of
these two channels is due to the symmetry of the geometry considered [see
Fig. 11.10(a)], and it can be lifted by changing the local environment for
the central atom. The fourth possible channel, an antisymmetric combina-
tion of s and pz , is found to have a negligible transmission probability.13
Thus, these calculations explained the experimental observation by Scheer
et al. [77] that three channels contribute to the conductance for a single

11 Strictly speaking, this is only true if the hopping elements in the TB Hamiltonian are

restricted to first nearest-neighbors. In general, it is a good approximation as long as

the direct coupling between atoms on the left and on the right of the central atom is
weak. The technical details concerning this discussion can be found in section 8.1.
12 The conduction channels, defined as the eigenfunctions of t† t, can be expressed in the

approach of Ref. [263] as a linear combination of the atomic orbitals of the central atom.
13 In simple terms, this antisymmetric combination in the central atom is almost orthog-

onal to the incoming states from the leads which results in a very weak effective coupling
and the corresponding negligible contribution to the total conductance.
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310 Molecular Electronics: An Introduction to Theory and Experiment

(a) (c) 3 Ttotal

T1
fcc [111]

Transmission
2.5 T 2 = T3
2 T 4 = T5
T6
1.5
1
0.5
0
-5 -4 -3 -2 -1 0 1 2 3 4 5
(b) (d) E-EF (eV)
3 1 s
Bulk DOS (1/eV)

px = py
LDOS (1/eV)

pz
0.8 d3z2-r2
2 dxy = dx2-y2
4d 0.6
5s dyz = dzx
5p 0.4
1
0.2

0 0
-5 0 5 10 15 -5 -4 -3 -2 -1 0 1 2 3 4 5
E-EF (eV) E-EF (eV)
Fig. 11.10 (a) Ideal geometry of a single-atom Ag contact grown along the [111] di-
rection (taken as the z-axis). The distances are set to bulk distances and the last two
layers on both sides correspond to those atoms in the infinite surfaces used to model
the leads that are coupled to the atoms in the constriction. (b) Bulk density of states
(DOS) projected onto the s, p and d orbitals as a function of energy (measured with
respect to the Fermi energy EF ). (c) Total transmission and transmission coefficients
of the contact of panel (a) as a function of energy. (d) Local density of states (LDOS)
at the central atom projected onto the different atomic orbitals as a function of energy.
Courtesy of Michael Häfner [331].

aluminum atom. Moreover, the results for the number of channels were
shown to be robust against changes in the atomic configuration, whereas
the total conductance was found to vary depending on the exact atomic
geometry. Finally, this analysis was extended to the case of transition met-
als (in particular Nb) showing that for these metals up to 5 channels can
be expected for a single-atom contact. Again, the sixth channel that could
potentially contribute in a transition metal is actually closed for symmetry
reasons.
Before turning to the analysis of the experiments that confirmed these
ideas, we now want to illustrate them in more detail. In what follows, we
shall make use of the NRL tight-binding method of section 9.6 and the
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The conductance of a single atom 311

formulas derived in section 8.1. The NRL method provides a very accurate
TB parameterization of the bulk properties of elementary solids that is also
well suited for low-dimensional structures (see discussion below). More-
over, this parameterization takes into account long range hopping matrix
elements and it includes up 9 orbitals in the basis set (the s, p and d closest
to the Fermi energy). Thus, this parameterization is more accurate than
that used in Ref. [263] and it serves us to test the conclusions drawn above.
Let us start by analyzing the conductance of an ideal single-atom contact
of Ag. The geometry of this ideal contact is shown in Fig. 11.10(a). It
is constructed by starting from a central atom and including the nearest
neighbors in the successive layers of the fcc lattice along the [111] direction.
The leads are modeled as two infinite surfaces grown along the same direc-
tion. The bulk density of states (DOS) of this metal computed from this
TB parameterization is shown in Fig. 11.10(b). Notice that the d bands
are filled, while the p bands have little weight at the Fermi energy. There-
fore, one expects the s band to dominate the transport properties of this
monovalent metal. Moreover, from the arguments above, one also expects
to have a single conduction channel in the case of one-atom contacts. This
is indeed confirmed by the calculations, as one can see in Fig. 11.10(c).
This figure shows both the total transmission and individual transmission
coefficients as a function of energy for this geometry. Notice in particular
that the transmission at the Fermi energy, which determines the conduc-
tance, is largely dominated by a single channel. One can get insight into
the nature of the conductance channels of single-atom contacts by analyz-
ing the corresponding local density of states (LDOS) at the central atom.
This LDOS projected onto the different atomic orbitals for the geometry
of panel (a) can be seen in Fig. 11.10(d). The first thing to notice is the
presence of true energy bands that, although are narrower than those of
the bulk solid, have widths of several electronvolts. This illustrates the
fact that the central atom is strongly coupled to the electrodes and there
is huge hybridization between its orbitals and those of the leads. Notice
also that there is a clear correlation between the energy dependence of the
transmission and that of the LDOS. In particular, one can see that the
transmission at the Fermi energy arises from a resonance of the s band, as
expected from the arguments above.
On the other hand, we can use this example to anticipate the results for
single-atom contacts of other metals in the periodic table. The idea goes
as follows. Most metals have similar energy bands and the main difference
is the position of the Fermi level, which is determined by the number of
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312 Molecular Electronics: An Introduction to Theory and Experiment

valence electrons. Thus, the energy dependence of the transmission shown

in Fig. 11.10(c) for an Ag contact can be used to understand what happens
in the case of other metals by simply imagining that the Fermi level is lo-
cated in a different position. To figure out what to expect for a transition
metal, we can concentrate on energies 3 eV below the Fermi energy, which
is the region dominated by the d bands, see Fig. 11.10(d). In this case
one can see that up to 5 channels can give a significant contribution to the
total transmission depending on the energy. This agrees with the predic-
tions described above for transition metals. On the other hand, the region
dominated by the p orbitals, a few eV above EF in Fig. 11.10(c), can give
us a hint about the expectations for metals like Al or Pb. In this region
one can see that three channels dominate the transport and their relative
contribution depends on the energy. Two of the channels are degenerate as
a result of the symmetry of the contact. This degeneracy is also reflected
in the LDOS, see Fig. 11.10(d), where the px and py bands are identical,
while the pz one has been shifted down due to the stronger hybridization of
the pz orbitals with the states in the leads. These results again agree with
arguments described above.
To confirm these ideas, we present in Fig. 11.11 the results for the
transmission coefficients for six different metals. This time we have chosen
a geometry for one-atom thick contacts that contains a dimer in its central
part. Different molecular dynamics simulations suggest that this type of
geometry is the most frequently realized in the last stages of the breaking
of the contacts [332, 333, 342, 343]. The results for Ag are similar to
those of Fig. 11.10(c), the main difference being that the channels arising
from the the px and py orbitals have been partially suppressed due to
the reduction of the effective coupling of these orbitals to those in the
leads. In the case of Au (another monovalent metal) the transmission is
also dominated by a single channel, although the second and third one give
a larger contribution than in Ag. Turning now to the sp-like metals Al
and Pb, we see that the three channels give a significant contribution at
the Fermi energy. Notice that two of them are degenerate in this highly
symmetric contact due to the reasons explained above. As anticipated in
the previous paragraph, the total conductance for Pb (1.67 G0 ) is higher
than for Al (1.16 G0 ) because the former has one more valence electron and
therefore the Fermi level lies in the middle of the p bands, rather than in
their tails as in the case of Al. In the case of the transition metals Nb and
Pt the d orbitals play a crucial role (see their bulk DOS in Fig. 9.7) leading
to the opening of additional channels and conductances well above G0 . In
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The conductance of a single atom 313

bcc [001]
fcc [111] T1
T2 = T3
T4
T5 = T6
T7

1 1
Transmission

0.8 Ag 0.8 Au
0.6 0.6
0.4 0.4
0.2 0.2
0 0
-5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5
1 1
Transmission

0.8 0.8 Pb
0.6 Al 0.6
0.4 0.4
0.2 0.2
0 0
-5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5
1 1
Transmission

0.8 Nb 0.8 Pt
0.6 0.6
0.4 0.4
0.2 0.2
0 0
-5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5
E-EF (eV) E-EF (eV)
Fig. 11.11 Individual transmission coefficients as a function of energy for dimer contacts
of Ag, Au, Al, Pb, Nb, and Pt computed with the NRL TB parameterization. The
structures of the contacts are shown in the upper part of the graph. For all the cases the
geometries are grown along the [111] direction of a fcc lattice (see left structure), except
for Nb, which is grown along the [001] direction of a bcc lattice. Courtesy of Michael
Häfner [331].
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314 Molecular Electronics: An Introduction to Theory and Experiment

4
Pb
3
6 3 3 1
2
1
0
-1.0 -0.8 -0.6 -0.4 -0.2 0.0
6
Al
conductance (2e /h)

4
5
2

2 ≥8 6 3 21

0
-1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0
6
Nb
4
≥7 5
2 3 1

0
-0.2 -0.1 0.0 0.1 0.2
4
5 4 Au
3
8 7 6 3 2 1 1
2
1
0
-2.0 -1.5 -1.0 -0.5 0.0
∆x (nm)

Fig. 11.12 Conductance curves measured as a function of contact elongation for Al, Pb,
Nb and Au. The number of channels contributing to the conductance was determined at
each point in the curves by recording the current-voltage relation and fitting the curves
with the theory for superconducting subgap structure. The numbers along the curves in
the figure indicate the number of channels obtained in this way. The number is constant
over a plateau, and usually jumps to a smaller value at the steps in the conductance.
Reprinted by permission from Macmillan Publishers Ltd: Nature [335], copyright 1998.

the case of Nb the total conductance (1.56 G0 ) is due to the contribution of

approximately 5 partially open channels. In the case of Pt, however, three
channels contribute to its conductance (2.6 G0 ). The difference between
these two transition metals is due to the fact that Pt has five more valence
electrons than Nb and the Fermi level lies on the edge of the d bands, while
for Nb is almost in the center of those bands, see Fig. 9.7.
Turning now to the experiments, the method described in the previous
section to extract the information on the conduction channels based on
the analysis of the superconducting I-V curves has been extended to other
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

The conductance of a single atom 315

metals. In particular, in a collaboration between three different laborato-

ries atomic contacts of Pb, Al, Nb and Au were analyzed [335]. Fig. 11.12
shows conductance curves for Pb, Al, Nb and Au, where at each point in
the figure I-V curves as in Fig. 11.9 were recorded and fitted in order to
determine the number of channels involved. The number is constant over
a plateau in the conductance, where the transmission probability for every
mode changes gradually. At the steps in the conductance the number of
channels involved is usually found to jump to a smaller number. In tunnel-
ing range, when the contact is broken and the distance is larger than 0.2 nm,
the I-V characteristics can in all cases be described by a single channel, with
a transmission probability which is given by the tunneling resistance. The
number of channels found for the smallest contacts, just before the jump
to tunneling is 1 for Au, 3 for Al and Pb, and 5 for Nb. The case of gold
deserves a special comment. This metal is not a superconductor, and a
special device was fabricated which allowed the use of proximity induced
superconductivity [335, 334]. The device is a nanofabricated MCBJ hav-
ing a thick superconducting Al layer forming a bridge with a gap of about
100 nm. This small gap was closed by a thin Au film in contact with the
aluminum. Superconducting properties were thereby induced in the Au
film, and by breaking this film and adjusting an atomic-size contact, the
same subgap analysis could be performed.14 Both the Al and Au junctions
were measured at temperatures of 100 mK, far below the superconducting
transition temperatures. Pb and Nb were measured at 1.5 K.
The number of channels and the total conductance at the last plateau
before breaking found in the experiment agree very well with the theory
detailed above and this can be seen as the confirmation of the fact that
the number of conduction channels in single-atom contacts is mainly de-
termined by the number of valence orbitals. Notice also that the relative
conductances of Al and Pb, which are metals of the same group, also agrees
with the theoretical predictions that say that Pb should be more conductive
because of the larger contribution of the px -py channels (see also the con-
ductance histograms of Fig. 11.6). Another important conclusion from the
work of Ref. [335] was that the smallest contacts produced by the different
experimental techniques are indeed one-atom thick contacts since the num-
ber of channels in the last conductance plateau never exceeds the number
of valence orbitals. That was an important conclusion at that time since
14 Notice that in Fig. 11.12 the conductance for Au in the last plateau is a factor 2–3

smaller than usually found, see Figs. 11.1 and 11.6. This was tentatively attributed to
the strong scattering in the nanofabricated device.
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316 Molecular Electronics: An Introduction to Theory and Experiment

there were no means to obtain direct information about the contact ge-
ometries. Later, it became possible to directly image atomic-size contacts
by means of high resolution transmission electron microscopy (HR-TEM),
see e.g. Refs. [63, 336, 62, 65]. This technique has allowed to confirm the
existence of single-atom contacts and the formation of monoatomic chains
(see below).

11.6 Some further issues

The ensemble of results presented in the previous section illustrates the

very good level of understanding achieved in this field. Anyway, there are
aspects of the transport properties of atomic contacts that deserve further
discussion. From the theory side, in spite of excellent overall agreement
with the experiments, one may wonder whether TB models based on pa-
rameterizations of bulk properties can provide a quantitative description of
the conductance of atomic contacts. In recent years, many different groups
have applied ab initio methods to the description of the transport prop-
erties of these metallic nanowires, see e.g. [337, 290, 287, 338, 339, 247]
and references therein. Most of these methods are based on the density
functional theory (DFT) and are described in Chapter 10. As an illustra-
tive example, we present in Fig. 11.13 a comparison of the transmission of
a single-atom Al contact computed with the NRL-TB method and with the
cluster-based DFT method of Ref. [247]. As one can see, the agreement on
the total conductance and transmission channels is quite satisfactory. Such
an agreement is very important because for many problems involving large
contacts or the analysis of a great number of configurations, the ab initio
calculations are either extremely time-consuming or simply not possible.
However, those problems can nowadays be tackled with, for instance, the
relatively inexpensive NRL-TB method.
A more important issue is the role of the mechanical properties in the
conductance of atomic contacts. Since these properties determine the geom-
etry of these metallic wires, they obviously have a major impact in the trans-
port properties. In this sense, a complete theoretical description should
ideally determine also the possible contact geometries realized in the exper-
iments. For this reason, different authors have combined conductance cal-
culations with geometry optimizations or molecular dynamics simulations
of different levels of sophistication, see for instance [332, 333, 340–344] and
references therein. Such combinations have allowed to tackle the following
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

The conductance of a single atom 317

Ttotal Ttotal
(a) TB calculation (b) DFT calculation
T1 T1
3.5
T 2 = T3 T 2 = T3
3
Transmission

T4 T4
2.5 T5 T 5 = T6
2 T6 T7
1.5 T 8 = T9
1
0.5
0
-8 -6 -4 -2 0 2 4 -8 -6 -4 -2 0 2 4
E-EF (eV) E-EF (eV)

Fig. 11.13 Total transmission and individual transmission coefficients as a function of

energy for an ideal single-atom Al contact in [111]-direction, as depicted in Fig. 11.10(a).
(a) Calculation done with the NRL-TB method [331]. Courtesy of Michael Häfner. (b)
DFT calculation from Ref. [247].

two important problems that we now proceed to describe.

The first one concerns the different slopes observed in the conductance
traces of different metals (see section 11.2). This issue was addressed in
Ref. [264], where it was argued that the slopes depend primarily on the evo-
lution of the local density of states at the contact region upon stretching.
This was further investigated with the help of first-principle simulations for
the case of Al by Jelı́nek et al. [332]. Their main results are reproduced
in Fig. 11.14 where one can see a typical evolution of an Al wire in their
simulations of the stretching process and the corresponding total conduc-
tance and the contribution of the individual channels. As one can see, these
results nicely reproduce the main findings of the experiments of Ref. [77]:
(i) the anomalous positive slope of the conductance plateaus, (ii) the fact
that three channels contribute to the conductance in the last plateau and
(iii) the fact that this last plateau has a conductance below 1 G0 , but it
raises to almost 1 G0 on the verge to break.
Maybe, the main problem that remains without a fully satisfactory so-
lution is that of the origin of the peaks in the conductance histograms of the
different metals, see Fig. 11.6. From the discussion in the last section, it is
clear that those peaks cannot longer be interpreted as signatures of conduc-
tance quantization, even if they appear close to integers of G0 as in the case
of Al (see Fig. 11.6). In some cases it has been understood that these peaks
are the result of the interplay between mechanical and electrical properties.
Thus for instance, in the case of alkali metals it has been understood that
the peaks are associated to the existence of exceptionally stable configura-
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

318 Molecular Electronics: An Introduction to Theory and Experiment

A B C

5 5 4
5 4 4
1 1 2
1 3 2 3
2
3
3.5
Total conductance
A st
1 channel
3.0 nd
2 channel
rd
B 3 channel
th
2.5 4 channel
D E F th
5 channel
2.0 D th
G/G0

6 channel

5 5 1.5 C
5 4 4 4
1 1
1 2 2 2 1.0 F
3 E
3 3 0.5

0
0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2
stretching displacement (Å)

Fig. 11.14 Left panel: Snapshots of the structure of an Al nanowire during a stretching
process. The atoms involved in the important bonding rearrangements related to dis-
continuous changes in total energy, force, and conductance are labeled 1-5. Right panel:
Total conductance in units of G0 and the channel contribution along the stretching path.
Adapted with permission from [332]. Copyright 2003 by the American Physical Society.

tions due to both electronic and atomic shell effects [72, 345, 346].15 In the
case of gold, it has become clear that the pronounced peak close to 1 G0 is
related to the formation of monoatomic chains which sustain a single almost
fully open channel (see section 11.8). Several suggestions for the origin of
the peaks in the low-temperature histograms of some multivalent metals
have been made [347].16 An interesting idea was put forward by Hasmy et
al. [351] who performed molecular dynamics simulations to study the his-
tograms of the minimum cross section for Al contacts. At low temperatures
they obtained peaks at multiples of the cross section of a single atom, which
led them to an interpretation of the conductance histogram peaks based on
preferential geometrical arrangements of nanocontact necks. Dreher et al.
[342, 343] have corroborated the existence of well-defined peaks in the min-
imum cross section histograms. However, they have shown that those peaks
15 This is a very interesting topic that will not be further discussed here because our

interest is focused on the smallest contacts. For a detailed discussion of the shell effects
we recommend Refs. [15, 315].
16 Let us mention that more recently room-temperature conductance histograms of Al

and noble metals have been interpreted as an evidence of electronic and atomic shell
effect in these metals [348–350].
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

The conductance of a single atom 319

are not necessarily reflected in the corresponding conductance histograms.

Their theoretical analysis of conductance histograms of Au, Ag, Pt and Ni
contacts shows that the lowest peak is related to the formation of single-
atom contacts and monoatomic chains in the cases of Au and Pt. The
origin of the multi-peak structure of metals like Al or Zn remains however
to be understood.

11.7 Conductance fluctuations

The method described in section 11.4 to extract the set of transmission

coefficients of an atomic contact can only be easily applied in the case
of superconducting metals. In this sense, it is important to have other
methods that give access to the conduction channels. As already mentioned,
the shot noise and the thermopower are two valuable transport properties in
this respect and we shall discuss them in detail in Chapter 19. In this section
we shall focus on the analysis of the so-called conductance fluctuations.
This discussion is based on Ref. [302].
The elastic scattering of conduction electrons on defects and/or im-
purities near atomic contacts leads to interference effects that are clearly
visible in the second derivative of the current with respect to bias voltage,
i.e. in dG/dV . This is similar to the universal conductance fluctuations
in diffusive mesoscopic conductors [50]. This phenomenon is well-known
in point contacts with dimensions much larger than those of an atomic
contact [352–355]. The conductance fluctuations on atomic contacts were
studied systematically by Ludoph et al. [312]. In these experiments lock-in
amplifiers were used to measure simultaneously the conductance and its
derivative during the breaking of the nanowires. By repeating this opera-
tion many times one can construct a conductance histogram together with
the average properties of dG/dV . Typical results for gold contacts can be
seen in Fig. 11.15, where the upper panel shows the standard deviation of
the derivative of the conductance with bias voltage σGV = h(dG/dV )2 i as
a function of the conductance. The conductance histogram for the same
set of data is shown in the lower panel. As one can see in this figure, the
data for σGV display pronounced minima for G near multiples of G0 .
The authors of Ref. [312] offered a very appealing explanation for this
quantum suppression of the conductance fluctuations, which is sketched in
the inset of Fig. 11.15. The idea goes as follows. The atomic contact is
modeled by a ballistic central part, which is described by a set of trans-
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320 Molecular Electronics: An Introduction to Theory and Experiment

mission coefficients, sandwiched between diffusive banks, where electrons

are scattered by defects characterized by an elastic scattering length le .
An electron wave of a given mode impinging on the contact is transmit-
ted with probability amplitude t and it is partially reflected back to the
contact by the diffusive medium, into the same mode, with probability am-
plitude an ≪ 1. This back-scattered wave is then reflected again at the
contact with probability amplitude rn , where Tn = |tn |2 = 1 − |rn |2 . The
latter wave interferes with the original transmitted wave. This interference

2.0
a

1.5
σGV (Go/V)

1.0

0.5

0.0
b
# points (x 10 )

20 53000
3

10

0
0 1 2 3 4
2
G (2e /h)
Fig. 11.15 (a) Standard deviation of the voltage dependence of the conductance versus
conductance for 3500 curves for gold measured with the notched-wire MCBJ technique
at 4.2 K. All data points in the set were sorted as a function of the conductance after
which the rms value of dG/dV was calculated from a fixed number of successive points.
The circles are the averages for 300 points, and the squares for 2500 points. The solid
and dashed curves depict the calculated behavior for a single partially-open channel and
a random distribution over two channels respectively. The vertical grey lines are the
corrected integer conductance values (see text). (b) Conductance histogram obtained
from the same data set. The peak in the conductance histogram at G0 extends to
53000 on the vertical scale. The insets shows a schematic diagram of the configuration
used in the analysis. The dark lines with arrows show the paths, which contribute to
the conductance fluctuations in lowest order. Reprinted with permission from [312].
Copyright 1999 by the American Physical Society.
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

The conductance of a single atom 321

depends on the phase difference between the two waves, and this phase dif-
ference depends on the phase accumulated by the wave during the passage
through the diffusive medium. The probability amplitude an is a sum over
all possible trajectories, and the phase for such a trajectory of total length
L is simply kL, k being the wave vector of the electron. The wave vector
can be influenced by increasing the voltage over the contact, thus launching
the electrons into the other electrode with a higher speed. The interference
of the waves changes as we change the bias voltage, and therefore the total
transmission probability, or the conductance, changes as a function of V .
This describes the dominant contributions to the conductance fluctuations,
and from this description it is clear that the fluctuations are expected to
vanish either when tn = 0, or when rn = 0.
Based on this model, Ludoph et al. [312] obtained the following analyt-
ical expression for σGV ,

³ ~/τ ´3/4 s
2.71 e G0 e
X
σGV = √ Tn2 (1 − Tn ) , (11.3)
~kF vF 1 − cos γ eVm n

where kF and vF are the Fermi wave vector and Fermi velocity, respectively,
τe = le /vF is the scattering time. The shape of the contact is taken into ac-
count in the form of the opening angle γ (see the inset in Fig. 11.15), and Vm
is the applied voltage modulation amplitude. The solid lines in Fig. 11.15(a)
are obtained from Eq. (11.3), assuming a single partially-open channel at
any point, i.e. assuming that channels open one-by-one as the conductance
increases. In agreement with the results discussed in previous sections, the
conductance for the smallest gold contacts is very well described by this
simple approximation. The amplitude of the curves is adjusted to fit the
data, from which a value for the mean free path is obtained, le = 5 ± 1 nm.
Similar experiments [312, 313] for copper and silver and for sodium also
show the quantum suppression of conductance fluctuations observed here
for gold. However, this suppression is not observed in the cases of alu-
minum or niobium [313], which clearly indicates that the transport in these
multivalent metals is governed by partially open channels. Thus, the con-
ductance fluctuation measurements confirm the overall picture described in
previous sections in which the transport in these nanowires is determined
by the valence orbitals of the corresponding material.
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322 Molecular Electronics: An Introduction to Theory and Experiment

11.8 Atomic chains: Parity oscillations in the conductance

As we have seen in previous sections, all evidence shows that for an one-
atom thick contact of monovalent metals the current is carried by a single
mode, with a transmission probability close to one. Guided by this knowl-
edge in experiments on gold Yanson et al. [356] discovered that during the
contact breaking process the atoms in the contact form stable chains of
single atoms, up to 7 atoms long. Independently, Ohnishi et al. [63] discov-
ered the formation of chains of gold atoms at room temperature using an
instrument that combines a STM with a transmission electron microscope,
where an atomic strand could be directly seen in the images. Similar results
were also obtained in Refs. [336, 357].
Some understanding of the underlying mechanism can be obtained from
molecular dynamics simulations. Already before the experimental observa-
tions, several groups had observed the spontaneous formation of chains of
atoms in computer simulations of contact breaking [358, 359]. The au-
thors argue that the interatomic potentials used in the simulation may not
be reliable for this unusual configuration. However, the stability of these
atomic wires has now been confirmed by various more advanced calculations
[360–364].
Only three metals are known to form purely metallic atomic chains,
namely Au, Pt, and Ir [365]. They are neighbors in the sixth period of the
periodic table of the elements and they share another property: they make
similar reconstructions of the surface atoms on clean [100], [110], and [111]
surfaces. A common origin for these two properties has been suggested in
terms of a relativistic contribution to the linear bond strength [365].
There are many interesting aspects of the physics of metallic atomic
chains that could be discussed in detail such as the formation mechanism,
their stability or the fundamental limits for their length. However, we
shall focus our attention here on the analysis of their transport properties
and, in particular, of the so-called parity oscillation of the conductance
because it nicely illustrates how the electrical conduction takes place in
these remarkable 1D systems.
Let us start our discussion by briefly describing the original observa-
tions of the parity oscillations reported by Smit et al. [366]. These authors
investigated the changes of conductance in the process of pulling atomic
chains of Au, Pt and Ir using a STM and MCBJs. In Fig. 11.16 we show a
typical conductance trace obtained during the breaking of an Au contact.
As we have discussed in previous sections, the last conductance plateau
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The conductance of a single atom 323

15

C o n d u c ta n c e [2 e /h ]
1 .1
2

1 .0

10
0 .9

0 .8
0. 0 0 .5 1 .0 1 .5

0
-1.5 -1.0 -0.5 0. 0 0 .5 1 .0 1 .5 2 .0

El ectrod e s epar a tio n [n m ]

Fig. 11.16 Evolution of the conductance while pulling a contact between two gold elec-
trodes (measured with the notched-wire MCBJ technique at 4.2 K). In the inset, an
enlargement of the plateau of conductance at ∼ 1 G0 is shown. Variations to lower
conductance and back up by about 10–15% can be noticed when the atomic chain is
stretched. Reprinted with permission from [366]. Copyright 2003 by the American
Physical Society.

before rupture is in general due to a single-atom contact. The formation of

an atomic wire results from further pulling of this one-atom contact, and
its length can be estimated from the length of the last conductance plateau
[356, 365, 367]. A histogram made of those lengths, see filled curves in
Fig. 11.17, shows peaks separated by distances equal to the inter-atomic
spacing in the chain. These peaks correspond to the lengths of stretching
at which the atomic chain breaks, since at that point the strain to incor-
porate a new atom is higher than the one needed to break the chain [368].
This implies that a chain of atoms with a length between the position of
the n-th and (n + 1)-th peak consists typically of n + 1 atoms.
As we learned in previous sections, the valence of the metal determines
the number of electronic channels through the chain, and each channel con-
tributes to the conductance with a maximum of G0 . For gold, a monova-
lent metal, both the one-atom contact and the chain have a conductance of
about 1 G0 with only small deviations from this value (see Fig. 11.16) sug-
gesting that the single channel has a nearly perfect coupling to the banks.
The small changes of conductance during the pulling of the wire shown in
the inset in Fig. 11.16 are suggestive of an odd-even oscillation. The jumps
result from changes in the connection between the chain and the banks
when new atoms are being pulled into the atomic wire. In order to un-
cover possible patterns hidden in these changes the authors averaged many
conductance traces starting from the moment that a single-atom contact
is formed (defined here as a conductance dropping below 1.2 G0 ) until the
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324 Molecular Electronics: An Introduction to Theory and Experiment

1.1

1.05 Au
1

0.95
conductance (2e /h)
2

2
Pt

1.5

2.2
Ir
2

1.8

1.6
0 0.2 0.4 0.6 0.8 1 1.2 1.4
length (nm)

Fig. 11.17 Averaged conductance traces for chains of atoms of Au, Pt, and Ir (measured
with the notched-wire MCBJ at 4.2 K). Each of the curves are made by the average of
individual traces of conductance while pulling atomic contacts or chains. Histograms of
the plateau lengths for the three metals obtained from the same set of data are shown by
the filled curves. Reprinted with permission from [366]. Copyright 2003 by the American
Physical Society.

wire is broken (conductance dropping below 0.5 G0 ). In the upper panel

of Fig. 11.17 it can be seen that the thus obtained average plateau shows
an oscillatory dependence of the conductance with the length of the wire.
The amplitude of the oscillation is small and differs slightly between exper-
iments.17
The same procedure was repeated for Pt and Ir. These metals have
s and d orbitals giving rise to several conduction channels. Each channel
may have a different transmission that can be affected by the details of the
contact and therefore the average plateau conductance is expected to show
a more complicated behavior. A one-atom Pt contact has a conductance of
about 2 G0 while for a Pt atomic chain it is slightly smaller, ∼ 1.5 G0 with
variations during the pulling process that can be as large as 0.5 G0 . For
Pt similar oscillations to those for Au were observed, which were compared
17 Thisbehavior is clearly at variance with that found in molecular junctions, where the
conductance typically decays exponentially with the molecule length.
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The conductance of a single atom 325

to the peak spacing in the length histogram in Fig. 11.17. The latter is
obtained by taking as a starting point of the chain a conductance drop-
ping below 2.4 G0 . Ir shows a similar behavior although somewhat less
pronounced and it is more difficult to obtain good length histograms.
The simplicity of the atomic chains had stimulated numerical simula-
tions of their transport properties well before their experimental observa-
tion [369]. In particular, various groups [369–374] had found oscillations
in the conductance as a function of the number of atoms for calculations
of sodium atomic chains, where this metal was selected because it has the
simplest electronic structure. Sim et al. [370], using first-principles calcu-
lations and exploiting the Friedel sum rule, found that the conductance for
an odd number of atoms is equal to G0 , independent of the geometry of
the metallic banks, as long as they are symmetric for the left and right
connections. On the other hand, the conductance is generally smaller than
G0 and sensitive to the lead structure for an even number of atoms. The
odd-even behavior follows from a charge neutrality condition imposed for
monovalent-atom wires. These predictions agree nicely with the results
found for the Au chains.
As explained by the authors of Ref. [366], the odd-even behavior is
essentially an interference effect and it can be easily understood in the
frame of a simplified one-dimensional free-electron model, see Exercise 11.1
and Refs. [366, 375]. Instead, we shall provide here an argument from our
usual “atomistic” point of view. We shall analyze the parity effect in gold
chain with the help of the simple model described in Exercise 7.5, which
is represented schematically in Fig. 11.18. In this model we describe the
gold chain with a tight-binding Hamiltonian with a single orbital per atom
and with hopping elements, t, only between nearest neighbors. We assume
that the on-site energy, ǫ0 , is the same for all the atoms in the chain and
we set it to zero. We describe the leads by two identical semi-infinite linear
chains with, for simplicity, the same parameters as in the finite chain (bulk
hopping t and on-site energy ǫ0 ). Finally, the coupling between the chain
and the leads is described by the hoppings tL and tR that can be different
from the intra-chain hopping t.
The calculation of the transmission in this model, and therefore of the
zero-bias conductance, is a simple exercise that we proceed to sketch.18
From the general formula of Eq. 8.18, it is easy to show that the zero-bias

18 Itis not necessary to follow this calculation to understand the main conclusions that
will be drawn from this toy model.
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326 Molecular Electronics: An Introduction to Theory and Experiment

1111
0000
0000
1111 1111
0000
0000
1111
0000
1111
tL t t t t tR1111
0000
0000
1111
0000
1111 0000
1111
1111
0000
L
0000
1111 R
0000
1111
0000
1111
0000 1
1111 ε0 0000
1111
0000
1111
0000
1111 N 1111
0000
0000
1111
0000
1111 0000
1111
0000
1111
Fig. 11.18 Schematic representation of the simple tight-binding model used to analyze
the parity effect in gold chains. In this model the chain has N atoms with a single
orbital per site and with an on-site energy ǫ0 = 0. There is only coupling between
nearest neighbors inside the chain, t. The coupling to the leads is given by the matrix
elements tL,R . The leads are modeled in practice by two identical semi-infinite chains
with the same parameters as the finite central chain.

transmission in this model is given by (see also Exercise 7.5)

T (E) = 4ΓL (E)ΓR (E)|Ga1N (E)|2 , (11.4)
where the scattering rates are given by ΓL,R = Im{ΣaL,R } and the self-
energies can be by expressed as ΣL,R = t2L,R gL,R
a a
, where gL,R are the ad-
vanced Green’s functions of the last atom of the two semi-infinite chains
used to model the leads [see Eq. (5.46)]. Finally, we have to determine
Ga1N (E), which is simply the element (1, N ) of the following matrix
−1
Ga (E) = [E a − Hchain − ΣaL − ΣaR ] (11.5)
 a −1
E − ǫ0 − ΣL −t 0 ···
a
 −t E − ǫ0 −t 0 
.. .. .. ..
 
= ,
 
 . . . . 

 0 t E a − ǫ0 −t 
··· 0 −t E a − ǫ0 − ΣR
where E a = E − i0+ . This tridiagonal matrix of dimension N can be
inverted numerically or even analytically (see Exercise 11.2).
Let us illustrate the results of this simple model. In Fig. 11.19 we show
the transmission as a function of energy for two chains with 4 and 5 atoms.
Let us remind that the conductance is determined by the value of the trans-
mission at the Fermi energy, which in this model is zero due to the inherent
electron-hole symmetry (we have a single electron per atom). The different
curves in both panels correspond to different values of the interface hop-
ping tL,R . Notice that if tL,R = t the system becomes an ideal infinite chain
where there is no backscattering, which leads to a perfect transparency in
the whole band (|E| < 2t). If the interface hopping is different from the
intra-chain hopping (interface mismatch), the backscattering builds Fabry-
Perot-like resonances in the transmission. In the case of the 4-atom chain
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The conductance of a single atom 327

(a) 4-atom chain (b) 5-atom chain

1 1
Transmission

0.8 0.8
0.6 tL = tR = t
0.6
0.4 tL = tR = 0.9t 0.4
tL = tR = 0.8t
0.2 tL = tR = 0.7t 0.2
0 0
-2 -1 0 1 2 -2 -1 0 1 2
E/t E/t
Fig. 11.19 Transmission as a function of energy (normalized by the intra-chain hopping
t) for two chains with 4 (a) and 5 (b) atoms. The different curves correspond to different
values of the interface hopping tL,R , as indicated in the legend. The vertical dotted lines
indicate the position of the Fermi energy, which is zero in this case.

(and in any chain with an even number of atoms), those resonances produce
a minimum of the transmission at the Fermi energy, whereas they lead to a
maximum for the chains with an odd number of atoms. This result explains
qualitatively the parity effect discussed above for a monovalent metal.
The presence of transmission maxima at the Fermi energy for odd num-
ber of atoms in the chain and the minima for the chains with even number
of atoms can be understood as follows (see Exercise 11.1). The maxima of
the transmission appear at the position of the levels of the decoupled chain.
In the case of odd N there is always a level in the chain spectrum exactly
at the Fermi energy (E = 0) for symmetry reasons, which together with
the charge neutrality leads to a maximum of the linear conductance. On
the contrary, when N is even there is no chain level at the Fermi energy
and therefore these chains exhibit a lower conductance. To conclude this
discussion, we show in Fig. 11.20 the transmission at the Fermi energy as
a function of the number of chain atoms. As one can see, the amplitude of
the even-odd oscillations depends on the quality of the interfaces.19
The simple explanation presented above can account qualitatively for
the experimental behavior in the case of Au, characterized by a full 5d
band and a nearly half-filled 6s band. However, for the case of Pt and Ir, in
which the contribution of 5d orbitals to the conductance is important, there
19 The conductance does not decay with length in this case because the Fermi energy
lies inside the “band” formed by the states of the finite chain. For energies outside this
energy window, the conductance decays exponentially with length. This is what happens
in the case of molecular junctions (see discussion in section 13.4).
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328 Molecular Electronics: An Introduction to Theory and Experiment

1
Transmission (EF)

0.8

0.6

tL = tR = t
0.4
tL = tR = 0.9t
tL = tR = 0.8t
0.2 tL = tR = 0.7t

0
2 3 4 5 6 7 8 9 10
Number of chain atoms

Fig. 11.20 Transmission at the Fermi energy as a function of the number of atoms in
a linear chain. Notice the even-odd effect. The different curves correspond to different
values of the interface hopping tL,R , as indicated in the legend.

is no reason why this simple picture should still hold. The experiments of
Smit et al. [366] triggered off new the theoretical analyses of the conduc-
tance of these monoatomic wires, see for instance Refs. [342, 343, 376–381].
In particular, de la Vega et al. [376] presented an appealing comparative
study that we now proceed to describe. These authors studied the con-
ductance of ideal chain geometries of Au, Pt and Ir, in which the atomic
chain is connected to bulk electrodes represented by two semi-infinite fcc
perfect crystals along the (111) direction. Using the Green’s function tech-
niques detailed in Chapters 7 and 8 and a parameterized self-consistent
tight-binding model, they obtained the evolution of the conductance with
the number of atoms in the chain depicted in Fig. 11.21(a). Notice that
this evolution is rather sensitive to the elongation, especially in the case
of Pt and Ir (for Au the conductance exhibits small amplitude even-odd
oscillations, which remain practically unaffected upon stretching).
The main features and the differences between Au, Pt and Ir are more
clearly understood by analyzing the local density of states and the energy
dependence of the transmission, shown in Fig. 11.21(b) for a N = 5 chain of
these metals at an intermediate elongation. The Au chains are characterized
by a single conduction channel around the Fermi energy with predominant
s character. The transmission of this channel lies close to one and exhibits
small oscillations as a function of energy resembling the behavior of the
single band TB model discussed above.
In the case of Pt the contribution from the almost filled 5d bands be-
comes important for the electronic properties at the Fermi energy. There
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The conductance of a single atom 329

(a) a=2.70 Å
(b)
1.02 0.8 3
a=3.0 Å
Au
1.00 0.6 Au Au
2
0.98 0.4
1
0.96 0.2

0.94 0.0 0

Total transmission
2.5 0.8 3
Pt
2.0 0.6 Pt Pt
G(EF)/G0

DOS
2
1.5 0.4
a=2.70 Å
1
1.0 a=2.80 Å 0.2
a=2.90 Å
0.5 0.0 0
Ir a=2.50 Å 3
a=2.75 Å 0.6
4 Ir Ir
a=2.85 Å
0.4 2
3

0.2 1
2

0.0 0
1 2 3 4 5 6 7 8 0 5 10 0 5 10
N Energy (eV) Energy (eV)

Fig. 11.21 (a) Evolution of the conductance with N for different values of the inter-
atomic distance a. (b) Local density of states (LDOS) at the central atom and total
transmission for Au, Pt and Ir chains N = 5 at an intermediate elongation. The LDOS
is decomposed in s (full line), d (dotted line) and p (dashed line) orbitals with the same
normalization in the three cases. Reprinted with permission from [376]. Copyright 2004
by the American Physical Society.

are three conduction channels with significant transmission at EF : one due

to the hybridization of s-pz and dz2 orbitals, and another two almost degen-
erate with px -dxz and py -dyz character, respectively (here z corresponds to
the chain axis). The contribution of the 5d orbitals is even more important
in the case of Ir where a fourth channel exhibits a significant transmission.
As discussed in Ref. [376], more insight into these results can be ob-
tained by analyzing the band structure of the infinite chains. Fig. 11.22(a)
shows the bands around the Fermi energy for Pt obtained from ab-initio
calculations. Two main features are worth commenting: (i) Symmetry con-
siderations allow to classify the bands according to the projection of the
angular momentum along the chain axis, m. (ii) Close to EF there is an
almost flat filled two-fold degenerate band with dxy and dx2 −y2 (m = ±2)
character. The other partially filled and more dispersing bands have s-
pz -dz2 (m = 0) and px -dxz or py -dyz (m = ±1) character (see labels in
Fig. 11.22).
The close connection between this band structure and the conduction
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330 Molecular Electronics: An Introduction to Theory and Experiment

(a) (b) 1
0.8
4 0.6
a = 2.33 Å m=0 m = 0 up
2 Pt 0.4
E-EF (eV)

1 m = -1,1

T(EF)
0
0.8
-2 m = -2,2
0.6
-4 0.4
m = -1,1
-6 m=0 0.6
-8 0.4 m = 0 down
0 0.2 0.4 0.6 0.8 1
0.2
ka/π 0
0 5 10 15 20 25 30
N
Fig. 11.22 (a) Band structure of the infinite Pt chain. The bands are classified by the
quantum number m corresponding to the projection of the angular momentum on the
chain axis. The arrows indicate the crossing of the Fermi level for the m = 0 and the
m = ±1 bands. (b) Channel decomposition for Pt chains as a function of N . The legends
indicate the symmetry of the corresponding bands in the infinite chain. Courtesy of A.
Levy Yeyati.

channels of the chains is realized when analyzing the evolution of the con-
ductance and its channel decomposition for even longer chains than in
Fig. 11.21 (N > 8). This is illustrated in Fig. 11.22(b). As it can be
observed, the decrease of the total conductance of Pt for N < 7 − 8 cor-
responds actually to a long period oscillation in the transmission of the
two nearly degenerate channels associated with the m = 1± bands. This
period can be related to the small Fermi wave vector of these almost filled
d bands, as indicated by the arrows in Fig. 11.22(a). In addition, the up-
per m = 0 band crossing the Fermi level is close to half-filling giving rise
to the even-odd oscillatory behavior observed in the transmission of the
channel with predominant s character. The lower m = 0 band tends to be
completely filled and the corresponding channel is nearly closed for short
chains. However, one can appreciate a very long period oscillation in its
transmission, rising up to ∼ 0.5 G0 , for N ∼ 13-14.
The general rule that emerges from the above analysis is that the
transmission corresponding to each conduction channel oscillates as ∼
cos2 (kF,i N a), where kF,i is the Fermi wave vector of the associated band in
the infinite chain. In the case of Pt the total conductance for short chains
(N < 7-8) exhibits an overall decrease with superimposed even-odd oscil-
lations in qualitative agreement with the experimental results. For even
longer chains (not yet attainable in experiments) these calculations pre-
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The conductance of a single atom 331

dict an increase of the conductance due to the contribution of conduction

channels with dxz , dyz character.

11.9 Concluding remarks

As we have shown in this chapter, thanks to a close interaction between ex-

periment and theory it has become possible to establish a coherent picture
of the transport in metallic atomic contacts. Moreover, we have learned a
few important lessons that are very useful for the field of single-molecule
conduction. First of all, we now understand the close relation between
the quantum properties of individual atoms used as building blocks and
the macroscopic transport properties of the circuits in which they are em-
bedded. This relation is nicely summarized in the connection between the
number of channels of a single-atom contact and the valence orbitals of the
corresponding atom. It has also become clear that a deep understanding
of the electrical conduction in these nanocircuits can only be achieved by
combining different experimental techniques and by studying a variety of
transport properties.
Let us emphasize again that in this chapter we have addressed only a
few basic issues concerning the very rich physics of (non-magnetic) atomic
contacts. We have left out many important topics, a discussion of which can
be found in the review of Ref. [15]. On the other hand, it is worth stress-
ing that there are still many basic issues to be resolved. We have already
mentioned some of them, like the problem of the origin of the peaks in the
conductance histograms of multivalent metals, but there are many others.
For instance, some materials like the semi-metals [382, 311, 384] exhibit
very peculiar transport properties that are not yet fully understood. Con-
cerning atomic chains, it has been recently discovered that the presence of
impurity atoms can facilitate the formation of atomic chains, even in met-
als in which they are not formed in the pure state [385]. The fundamental
limits for the lengths of these hybrid chains as well as their mechanical and
transport properties still need to be investigated in further detail. With re-
spect to the optical properties of atomic contacts, experiments measuring
the laser-assisted transport are just beginning to be reported (see Chapter
20). On the other hand, experiments on light emission from atomic contacts
are starting to reveal new information about the physics of these contacts
[386]. So in short, atomic contacts will continue to be a marvelous source
of new and fascinating physics.
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332 Molecular Electronics: An Introduction to Theory and Experiment

11.10 Exercises

11.1 1D model for the parity effect in gold chains:

Let us model the gold monoatomic chains by the simple 1D potential shown
in Fig. 11.23. The regions on the left and right of the potential step represent the
electrodes, while the chain corresponds to the step region. The scattering at the
chain-leads interfaces is taken into account via a mismatch in the wave vectors:
k1 = k3 6= k2 .

V2

k1 k2 < k1 k3 = k1

V1 V3
x=0 x=L x
Fig. 11.23 One-dimensional model for the potential landscape describing an atomic
chain of length L.

(i) Compute the transmission through this potential barrier for energies higher
than the step height as a function of k1 and k2 . Hint: the solution is given in
Eq. (4.16).
(ii) Show that the transmission exhibits oscillations as a function of the chain
length where the maxima are given by Tmax = 1 and minima by Tmin = 4γ 2 /(1 +
γ 2 )2 , where γ = k2 /k1 .
(iii) To determine the value of k2 relevant for the transport, one might be
tempted to fix it to kF of an infinite chain. Show that assuming that there is
an electron per atom this Fermi wave vector is given by kF = π/(2a), where
a = L/N is the interatomic distance, N the number of atoms in the chain and
L its length. Show also that with this choice for k2 this 1D model predicts that
the conductance maxima should appear for chains with even number of atoms,
contrary to the model explained in section 11.8.
(iv) The problem found in (iii) can be solved by computing k2 in the following
more appropriate manner. Since the chain is finite, there is a limited set of
possible values for k2 , namely k2i = (π/a)i/N with i = 1, ..., N (can you explain
why?). Then, imposing charge neutrality one obtains k2 = (π/2a)N/(N + 1) for
the Fermi wave vector. Use this value for k2 in the expression of the transmission
to show that now the model reproduces the correct phase of the conductance
oscillations.
11.2 Even-odd effect in gold atomic chains:
Let us consider the chain model discussed in section 11.8 to explain the even-
odd effect in the conductance of gold atomic wires.
(i) Reproduce the results of Figs. 11.19 and 11.20.
(ii) Diagonalize the Hamiltonian of the uncoupled finite chain for different
number atoms, N , to obtain its energy spectrum. Show that for N odd there is
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

The conductance of a single atom 333

always an energy level of the chain at E = 0.

(iii) Compute the local density of states inside the chain for N = 4, 5 and
study its relation with both level spectrum obtained in (iii) and the transmission
function of Fig. 11.19.
(iv) Demonstrate that the transmission at the Fermi energy (EF = 0) is given
by the following analytical expression
 2
4t ΓL ΓR /(t2 + ΓL ΓR )2 for even N
T (EF ) = (11.6)
4ΓL ΓR /(ΓL + ΓR )2 for odd N.

Here, ΓL,R are the scattering rates at the Fermi energy given by ΓL,R = tL,R ,
where we have assumed that the semi-infinite chains describing the leads have
the same hopping as the finite central chain.
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334 Molecular Electronics: An Introduction to Theory and Experiment

January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Chapter 12

Spin-dependent transport in
ferromagnetic atomic contacts

The use of the spin degree of freedom of the electron in conventional charge-
based electronic devices has lead to the discovery of many fundamental
effects and, in some cases, to new technological applications [387, 388].
The emblematic physical effects in this new field, already known as spin-
tronics, like the giant magnetoresistance (GMR), tunneling magnetoresis-
tance (TMR) or anisotropic magnetoresistance (AMR) stem from the spin-
sensitivity of the scattering mechanisms that dominate the transport prop-
erties in electronic devices made of magnetic materials.1 In recent years,
a great effort has been devoted to understand how these fundamental ef-
fects are modified when the dimensions of a magnetic device are reduced of
the way down to the atomic scale. Contrary to the case of non-magnetic
atomic contacts, the physics of their ferromagnetic counterparts is not so
well established and there are still basic open problems. The goal of this
chapter is to provide a brief introduction to the transport properties of
ferromagnetic atomic-size contacts and to draw the attention to problems
that could be soon analyzed in the context of molecular junctions.
There are many different topics in this field that one could address. In
order to illustrate the interesting physics of ferromagnetic atomic contacts,
we have chosen to discuss three issues that are attracting a lot of atten-
tion. The first one concerns the conductance of these atomic contacts in
the absence of domain or external magnetic fields and, in particular, the
possibility of observing conductance quantization. The second problem is
related to the magnetoresistance of these atomic-scale conductors, which
has been shown to be enormous in comparison with the one found in larger
devices made of the same materials. Finally, we shall address the issue of

1 Fora basic explanation of all these magnetoresistive effects, see Ref. [388] or chapter
15 in Ref. [389].

335
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336 Molecular Electronics: An Introduction to Theory and Experiment

the anisotropic magnetoresistance in ferromagnetic atomic contacts, which

again is very different from the one found in bulk systems. After discussing
these topics in the next three sections, we shall conclude this chapter with
some final remarks and a brief discussion about the challenges and open
problems.

12.1 Conductance of ferromagnetic atomic contacts

The first issue that we want to address is the conductance of ferromagnetic

atomic-size contacts.2 This question has been experimentally investigated
by numerous groups [311, 313, 315, 390–406] which, in particular, have
studied the conductance histograms of atomic contacts made of the 3d
ferromagnetic metals (Ni, Co and Fe). To make a long story short, let us
say that two type of contradictory results have been reported. On the one
hand, several groups have observed peaks in the conductance histogram
at half-integer multiples of G0 [397–402]. This has been interpreted as a
manifestation of half-integer conductance quantization [400], implying that
only fully open channels contribute to the conductance. In this sense, a
peak at 0.5 G0 would then additionally mean the existence of a fully spin-
polarized current. Furthermore, some authors have reported conductance
histograms that are very sensitive to an external magnetic field [394].
On the other hand, another group of experiments, see e.g.
Refs. [311, 313, 390, 403], show that the conductance histograms are either
featureless at room temperature or they exhibit a single peak at conduc-
tances well above 1 G0 at low temperatures. Let us mention in particular
the work of Untiedt et al. [403] in which conductance histograms of Fe, Co,
and Ni using notched-wire MCBJs under cryogenic vacuum conditions.
These histograms are reproduced in Fig. 12.1. Notice the absence of frac-
tional conductance quantization, even when a high external magnetic field
was applied. Notice also that the histograms show broad peaks above 1 G0 ,
with only little weight below it. Furthermore these authors suggested that
the differences between these two groups of experiments could be due to
the fact that all room temperature experiments are performed under at-
mospheres that are considerably less pure than that provided by cryogenic
vacuum and thus, one cannot disregard the possibility of atomic-scale con-
tamination of the contact by foreign atoms or molecules. Indeed, these
2 Here,we have in mind situations where there is a homogeneous magnetization in the
junctions, i.e. no domains.
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Spin-dependent transport in ferromagnetic atomic contacts 337

3000
Fe
2000

1000

0
0 1 2 3 4 5
3000
Co
2000
Counts

1000

0
0 1 2 3 4 5
4000
3000 Ni

2000
1000
0
0 1 2 3 4 5
2
Conductance (2e /h)

Fig. 12.1 Conductance histograms for Fe, Co and Ni atomic contacts obtained with the
notched-wire MCBJ technique without magnetic field (thin curve) and when a magnetic
field of 5 T parallel to the current direction was applied (thick curve). The conductance
was measured using a dc bias voltage of 20 mV and a temperature of 4.2 K. Reprinted
with permission from [403]. Copyright 2004 by the American Physical Society.

authors showed that the inclusion of hydrogen molecules significantly mod-

ifies the histograms of Fe, Co, and Ni.
The observation of any kind of quantization of ferromagnetic materials
like Fe, Co or Ni is certainly surprising. These materials are transition
metals in which, according to our discussions in the previous chapter, the d
bands are expected to play a fundamental role in the transport properties
contributing with partially open channels. The conductance of ferromag-
netic atomic contacts has been analyzed theoretically by many different
groups using a variety of methods, see e.g. Refs. [407–427]. The general pic-
ture that emerges from these works confirms the naive picture and clearly
suggests that conductance quantization is not really expected in these fer-
romagnetic nanowires.
In what follows, we shall discuss in certain detail the results of Ref. [427],
which illustrate the most commonly accepted picture of the transport in
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338 Molecular Electronics: An Introduction to Theory and Experiment

ferromagnetic contacts. In this work the calculations are based on the com-
bination of the NRL tight-binding method of section 9.6 (see also Ref. [428])
and nonequilibrium Green’s function techniques, which was already used
in the last chapter.3 Let us stress that in this discussion we shall neglect
both the spin-orbit interaction and we shall assume that there are no do-
main walls present in the contacts.4 With these assumptions, the transport
properties of ferromagnetic contacts can be described in terms of two inde-
pendent contributions coming from both spin bands. In particular, in the
framework of Landauer’s approach the linear conductance at low tempera-
ture can be expressed as follows
e2 X
G= Tσ (EF ), (12.1)
h σ

where Tσ (E) is the total transmission for spin σ =↑, ↓ at energy E and EF
is the Fermi energy. We also define the spin-resolved conductances Gσ =
(e2 /h)Tσ (EF ), such that G = G↑ + G↓ . The transmissions are obtained as
follows
X
Tσ (E) = Tr[t†σ (E)tσ (E)] = Tn,σ (E), (12.2)
n

where tσ (E) is the transmission matrix and Tn,σ (E) are the individual
transmission eigenvalues for each spin σ (see section 8.1.3 for more details
on the calculation of these transmission matrices).
An important quantity in our discussion will be the spin polarization P
of the current, which we define as
G↑ − G↓
P = × 100%. (12.3)
G↑ + G↓
Here, we shall assume that spin up denotes the majority spins, while spin
down corresponds to the minority ones.
In order to understand the results described below, it is instructive to
first discuss the bulk density of states (DOS). The spin- and orbital-resolved
bulk DOS of these materials around EF , as calculated from the NRL tight-
binding method, is shown in Fig. 12.2. The common feature for the three
ferromagnets is that the Fermi energy for the minority spins lies inside the
d bands. This fact immediately suggests that the d orbitals may play an
important role in the transport. For the majority spins the Fermi energy lies
3 The technical details of the calculation of the current in ferromagnetic contacts have

been discussed in section 8.1.3.

4 Later in this chapter we shall discuss the role of these two factors.
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Spin-dependent transport in ferromagnetic atomic contacts 339

(a) Fe (b) Co (c) Ni

Bulk DOS (1/eV) Bulk DOS (1/eV)

3d
1.5 1.5 1.5 4p
4s
1 1 1

0.5 0.5 0.5

0 0 0
1.5 1.5 1.5

1 1 1

0.5 0.5 0.5

0 0 0
-6 -4 -2 0 2 -6 -4 -2 0 2 -6 -4 -2 0 2
E (eV) E (eV) E (eV)
Fig. 12.2 Bulk density of states (DOS) of Fe, Co, and Ni, resolved with respect to the
individual contributions of 3d, 4s, and 4p orbitals, as indicated in the legend. The upper
panels show the DOS for the majority spins (spin up) and the lower ones the DOS for
minority spins (spin down). The vertical dotted lines indicate the Fermi energy, set to
zero. Reprinted with permission from [427]. Copyright 2008 by the American Physical
Society.

close to the edge of the d band. The main difference between the materials
is that for Fe there is still an important contribution of the d orbitals, while
for Ni the Fermi level is in a region where the s and p bands become more
important. The calculated values of the magnetic moment per atom (in
units of the Bohr magneton) of 2.15 for Fe, 1.3 for Co, and 0.45 for Ni are
reasonable agreement with the literature values [429].
Let us turn now to the analysis of the conductance of Fe, Co and Ni
contacts. We consider ideal one-atom thick contact geometries with a cen-
tral dimer as shown in the upper part of Fig. 12.3. In this figure we also
present the total transmission for majority spins and minority spins as a
function of energy as well as the individual transmission coefficients for
those geometries. As one can see in Fig. 12.3(a), for the case of Fe one
finds 3 channels for the majority spins, yielding G↑ = 1.24e2 /h, while for
the minority spins 3 channels contribute to G↓ = 0.70e2 /h. The total con-
ductance is 0.94 G0 and the polarization P = +28%. For the Co contact,
see Fig. 12.3(b), one finds G↑ = 0.90e2 /h and G↓ = 2.23e2 /h, summing up
to a total conductance of 1.6 G0 . The transmission is formed by 3 channels
for the majority spins (with one clearly dominant) and 6 channels for mi-
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340 Molecular Electronics: An Introduction to Theory and Experiment

Fig. 12.3 Transmission for the three single-atom contacts of Fe, Co, and Ni containing a
dimer in the central part of the contact. The geometries are shown in the upper graphs.
The distances are set to bulk distances and the atoms of the last two layers correspond
to the atoms of the leads (semi-infinite surfaces) that are coupled to the central atoms in
the model. We present the total transmission (black solid line) for both majority spins
and minority spins as well as the transmission of individual conduction channels that give
the most important contribution at Fermi energy, which is indicated by a vertical dotted
line. The channels corresponding to τ1 , τ2 , and τ3 are two-fold degenerate. Reprinted
with permission from [427]. Copyright 2008 by the American Physical Society.

nority spins and polarization is P = −42%. Finally, for the Ni contact in

Fig. 12.3(c), a single channel contributes to G↑ = 0.86e2 /h and 4 channels
add up to G↓ = 2.66e2 /h. This means that one has a total conductance of
1.8 G0 , while the current polarization adopts a value of P = −51%.
From the analysis of Fig. 12.3 and many other one-atom thick geome-
tries, the following basic conclusions were drawn in Ref. [427] concerning
the conductance of a ferromagnetic single-atom contact. First, both spin
bands contribute significantly to the transport. Second, the d orbitals give
a very important contribution to conductance of the minority spins and
they give rise to several channels (from 3 to 5 depending on the material).
Third, for the majority spins there is a smaller number of channels ranging
from 3 for Fe to 1 for Ni. This contribution is dominated by the d and
s orbitals for Fe and only by the s orbitals for Co and Ni. The relative
contribution and number of channels of the two spin species is a simple
consequence of the position of the Fermi level and the magnitude of the
spin splitting, see Fig. 12.2. In particular, notice that as we move from
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Spin-dependent transport in ferromagnetic atomic contacts 341

Fe to Ni, the Fermi energy lies more and more outside of the d band for
the majority spins, which implies that the number of channels is reduced
for this spin species. In particular, for Ni a single majority spin channel
dominates and in some sense, this material behaves as a monovalent and
a transition metal combined in parallel. Finally, the conductance values
for single-atom contacts lie typically above 1 G0 , in agreement with the
experimental results of Fig. 12.1.
The analysis of Häfner et al. [427] also put forward two additional im-
portant conclusions. First, as a consequence of the contribution of the d
bands, the value of the conductance and the current polarization are very
sensitive to the contact geometry and to disorder. Second, in the tunneling
regime one can have a much higher current polarization reaching in some
cases values close to 100%.
These ideas and conclusions can be further illustrated with the results
of Ref. [343] where conductance calculations were combined with classical
molecular dynamics simulations to determine the contact geometries. In
Fig. 12.4 we show the formation of a single-atom Ni contact containing a
dimer in its central part just before rupture. Moreover, this figure shows the
corresponding conductance and channel transmissions for both spin compo-
nents, the strain force necessary to break the contact, the spin polarization
of the current and the contact geometries. As one can see in this figure, in
the last stages of the stretching the conductance is dominated by a single
channel for the majority spins, while for the minority spin band there are
still up to 4 open channels. In particular, in the very final stages (regions
of 3 or 1 open channels for G↑ ) the spin-up conductance lies below 1.2e2 /h,
while for spin down it is close to 2e2 /h, adding up to a conductance of
around 1.2-1.6 G0 .
It is worth discussing the behavior of the spin polarization of the current,
P . Notice that at the beginning of this contact evolution it takes a value
around −40%, which is indeed close to the spin polarization of the bulk DOS
at the Fermi energy (−40.5% in these model calculations). However, as the
contact evolves, P fluctuates and even increases to positive values, which
cannot be simply explained in terms of the bulk DOS. Notice also that
P reaches the value of +80% in the tunneling regime, when the contact
is broken. Such a huge value in this regime is due to the fact that the
couplings between the d orbitals of the two Ni tips decrease much faster
with distance than the corresponding s orbitals. As a result there is a great
reduction of the spin-down conductance and in turn in a large positive value
of P .
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342 Molecular Electronics: An Introduction to Theory and Experiment

8
force (nN)

6
4
2
0
0.2 0.4 0.6 0.8

rad. of the min. cross-section (Å)

4
conductance (e /h)
2

3 (7) (6) (3)

(1)

2
G↑ (5) (4)
MCS
1

0
80
spin polarization (%)

9 60
rad. of the min. cross-section (Å)

40
8 20
0
7
conductance (e /h)

-20
6 -40
2

-60
5 (18) (17) -80
-100
0 0.2 0.4 0.6 0.8
4 elongation (nm) (7)
(6)
3 (5)
(4)
2 G↓
MCS (9) (8)
1
0
0 0.2 0.4 0.6 0.8
elongation (nm)

Fig. 12.4 Classical molecular dynamics simulation of the formation of a single-atom

Ni contact at 4.2 K ([001]-direction). The upper panel shows the strain force as a
function of the elongation of the contact. In the lower two panels the conductance G,
the MCS (minimum cross-section) radius and the channel transmissions are displayed
for the majority and minority spin components. Vertical lines separate regions with
different numbers of open channels ranging from 7 to 1 and 18 to 4, respectively. The
inset shows the evolution of the spin polarization of the current. Above and below these
graphs snapshots of the stretching process are shown. Reprinted with permission from
[343]. Copyright 2006 by the American Physical Society.
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Spin-dependent transport in ferromagnetic atomic contacts 343

As a final comment on these results, let us to point out that the contri-
bution of the minority spin component to the conductance is more sensitive
to changes in the contact geometry, as one can see in Fig. 12.4. Again, this
is a consequence of the fact that the minority spin contribution is domi-
nated by the bands arising from the d orbitals, which are anisotropic and
therefore more sensitive to disorder than the s states responsible for the
conductance of the majority spins.

12.2 Magnetoresistance of ferromagnetic atomic contacts

Another aspect of the transport through ferromagnetic atomic contacts that

has been extensively investigated in recent years is the magnetoresistance.
In this case the resistance (or conductance) of a junction is measured for
the two possible relative orientations (parallel and antiparallel) of the mag-
netization of the electrodes.5 The way to quantify the resistance change is
by means of the magnetoresistance defined as
R(AP ) − R(P )
MR = × 100%, (12.4)
R(P )
where R(AP ) is the resistance with an antiparallel orientation for the mag-
netizations in the electrodes and R(P ) is the resistance with parallel magne-
tizations. Normally, the antiparallel orientation exhibits a higher resistance
and with this definition the magnetoresistance has not upper bound and,
in particular, it can be larger than 100%.6
In the AP orientation there must be a domain wall somewhere in the
contact and it can play an important role in the resistance of the sam-
ple. For large contacts (with diameter greater than tens of nm), the main
contribution to the domain wall resistance is expected to come from the
anisotropic MR, a difference in the resistivity of a magnetic material de-
pending on whether the magnetic moment is oriented parallel or perpendic-
ular to the current.7 This contribution is relatively small, typically giving
MR values of a few percent [430].
As the contact diameter is reduced, the width of the domain wall can
be constrained by the geometry and decreases in proportion to the contact
width [431]. Eventually a new mechanism of MR may become dominant if
5 This requires to design the geometry of the magnetic electrodes so that their moments

can be controlled between reliably antiparallel and parallel configurations.

6 Other definitions are also used in the literature as, for instance, MR = [R(AP ) −

R(P )]/R(AP ) × 100%, which in the usual situations has an upper bound of 100%.
7 The anisotropic MR will be the subject of the next section.
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344 Molecular Electronics: An Introduction to Theory and Experiment

Fig. 12.5 (A) Scanning electron micrograph of a device where gold electrodes are used
to contact two permalloy thin-film magnets (inset) on top of an oxidized aluminum
gate. (B) Micromagnetic modeling showing antiparallel magnetic alignment across the
tunneling gap in an applied magnetic field of H= 66 mT. Reprinted with permission
from [438]. Copyright 2006 American Chemical Society.

a domain wall is sufficiently narrow that the spin of a conduction electron

cannot follow the direction of the local magnetization adiabatically [432].
In that case the domain wall can enhance the electron scattering in a similar
way to what happens in the giant magnetoresistance effect in magnetic mul-
tilayers [433]. For contacts approaching the single-atom diameter regime,
values of MR as large as 200% [434] to 100000% [435] have been reported,
and ascribed to a “ballistic magnetoresistance” effect involving scattering
of electrons from an atomically-abrupt domain wall. However, these large
effects have not been reliably observed in well-controlled mechanical break
junctions [404], and it has been argued that the very large changes in re-
sistance are due to the effects of magnetostriction or magnetostatic forces
that cause the contact to break and reform as the magnetic field is varied
[436].
Finally, if the contact diameter is reduced beyond the single-atom limit,
it enters the tunneling regime. MR in that regime reflects the spin po-
larization of tunneling electrons and it is also expected to depend on the
geometry of the contacts [437, 415].
As a representative example, we shall now describe the experiments of
Bolotin et al. [438]. These authors fabricated two thin-film ferromagnets
connected by a small magnetic constriction made of permalloy which can
be controllably narrowed by electromigration from about 100 × 30 nm2
to the atomic scale and finally to a tunnel junction, see Fig. 12.5. This
allowed them to study the MR as the contact region between the two fer-
romagnets is progressively narrowed in a single sample. One additional
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Spin-dependent transport in ferromagnetic atomic contacts 345

2
40
(a) (b)
∆R/RP (%)

1.5 30
20
1
10
0.5 0
-10
100 200 300 10
3
10
4
Resistance (Ω) Resistance (Ω)

Fig. 12.6 (a) Magnetoresistance as a function of resistance in the range less than 400
Ω (device I). (b) Magnetoresistance as a function of resistance in the range 60 Ω - 15 kΩ
(device II). Adapted with permission from [438]. Copyright 2006 American Chemical
Society.

advantage of this device geometry is that the magnets are attached rigidly
to a non-magnetic substrate with no suspended parts, so that the influence
of magnetostriction and magnetostatic forces on the contact are expected to
be negligible. Moreover, these experiments were conducted at low temper-
atures (4.2 K) to have the required thermal stability. Although it cannot
be taken for granted that electromigration of alloys would maintain the
stoichiometry down to the atomic scale, the magnetic properties seem not
to have changed during the final phase of the electromigration process.
Let us now summarize the main findings of this work. When the resis-
tance of a device is low (< 400 Ω), it increases smoothly as electromigration
proceeds. The cross-section of the constriction varies from 100 × 30 nm2
(60 Ω) to approximately 1 nm2 (400 Ω). In this regime small (< 3%) pos-
itive MR was found which increases as the constriction is narrowed, see
Fig. 12.6(a), as expected from the semiclassical theory of Levy and Zhang
[432]. In this theory, the resistance of the domain wall scales inversely with
its width and the MR ranges typically from 0.7% to 3% for bulk ferromag-
nets.
In the resistance range from 400 Ω to 25 kΩ, corresponding to a crossover
between transport through just a few atoms and tunneling, the value of
MR exhibits pronounced dependence on the resistance of the device, see
Fig. 12.6(b). The MR has a minimum for resistances above 1 kΩ, and
typically changes sign here to give negative values. As the resistance is
increased further into the kΩ range, the MR increases gradually to positive
values of 10-20%. The observed MR values in the point contact regime are
smaller than expected from scaling results of the semiclassical theory [432],
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346 Molecular Electronics: An Introduction to Theory and Experiment

which is not surprising since the current is transmitted through just a few
channels.
Finally, in the tunnel regime, when the resistance of a device becomes
greater than tens of kΩ, MR values in the range from -10% to a maximum
of 85% where observed. These large fluctuations clearly indicate that the
MR is sensitive to the details of the atomic structure near the tunnel gap.
The tunneling current is flowing through just a few atoms on each of the
electrodes, and the electronic structure at these atoms does not necessarily
reflect the same degree of spin polarization as in the bulk of the ferromagnet.
Experiments like the ones just described raise several basic questions,
most of them related to the role of a domain wall scattering in the mag-
netoresistance of atomic-size contacts and whether or not it can be re-
sponsible for the huge MR values reported in some experiments. These
questions have been addressed theoretically by numerous authors, see e.g.
Refs. [439–441, 409, 410, 413, 412, 414, 415]. In order to elucidate these
issues a theory should incorporate three basic ingredients: (i) a proper de-
scription of the electronic structure of ferromagnetic atomic contacts, (ii)
an adequate description of the domain wall or magnetization profiles that
can appear in atomic-scale junctions and (iii) an analysis of realistic atomic
geometries. One of the few works that meets these requirements is that of
Jacob et al. [415] in which the authors studied the magnetoresistance of Ni
atomic contacts using ab initio transport calculations. In Fig. 12.7 we repro-
duce results from this work for the transmission as a function of energy for a
single-atom Ni contact for both P and AP orientations. These results were
obtained using local spin density approximation (LSDA). In the AP case
the self-consistent magnetization reverses abruptly between the tips atoms,
i.e. this calculation confirms the possibility of having atomically-abrupt do-
main walls. However, despite this fact, the MR acquires a moderate value
of 23%, which suggests that the domain wall scattering does not account
for the large MR in Ni single-atom contacts.
The quantitative result above was found to be very sensitive to the func-
tional used in the DFT calculations, but in no case very large MR values
were found. According to the authors, the reason for the moderate MR val-
ues is two-fold. First, in the AP configuration the resistance is never too low
because of the robust contribution of the s orbitals, which is of the order of
G0 for a single-atom contact. Second, in the P configuration the resistance
never reaches the minimum value of the ballistic case because, as we saw in
the previous section, the transport in these ferromagnetic contacts is not
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Spin-dependent transport in ferromagnetic atomic contacts 347

Fig. 12.7 Conductance per spin channel in the P configuration for the model nanocon-
tact shown in the inset calculated with the local spin density approximation. (b) Same
as in (a), but for the AP configuration. Reprinted with permission from [415]. Copyright
2005 by the American Physical Society.

really ballistic8 and the d bands contribute with partially open channels.
Another interesting finding of this work is the fact that the MR in atomic
contacts can become negative, as in the experiments described above. This
shows once more that the usual classical or semiclassical arguments do not
apply to the transport in ferromagnetic atomic-size contacts.

12.3 Anisotropic magnetoresistance in atomic contacts

Lord Kelvin discovered in 1857 that the resistivity of bulk ferromagnetic

metals depends on the relative angle between the electric current and the
magnetization direction.9 The importance of this phenomenon, known as
anisotropic magnetoresistance (AMR), was recognized in the 1970’s when
AMR of a few percent at room temperature was found in a number of alloys
based on Fe, Co, and Ni. This fact stimulated the development of AMR
sensors for magnetic recording (for reviews on AMR see Refs. [389, 443]).
In the usual AMR effect, the resistivity of a ferromagnetic metal reaches
a maximum when the current is parallel to the magnetization direction, ρk ,
and a minimum when the current is perpendicular to the magnetization
8 By ballistic transport we mean a situation where all the open conduction channels
have a transmission equal to one.
9 The experimental study of the anisotropic magnetoresistance requires the application

a magnetic field high enough to saturate the magnetization of the system.

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348 Molecular Electronics: An Introduction to Theory and Experiment

direction, ρ⊥ . The magnitude of AMR can be defined by

ρk − ρ⊥
AMR = . (12.5)
ρ⊥
As a function of the angle, θ, between the current and magnetization, the
resistivity of a polycrystalline sample can often be described by
ρ(θ) = ρ⊥ + (ρk − ρ⊥ ) cos2 θ. (12.6)
The origin of AMR stems from the anisotropy of scattering produced by
the spin-orbit interaction [444]. The stronger scattering is expected for
electrons traveling parallel to magnetization, resulting in larger resistivity
ρk as compared to ρ⊥ (see Refs. [389, 443, 444] for more details).
As usual, when the dimensions of a metallic wires are shrunk to the
atomic scale, its transport properties (including its AMR) are significantly
altered. Indeed, inspired by the work of Ref. [445] on Ni contacts, Bolotin et
al. [446] investigated the AMR of permalloy electromigrated junctions and
found that it can be considerably enhanced as compared with bulk samples
and that it exhibits an angular dependence that clearly deviates from the
cos2 θ law of Eq. (12.6). These results are illustrated in Fig. 12.8. In panel
(a) one can see, as a reference, the AMR signal of a large device exhibiting
the cos2 θ-behavior. Panel (b) shows the AMR signal of a device as its
cross section is reduced approaching atomic dimensions. Notice how the
AMR signal progressively deviates from the bulk behavior. Finally, panel
(c) shows results for the tunneling regime (when the contacts are already
broken) exhibiting an amplitude of more than 10% to be compared with
typical amplitudes of the bulk samples of less than 1%.
Additionally, these authors found a significant voltage dependence on
the scale of millivolts, which led them to interpret the effect as a con-
sequence of conductance fluctuations due to quantum interference [447].
Independently, Viret and coworkers [448] reported similar results in Ni con-
tacts, but also the occurrence of conductance jumps upon rotation of the
magnetization. Similar stepwise variations of the conductance have been
found in Co nanocontacts [449], see Fig. 12.9.
These jumps have been interpreted as a manifestation of the so-called
ballistic AMR (BAMR), a concept that we now proceed to explain. In
2005 Velev et al. [450] predicted that the conductance of a ferromagnetic
ballistic conductor can change abruptly with the direction of magnetization.
This prediction was based on ab initio calculations of the electronic band
structure of infinite chains of Ni and Fe. One of those calculations for Ni
is reproduced in Fig. 12.10, where one can see the band structure of an
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Spin-dependent transport in ferromagnetic atomic contacts 349

(a)

(b) (c)

Fig. 12.8 (a) Zero-bias differential resistance vs angle of applied magnetic field at dif-
ferent field magnitudes at 4.2 K, illustrating bulk AMR for a permalloy constriction size
of 30×100 nm2 and resistance R0 = 70 Ω. The inset shows a scanning electron micro-
graph of a typical device. (b) Evolution of AMR as the device resistance R0 is increased
from 56 to 1129 Ω. (c) AMR for a device with R0 = 6 kΩ exhibiting 15% AMR, and a
R0 = 4 MΩ tunneling device, exhibiting 25% AMR. All measurements were made at a
field magnitude of 800 mT at 4.2 K. Inset in panel (b): AMR magnitude as a function
of R0 for 12 devices studied into the tunneling regime. Adapted with permission from
[446]. Copyright 2006 by the American Physical Society.

infinite chain in the absence of spin-orbit interaction [panel (a)] and in the
presence of spin-orbit interaction for magnetizations both parallel to the
chain axis [panel (b)] and perpendicular to it [panel (c)]. The key idea is
that by rotating the magnetization one can change the number of bands
crossing the Fermi energy, EF . Since in a ballistic conductor the number of
bands at EF is equal to the number of conduction channels (all of them with
perfect transparency), this change is reflected in an abrupt change of the
corresponding linear conductance. In the particular example of Fig. 12.10,
the conductance would change from 6 e2 /h to 7 e2 /h and back upon rotation
of the magnetization.
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350 Molecular Electronics: An Introduction to Theory and Experiment

Fig. 12.9 Angular dependence of conductance of Co nanocontacts. (a-d) The angle

Θ between the magnetic field and the sample plane changes from 0o to 180o . Results
for four different samples exhibiting different sign and magnitude of AMR. Reprinted
by permission from Macmillan Publishers Ltd: Nature Nanotechnology [449], copyright
2007.

The results of Fig. 12.9 definitively resemble the expected BAMR be-
havior described above, although in general the conductance jumps do not
occur between quantized values, as can easily be seen in the representative
traces However, as we have seen in previous sections, realistic ferromagnetic
contacts made of transition metals are not ballistic and thus, the interpre-
tation of the conductance jumps in terms of BAMR is at least question-
able. Indeed, Shi and Ralph [451] have suggested that these jumps might
originate from two-level fluctuations due to changes in atomic configura-
tions [452].
From the above discussion, one can see that at present there is still a
controversy about AMR in atomic contacts, concerning the origin of the
enhanced amplitude, the anomalous angular dependence, the occurrence
of conductance jumps and the voltage dependence. Different theoretical
groups have tried recently to shed new light on this problem. Thus for
instance, it has been proposed that the presence of resonant states local-
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Spin-dependent transport in ferromagnetic atomic contacts 351

Fig. 12.10 Calculated electronic structure of monoatomic Ni chain with equilibrium

interatomic distance in the absence of spin-orbit interaction (a) and in the presence
of spin-orbit interaction for magnetization lying along the wire axis M k ẑ (b) and
perpendicular to the wire axis M ⊥ ẑ (c). The solid and dashed lines in (a) show
the minority-spin and majority-spin bands, respectively. The labels stand for the irre-
ducible representation of the group C∞ν and are displayed for minority-spin bands only.
Reprinted with permission from [450]. Copyright 2005 by the American Physical Society.

ized in the electrodes near the junction break could give rise to a strong
dependence of the conductance on the magnetization direction [453, 454].
This is an appealing explanation, but as we have seen in section 12.1, the
transmission of ferromagnetic contacts is usually very smooth around the
Fermi energy on the scale of a few meV. On the other hand, Autes et al.
[455] have proposed an alternative explanation of the conductance jumps
in terms of the existence of giant orbital moments in the contacts.
More recently, Häfner et al. [456] have put forward a simple explanation
for the anomalous AMR in terms of the reduced symmetry of the atomic
contacts as compared with bulk samples. The idea goes as follows. The
AMR stems from the scattering between the s and d energy bands induced
by the spin-orbit interaction and therefore, the AMR signal may reflect
the symmetry of the lattice. In bulk samples the final signal is a result of
the average over many impurities, but in the extreme case of an atomic-
scale contact, such signal strongly depends on the local geometry. This is
particularly clear in the case of a single-atom contact where all the current
must flow through a single bond. Thus, it is not so strange to observe,
depending on the contact realization, a large amplitude or an anomalous
angular dependence as compared with bulk samples. This idea is illustrated
in Fig. 12.11, where we reproduce the results of Ref. [456]. Here, one can
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

352 Molecular Electronics: An Introduction to Theory and Experiment

Fig. 12.11 Contact evolution of a Ni junction grown in the fcc [001] direction as ob-
tained from classical molecular dynamics simulations. (a) Spin-projected, G↑,↓ , and
total conductance in the absence of spin-orbit interaction and total conductance aver-
aged over θ, φ in the presence of spin-orbit interaction. Vertical lines correspond to the
contact geometries in (b). Inset: relative AMR amplitude ∆G/hGiθ = (Gmax,θ (φ) −
Gmin,θ (φ))/hG(θ, φ)iθ vs. inverse averaged conductance. (c) Conductance vs. θ for the
geometries in (b) and with φ in steps of π/6. (d) Same as (c) for the thick contact with
324 atoms shown in this panel. Reprinted with permission from [456]. Copyright 2009
by the American Physical Society.

see the evolution of the conductance and AMR signal10 as a function of the
polar angle θ and azimuthal angle φ during the formation of a Ni atomic
contact. This formation was simulated by means of molecular dynamics
(see Ref. [456] for technical details).
In Fig. 12.11(d) one can see that in the limit of thick contacts, these
model calculations recover the bulk behavior with an AMR amplitude of
0.45%. However, in the case of small contacts, one can observe clear devi-
ations from the cos2 θ-behavior and an enhancement of the amplitude [see
Fig. 12.11(a-c)]. Notice in particular that the signal in this case also de-
pends strongly on the azimuthal angle φ, contrary to the bulk case. Finally,
the statistical analysis of the data of these simulations reveals strong fluc-
tuations in the AMR signal and an increase to 2% on average in the last
steps before breaking, see inset of Fig 12.11(a).
On the other hand, in the analysis of these realistic geometries, Häfner
et al. [456] did not find signs of BAMR or the presence of pronounced
resonances in the local density of states of the electrodes. These authors
argued finally that the voltage dependence observed in the experiments
10 In this case the AMR signal is defined in terms of the conductance, see caption of
Fig. 12.11.
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Spin-dependent transport in ferromagnetic atomic contacts 353

of Refs. [446, 452] could be a result of the combination of the intrinsic

anomalous AMR of atomic contacts with conductance fluctuations origi-
nating from impurities near the contacts [447].

12.4 Concluding remarks and open problems

As we have seen in the previous sections, the ferromagnetic atomic con-

tacts exhibit a very rich phenomenology. Moreover, in spite of progress
made in recent years, the level of understanding of the transport effects in
these systems is not comparable to the corresponding one in non-magnetic
contacts. With respect to the issues addressed in this chapter, there is an
increasing evidence (both experimental and theoretical) that the electronic
transport in these nanowires is not ballistic and phenomena like conduc-
tance quantization are not expected. Different theories show consistently
that in ordinary ferromagnetic metals, like Fe, Co or Ni, the d bands give
an important contribution to the transport, but the conduction channels
originating from these states are in general only partially open.
While the results for the properties of ferromagnetic atomic contacts
in the absence of field seem to be converging, there is not yet a similar
consensus about the magnetoresistive effects. With respect to MR, it is be-
coming clear that the huge values reported in some of the first experiments
are most likely due to magnetostriction. However, more controlled experi-
ments are needed to establish the values of the MR for different materials
as a function of the different system parameters (contact size, field, temper-
ature, etc.). From the theory side, more work is required to elucidate the
questions related to the existence and properties of domain walls in these
systems, as well as their influence in the transport characteristics. In some
cases, very sophisticated calculations have been performed for academic ge-
ometries like atomic chains, which often results in misleading conclusions
that do not apply to the systems explored experimentally.
The situation is very similar in the case of AMR of these magnetic
contacts. New experiments are needed to, in particular, identify the origin
of the abrupt steps observed in some experiments in the conductance as a
function of the angle between the magnetization and the current direction.
More theoretical calculations for realistic geometries are highly desirable to
find out the origin of the anomalous amplitude and angular dependence of
the AMR in these systems.
Finally, let us say the phenomena described in this chapter consti-
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354 Molecular Electronics: An Introduction to Theory and Experiment

tute the starting point for the investigation the spin-dependent transport
through single-molecule junctions. As we shall see in the next part of book,
experiments of that kind have been already reported. Some of them are
exploring the spin injection in molecules with the use of ferromagnetic elec-
trodes, while others investigate how the molecular magnetism is reflected
in the transport properties of molecular junctions with non-magnetic elec-
trodes.
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

PART 4

Transport through molecular

junctions

355
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

356
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Chapter 13

Coherent transport through

molecular junctions I: Basic concepts

As we have just seen in Part 3, the level of understanding achieved in the

field of metallic atomic-size contacts is certainly remarkable. However, it
is also clear that such metallic nanowires are not very “flexible” in many
respects. Thus for instance, their conductance can hardly be changed with
a gate voltage and often their current-voltage characteristics are simply
linear, which hinders the implementation of interesting electronic function-
alities. Thus, it seems natural to investigate the use of molecules as possible
building blocks of nanoscale circuits. Molecules are still small enough to
take advantage of their size, and the great variety of their physical prop-
erties make them ideal not only to mimic ordinary components of today’s
microelectronics, but also to provide new electronic functions.1 For these
reasons, the analysis of the transport properties of molecular junctions is
attracting a lot of attention and this will be the subject of the rest of this
book.
The study of the transport properties of molecular junctions constitutes
a formidable challenge. As we discussed in Chapter 3, there are still many
basic problems to be solved from the experimental side: reproducibility
of the results, stability of the contacts, external control, mass production,
etc. On the other hand, the theoretical description of the electrical con-
duction in molecular circuits is, in general, considerably more complicated
than in the case of atomic wires for various reasons. First, a molecule has
a more complicated electronic structure than an atom, simply because it
is composed of several atoms of, in general, different species. The accu-
rate description of the interaction between those atoms which leads, among
other things, to substantial charge transfer between them requires sophis-

1 The basic properties of the main molecules explored so far in molecular electronics, as
well as their possible functionalities, are described in section 3.2.

357
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358 Molecular Electronics: An Introduction to Theory and Experiment

ticated ab initio methods.2 Second, in the case of a molecular junction,

a molecule may have a weak chemical interaction with the metallic elec-
trodes, which implies that the charge carriers can spend a long time in
the molecule. This may in turn lead to the appearance of correlation ef-
fects well-known in mesoscopic physics such as the Coulomb blockade or
the Kondo effect. Third, molecules possess internal degree of freedom, in
particular vibrations modes, which can be excited by the transport elec-
trons leading to a modification of the current-voltage (I-V) characteristics.
Obviously, the probability to excite a vibration depends on various factors
like the strength of the electron-vibration interaction, the quality of metal-
molecule interfaces and, of course, the length of the molecule. Depending
on this latter factor, the vibrations can produce weak signals in the I-V
curves in the case of short molecules or they can completely dominate the
transport characteristics like in long DNA strands. Fourth, a molecule can
undergo conformation changes due to, for instance, the high electric fields
applied in the contacts, mechanical stress, an external field (electromagnetic
radiation) or the local environment (red-ox reactions).
Due to the very rich phenomenology of molecular transport junctions,
it is not an easy task to organize the existent material in the literature.
Since this is not merely a review, we shall not follow a chronological or-
der. Instead, we find didactic to organize the huge amount of results con-
cerning the physics and chemistry of molecular junctions according to the
dominant transport mechanism. Thus, we shall first discuss the coher-
ent transport through molecular wires, in which electrons flow elastically
through the molecules without exchanging energy. The main goal in this
case is to understand the relation between the electronic structure of in-
dividual molecules and the transport properties of the junctions in which
they are embedded. This discussion will be divided into two parts. In the
first one, which is covered in this chapter, we shall discuss several coherent
transport phenomena which can be understood in the light of simple toy
models or handwaving arguments. These phenomena cover issues like the
shape of the current-voltage characteristics, their temperature dependence,
their symmetry or the dependence of the conductance on the length of the
molecules. Then, in the Chapter 14, we shall address similar issues, but

2 Let us remind the reader at this stage that empirical methods like extended Hückel and

its descendents have played a fundamental role in quantum chemistry, but one cannot
expect these methods to give quantitative answers to the key questions in molecular
electronics, such as the position of the molecular levels, hybridization with the extended
states of the metallic leads, metal-molecule charge transfer, etc.
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Coherent transport through molecular junctions I: Basic concepts 359

this time from a more quantitative point of view. In particular, we shall

discuss the transport through short molecules which serve as test-beds for
molecular electronics and we shall try to establish to what extend their
electronic transport properties are quantitatively understood at present.
In Chapter 15 we shall discuss the transport through weakly coupled
molecules, where correlation effects such as the Coulomb blockade and the
Kondo effect play an essential role. Then, we present in Chapters 16 and 17
a thorough discussion of the role of vibration modes in the current through
short molecules, while the incoherent or hooping transport regime in long
molecules will be deferred until Chapter 18. Chapter 19 is devoted to the
analysis of transport properties (different from the electrical conductance)
that provide very valuable information about the transport in different
types of junctions. In particular, we address in that chapter the thermal
transport in molecular circuits. In Chapter 20 we shall discuss the optical
properties of current-carrying molecular junctions. Finally, in Chapter 21
we shall briefly mention some of the topics in molecular electronics that are
not addressed in this monograph.
We want to stress that, as the previous part of this book, the remaining
chapters have bee written in such a way that most sections are accessible
for both theorists and experimentalists. Our main goal has been to give a
didactic introduction to the basic concepts in molecular electronics, but at
the same time we have made an effort to review the most relevant contri-
butions to the different topics in this field. Let us finally say that, as stated
in the introductory part of this book, we shall mainly focus our attention
on single-molecule junctions.

13.1 Identifying the transport mechanism in single-

molecule junctions

As explained in the introduction, in this chapter we want to discuss the

coherent transport through molecular junctions. Let us stress that by co-
herent transport (or tunneling) we mean the transport regime in which the
information about the phase of the wavefunction of conduction electrons is
preserved along the molecular bridges and the inelastic interactions take
only place well inside the electrodes.3 The first question that we want to
3 In Chapter 4 we have presented an introduction to the scattering approach, which
is the most popular and appealing theoretical formalism for the description of phase-
coherent transport in nanoscale junctions. If you are not familiar with this approach, we
recommend you to read that chapter at least up to section 4.4.
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360 Molecular Electronics: An Introduction to Theory and Experiment

Table 13.1 Possible conduction mechanisms. Here, J is the current density, V is the bias
voltage, ϕB is the barrier height, d is the barrier length and T the temperature.

Conduction Characteristic Temperature Voltage

mechanism behavior dependence dependence

Direct “ √ ”
tunneling J ∼ V exp − 2d
~
2mϕB none J ∼V

Fowler-Nordheim „ √ 3/2
«
4d 2mϕB
tunneling J ∼ V 2 exp − 3q~V
none ln( VJ2 ) ∼ 1
V

Thermionic „ √ « “ ”
ϕ −q qV /4πǫd J 1
emission J ∼ T 2 exp − B k T ln T2
∼ T
ln(J) ∼ V 1/2
B

Hopping “ ” “ ”
conduction J ∼ V exp − kϕBT ln J
V
∼ 1
T
J ∼V
B

address is: How do we know that the transport in a particular junction is

coherent? Or more generally, how can we identify the transport mechanism
from the experimental results? There is no unique answer to these ques-
tions, but certainly both the shape of the I-V characteristics and, specially,
their temperature dependence are very useful in this respect. Following
the instructive work of Reed’s group (see Ref. [130]), we list in Table 13.1
some possible conduction mechanisms along with their characteristic tem-
perature and voltage dependence of the current.4 This list is by no means
complete and some other mechanisms will be discussed in later chapters,
but it constitutes a good starting point. The mechanisms listed in Table
13.1 have been extensively studied in the context of metal tunnel junctions
and semiconductor devices.
The first two conduction mechanisms, direct tunneling and Fowler-
Nordheim tunneling, are two manifestations of coherent tunneling through
a potential barrier. The explicity voltage and temperature dependence are
taken from the Simmons model that we discussed in detail in section 4.4.
Direct tunneling refers to what happens at low bias, when the voltage is
much smaller than the barrier height, whereas Fowler-Nordheim tunneling
occurs when voltage is larger than the average barrier height and it is sim-
ilar to field emission. Both mechanisms (or regimes, to be more precise)
4 This list was adapted from Ref. [457].
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Coherent transport through molecular junctions I: Basic concepts 361

have in common that the I-V’s are rather insensitive to temperature and
they only differ in the voltage dependence.
The third mechanism, thermionic emission, is a process that takes place
when the electrons are excited over a potential barrier, as opposed to tun-
neling through it. This clearly has a very strong temperature dependence,
and will become significant when the potential barrier is relatively small.
Notice that, strictly speaking, thermionic emission is also a coherent mech-
anism since the electrons proceed elastically through the barrier without
losing their phase memory.
Hopping conduction is a mechanism in which electrons are localized at
certain points within the molecule, and can hop between those points. This
will also be a thermally activated process.5 This mechanism dominates the
transport properties of long molecules, except in some remarkable cases
such as carbon nanotubes.
Based on whether thermal activation is involved, the conduction mecha-
nisms fall into two distinct categories: (i) thermionic or hopping conduction,
which has temperature-dependent I-V characteristics, and (ii) direct tunnel-
ing or Fowler-Nordheimer tunneling, which does not exhibit temperature-
dependent I-V curves. According to this slightly oversimplified discussion,
one can conclude that if the I-V curves are temperature independent, the
dominant conduction mechanism is (coherent) tunneling. Moreover, the
transport regime can be discriminated by the analysis of the shape of the
I-V characteristics. It is important to recall that most experimental tech-
niques, especially those designed to work with single molecules, are not
suitable for temperature-dependent measurements. Thus, it may not be
easy to carry out the test proposed above to elucidate the transport mech-
anism.
The working principle stated in the previous paragraph has been used
in many different investigations to establish the conduction mechanism.
In Fig. 13.1 one can see an example taken from Ref. [130]. In this case,
the authors studied the transport through thiolated alkanes of different
length using the nanopore technique (see section 3.5.1). In this experiment
the transport through a self-assembled monolayer (SAM) was investigated.
Although our main interest is on single-molecule junctions, this experiment
is specially illustrative and it will be used several times in this chapter.
As one can see in Fig. 13.1(b,c), the current is rather insensitive to the
temperature and thus it was concluded that the conduction mechanism

5 This transport mechanism will be discussed in detail in Chapter 18.

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362 Molecular Electronics: An Introduction to Theory and Experiment

(a) (b)

(c)

Fig. 13.1 (a) Schematics of a nanometer-scale device used in the experiments [130].
The structure of octanethiol is shown as an example. (b) Temperature-dependent I-V
characteristics of dodecanethiol (C12). I-V data at temperature from 300 to 80 K with
20 K steps are plotted on a logarithmic scale. (c) Arrhenius plot generated from the
I-V data in panel (b), at voltages from 0.1 to 1.0 V with 0.1 V steps. Reprinted with
permission from [130]. Copyright 2003 by the American Physical Society.

through alkanethiols is tunneling, i.e. the electronic transport is coherent.

Once it has been established that coherent tunneling is the dominant
transport mechanism, one can use, for instance, the Simmons model to un-
derstand the shape of the I-V characteristics. As we explained in section
4.4, see in particular Eq. (4.17), the current in this model is given in terms
of different parameters like the electron mass, m, the barrier width, d, the
barrier height, ϕB , and a dimensionless parameter called α. This parame-
ter is of the order of 1 for a rectangular barrier and bare electron mass. It
is sometimes used as a fitting parameter to account for the possibility of
non-rectangular barriers or an effective mass, m∗ , different from the bare
electron mass. Eq. (4.17), with ϕB and α as adjustable parameters, was
used in Ref. [130] to fit the I-V curves of different thiolated alkanes. An
example of such fits is shown in Fig. 13.2, where the I-V curve of a do-
decanethiol (C12) was fitted. The best nonlinear least-square fitted was
performed with ϕB = 1.42 eV and α = 0.65.
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Coherent transport through molecular junctions I: Basic concepts 363

Fig. 13.2 Measured current-voltage characteristics (circular symbols) of a nanopore

junction with dodecanethiols compared with calculations based on Simmons model (solid
line) using ϕB = 1.42 eV and α = 0.65. The calculated I-V from a simple rectangular
model (α = 1) with ϕB = 0.65 eV is also shown as dashed curve. Reprinted with
permission from [130]. Copyright 2003 by the American Physical Society.

In spite of the quality of the fit shown in Fig. 13.2, there are a few
things that are not very satisfactory. First, an attempt to fit the results
with a rectangular barrier fails to describe the high-bias regime, see dashed
line in Fig. 13.2. This conclusion has been drawn in several analyses of
the transport through alkanethiol [458, 459]. This is the reason why α
was used above as an adjustable parameter, although its physical meaning
is not really clear. Second, the value obtained for the barrier height is
certainly small as compared with the expectations. This height, ϕB , is
in principle the distance between the Fermi energy of the electrodes and
the nearest molecular energy level in the molecule. For the combination of
Au contacts and alkanes, this distance is expected to be between 4 and 5
eV [299]. A possible way out for these problems has been pointed out by
Akkerman et al. [460]. These authors have shown that the description of
the transport through SAMs of alkenedithiols can be improved by including
the effect of image charges in the Simmons model (see brief discussion of
the role of image charges in section 4.4). They were able to describe the
transport in their experiments up to 1 V by using a single effective mass
and a barrier height. The barrier heights found were in the order of 4-5 eV
and, irrespective of the length of the molecules, an effective mass of 0.28 m
was determined in agreement with theoretical predictions [299].
Simmons model has been used in many other examples in molecular
electronics to interpret the observed I-V characteristics. For instance, an-
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364 Molecular Electronics: An Introduction to Theory and Experiment

other beautiful example can be seen in Ref. [461], where the authors used
this model to explain the measured I-V curves in metal-molecule-metal
junctions formed from π-conjugated thiols, which were consistent with a
change in transport mechanism from direct tunneling to field emission.
Tunneling models, like Simmons one, borrowed from the field of metal-
lic tunnel junctions and semiconductor devices will continue to play an
important role in molecular electronics. However, their use is at least ques-
tionable. For instance, one may argue that one should use at least a double-
barrier model to describe a metal-molecule-metal junction since we have two
interfaces. Of course, such models are available, as we showed in Chapter
4. However, one could still argue that the bound states of a simple double-
barrier structure do not necessarily resemble those of a molecule. One could
go on trying to refine even further such barrier models, but it seems more
natural to use models that already incorporate the molecular features right
from the start. This is precisely the strategy that we are going to follow in
the rest of this chapter, where we shall introduce simple molecular-based
models to describe the transport in molecular junctions. In particular, we
shall start in the next section by studying the main conclusions that can
be drawn from the simple resonant tunneling model.

13.2 Some lessons from the resonant tunneling model

When the coherent transport through a metal-molecule-metal contact is

discussed, one typically thinks of the molecular orbitals of the molecule
within the junction. These orbitals are occupied up to the highest occu-
pied molecular orbital (HOMO), which for a characteristic molecule could
be roughly −7 eV. This has to be compared with the Fermi level of the
metal, which for a noble material is around −5 eV.6 Due to the inter-
action between the molecule and the metal electrodes, some charge flow,
charge rearrangements, and geometric reorganization will occur. After this
process, the simplest viewpoint is expressed by the level scheme depicted
in Fig. 13.3(a). Here, the Fermi energy of the electrodes lies somewhere
within the HOMO-LUMO (lowest unoccupied molecular orbital) gap of
the molecule. Moreover, due to the hybridization of the molecular orbitals
and the metallic states, the former ones acquire a finite broadening that
depends on the strength of the metal-molecule coupling, i.e. the original
molecular states have now a finite lifetime.
6 These energies are measured with respect to the vacuum level, which is set to zero.
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Coherent transport through molecular junctions I: Basic concepts 365

(a) (b)
11
00
00
11 11
00
00
11 1
0
0
1 1
0
0
1
00
11
00
11 00
11
00
11 0
1
0
1 0
1
0
1
00
11 00
11 0
1 0
1
00
11
00
11 00
11
00
11 ΓL 0
1
0
1 ε0 0
1
0 ΓR
1
00
11
00
11 00
11
00
11 0
1
0
1 0
1
0
1
00
11
00
11 00
11
00
11 0
1
0
1 0
1
0
1
00
11 00
11 0
1 0
1
00
11
00
11 00
11
00
11 0
1
0
1 0
1
0
1
00
11
00
11 00
11
00
11 L 0
1
0
1 0
1
0
1
L 00
11 00
11 R 0
1 0 R
1
00
11
00
11 00
11
00
11 0
1
0
1 0
1
0
1
00
11 00
11 0
1 0
1
00
11 00
11 0
1 0
1

Fig. 13.3 (a) Level scheme of a molecular junction. The molecule has a series of sharp
resonances corresponding to the different molecular orbitals, whereas the metal possess
a continuum of states that is filled up to the Fermi energy of the metal. (b) The same
as in panel (a) for a situation where the transport is dominated by a single level, ǫ0 .

In principle, different molecular orbitals can participate in the electron

transport simultaneously. However, there are many situations where one
level (HOMO or LUMO) lies closest to the Fermi level of the metal and
therefore dominates the transport in a certain voltage range. In this case,
the situation is better represented by the scheme of Fig. 13.3(b). This is
precisely the situation that we will be considering throughout this section.
Such situation can be described with the (single-level) resonant tunneling
model considered, for instance, in section 7.4.1.7 In this model, the level
position is denoted by ǫ0 and we measure it with respect to the Fermi en-
ergy of the electrodes, which we set to zero. At finite bias, this position
depends on the voltage applied across the junction (and on the way the
voltage drops at the interfaces) and to indicate it explicitly we shall write
ǫ0 (V ). The other key parameters of this model are the scattering rates
ΓL,R , which describe the strength of the coupling to the metal electrodes
(L, R). These parameters have dimensions of energy and they determine
the lifetime or broadening of the resonant level. Such broadening, to be
precise the half-width at half-maximum, is simply given by Γ = ΓL + ΓR .
The different parameters of the model will be considered as phenomenolog-
ical parameters, but they could in principle be obtained from a fit to the
experimental results or they can be calculated from ab initio methods.
As we have seen in the previous chapters, see Chapter 4 and section
7 The word “resonant” in the name of this model is maybe a bit misleading since it may
suggest that the transport takes place on resonance. This is actually not the case and,
as we shall see in this section, this model describes in a unified manner different regimes
within the coherent tunneling picture.
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366 Molecular Electronics: An Introduction to Theory and Experiment

7.4.1, following the spirit of the Landauer approach, the I-V characteristics
in this model can be computed from the following expression
2e ∞
Z
I(V ) = dE T (E, V ) [f (E − eV /2) − f (E + eV /2)] , (13.1)
h −∞
where the factor 2 is due to the spin symmetry of the problem, f (E) is the
Fermi function and T (E, V ) is the energy- and voltage-dependent trans-
mission coefficient given by the Breit-Wigner formula
4ΓL ΓR
T (E, V ) = . (13.2)
[E − ǫ0 (V )]2 + [ΓL + ΓR ]2
Here, the scattering rates are assumed to be energy- and voltage-
independent. This assumption can be easily relaxed, but it is usually a
good approximation for noble metals like gold with a rather flat density of
states around the Fermi energy. Notice also that we assume that the voltage
is applied symmetrically between the left and right electrode. Obviously,
this is irrelevant and the current only depends on the different of the chem-
ical potentials. The previous simple expressions will be our starting point
to discuss a few basic issues in the next subsections.

13.2.1 Shape of the I-V curves

The first obvious issue to be discussed is the shape of the I-V characteristics.
Let us assume for the moment that the voltage drops symmetrically in both
interfaces and therefore ǫ0 (V ) = ǫ0 . This is the situation expected when
the molecule is equally coupled to both electrodes (ΓL = ΓR ). In the zero-
temperature limit the integral of Eq. (13.1) can be done analytically and
the current adopts the following form
· µ ¶ µ ¶¸
2e 4ΓL ΓR eV /2 − ǫ0 eV /2 + ǫ0
I(V ) = arctan + arctan , (13.3)
h Γ Γ Γ
where Γ ≡ ΓL + ΓR . From this expression one can see that the cur-
rent at sufficiently large voltages saturates to a value given by Isat =
(2e/h)4πΓL ΓR /Γ, which one can show to be independent of the tempera-
ture. This simple result illustrates how the scattering rates determine the
order of magnitude of the current.
In order to have an idea about how the I-V curves looks like, we show
in Fig. 13.4 the current vs. bias voltage and the corresponding differen-
tial conductance (G = dI/dV ) for different values of the scattering rates
(symmetric situation), a level position of ǫ0 = 1 eV and room temperature.
Notice that the current is symmetric with respect to voltage inversion and it
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Coherent transport through molecular junctions I: Basic concepts 367

0.5
40 (a) (b)
0.4
20
I (µA)

G/G0
0.3
0
ΓL = ΓR = 0.1eV 0.2
-20 ΓL = ΓR = 0.05eV
ΓL = ΓR = 0.02eV 0.1
-40
0
-4 -2 0 2 4 -4 -2 0 2 4
V (V) V (V)
Fig. 13.4 (a) Current vs. bias voltage in the resonant tunneling model for a level position
ǫ0 = 1 eV (measured with respect to the Fermi energy of the electrodes) and at room
temperature (kB T = 0.025 eV). The different curves correspond to different values of the
scattering rates that are assumed to be equal for both interfaces. (b) The corresponding
differential conductance G = dI/dV normalized by G0 = 2e2 /h.

has a characteristic shape where one can distinguish three different regions.
We focus on the positive bias part. The first region is at low bias, when the
voltage is much smaller than |ǫ0 |, see Fig. 13.5(a). In this case the current is
quite low, specially if Γ is rather small. The second region is defined by the
resonant condition: eV /2 = ǫ0 (V ), i.e. eV = 2ǫ0 , see Fig. 13.5(b), where
the level is aligned with the chemical potential of one of the electrodes.
Here, when the voltage approaches this condition, the current is greatly
enhanced. Finally, when the voltage is larger than 2|ǫ0 | + Γ, the current
saturates to the value given by Isat obtained above, see Fig. 13.5(c).
As one can see in Fig. 13.4(b), the corresponding differential con-
ductance, G = dI/dV , exhibits two peaks at the resonant conditions

(a) (b) (c)

ε0(V)
ε0 ε0(V)
L
L
L R
R
R

Fig. 13.5 Voltage dependence of the level alignment in the resonant tunneling model
for symmetric coupling. (a) Zero bias region, (b) resonant situation where the level is
aligned with the chemical potential of one of the electrodes and (c) large bias region
where the current saturates. The level has a finite broadening given by Γ = ΓL + ΓR .
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368 Molecular Electronics: An Introduction to Theory and Experiment

(a) (b)

Fig. 13.6 (a) I-V curves of octanedithiol-based junctions measured with the appara-
tus shown in the inset. Here octanedithiol molecules are placed inside a monolayer of
octyl chains on top a gold surface. The molecules are contacted with a gold nanopar-
ticle, which in turn is contacted by the gold tip of a conducting AFM. From [125].
Reprinted with permission from AAAS. (b) I-V curves (solid lines) measured in molec-
ular junctions formed with the break junction technique at room temperature where a
trans-platinum(II) complex is contacted with gold electrodes (see inset). The curves
were fitted with a model for a rectangular barrier of height 2.5 eV (circles). Reproduced
with permission from [462]. Copyright Wiley-VCH Verlag GmbH & Co. KGaA.

eV = ±2ǫ0 . The width of these peaks is determined by the largest en-

ergy scale between Γ and kB T . In the example of Fig. 13.4, the width is
mainly determined by Γ and the conductance at low temperatures would
reach a value close to G0 at the resonant conditions. Then, in the plot of
the conductance vs. bias voltage one can read off at low temperatures the
parameter Γ, which determines the strength of the metal-molecule coupling.

13.2.2 Molecular contacts as tunnel junctions

In Fig. 13.2 one can see an example of the I-V characteristics of a molec-
ular junction that resembles those that typically are reported in tunnel
junctions. These type of curves are encountered quite frequently in the lit-
erature and we show two more examples in Fig. 13.6. The first one, see panel
(a), was obtained by measuring the current through an alkane thiol ad-layer
with a gold cluster attached to a conductive AFM tip [125]. In the second
example, see panel (b), the current through a trans-platinum complex was
measured making use of the microfabricated MCBJ technique [462]. As
discussed in the previous section, this type of curves can be described with
standard tunneling models. Indeed, in the example of Fig. 13.6(b), the
authors were able to fit quite well the I-V curves using the model of a
rectangular potential barrier of height 2.5 eV.
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Coherent transport through molecular junctions I: Basic concepts 369

0.15
6
4 (a) (b)
2 0.1
I (µA)

G/G0
0
-2 ΓL = ΓR = 0.1eV
ΓL = ΓR = 0.05eV
0.05
-4 ΓL = ΓR = 0.02eV
-6
0
-1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5
V (V) V (V)
Fig. 13.7 The same as in Fig. 13.4 for low bias (|eV | < ǫ0 (V )).

Such tunneling curves can also be described with the resonant tunneling
model. If in Fig. 13.4 we focus on the low bias regime, i.e. before the
resonant condition is reached, one obtains the current and conductance
shown in Fig. 13.7. The similarity with the experimental curves is rather
obvious and by adjusting the parameters ǫ0 and Γ, one can in principle
fit those I-V curves. Anyway, it is important to emphasize that what it
is usually called a tunnel-like curve is nothing but a cubic function of the
form: I(V ) = AV + BV 3 , where A and B are constants. Almost any
tunneling model that produces symmetric I-V curves gives rise to such a
voltage-dependence at low bias and therefore it is suitable for fitting the
I-V characteristics in this regime.8 For this reason, if the I-V curves have
no much structure, one must be careful in interpreting the fits and one
should make sure that the values of the parameters obtained from the fits
are sensible.

13.2.3 Temperature dependence of the current

As we discussed in the previous section, the temperature dependence of the
current is a key issue for identifying the transport mechanism. In particular,
we concluded that temperature-independent I-V curves are a signature of
coherent tunneling. In this subsection we shall show that this conclusion
is basically supported by the resonant tunneling model, although we shall
show that coherent tunneling can also give rise to temperature-dependent
I-V curves.
Quite generally, if the transport is coherent, the Landauer formula, see
8 We showed in section 4.4 that the I-V curves in Simmons model has this cubic depen-
dence in an intermediate voltage range.
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370 Molecular Electronics: An Introduction to Theory and Experiment

1 2
(a) 1.5 (b)
0.8 kBT = 2 meV 1 I (nA)
kBT = 10 meV 0.5
kBT = 25 meV
I (µA)

0.6 0
0 0.25 0.5 0.75 1
V (V)
0.4 30
25 (c)
20
0.2 15 G (µS)
10
5
0 0
1.5 2 2.5 1.8 1.9 2 2.1 2.2
V (V) V (V)

Fig. 13.8 (a) Current-voltage characteristics computed with the resonant tunneling
model for different temperatures. The parameter values are: ΓL = ΓR = 2 meV and
ǫ0 = 1 eV. (b) Blow-up of the low bias region. Notice that the current is independent of
the temperature. (c) The corresponding differential conductance vs. voltage.

Eq. (13.1), tell us that the temperature dependence of the current or of the
conductance is determined by the energy dependence of the transmission
coefficient, which is usually not very pronounced. Thus, the temperature
dependence in the coherent regime, if any, is typically a power law, which
is clearly at variance with, for instance, the exponential behavior in the
incoherent hopping regime that takes place in very long molecules. In the
particular case of the resonant tunneling model, it is easy to see that if the
transmission is fairly energy-independent in the energy window controlled
by the voltage, then the current is insensitive to the temperature. This is
precisely what occurs at low bias when the level lies well above (or below)
the equilibrium Fermi energy of the system. Therefore, we can conclude
that the current (and also the conductance) is temperature independent in
an off-resonant situation.
The situation changes when the transport takes place on resonance. In
this case, if the temperature is comparable or larger than Γ, the current
depends on temperature. This is illustrated in Fig. 13.8 where we show
the I-V curves and the corresponding differential conductance for temper-
atures larger than the width of the resonance. As one can see, the current
and conductance depend on temperature for voltages around the resonant
condition, while at low bias they are insensitive to its value, see Fig. 13.8(b).
To be more precise, let us now study the temperature dependence of
the conductance in the linear regime. From Eqs. (13.1) and (13.2), one can
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Coherent transport through molecular junctions I: Basic concepts 371

show that the linear conductance is given by

µ 2¶ Z ∞ · ¸
2e 1 4ΓL ΓR 1
G(T ) = dE , (13.4)
h 4kB T −∞ (E − ǫ0 )2 + Γ2 cosh2 (βE/2)
where β = 1/kB T . There are two limiting cases in which we can get a
simple analytical expression. First, if we are in an off-resonant situation,
where |ǫ0 | ≫ Γ, kB T , then the conductance is temperature independent and
it is given by
µ 2¶
2e 4ΓL ΓR
G= . (13.5)
h ǫ20
On the other hand, in a weak coupling situation (Γ ≪ kB T ) the linear
conductance can be expressed as
µ 2¶
2e πΓL ΓR 1
G(T ) = . (13.6)
h Γ kB T cosh2 (βǫ0 /2)
This means that in this limit the conductance increases as the temperature
decreases. Such temperature dependence is illustrated in Fig. 13.8(c).

13.2.4 Symmetry of the I-V curves

The symmetry of the I-V characteristics with respect to voltage inversion
has played a prominent role in the history of molecular electronics. As
we discussed in section 1.2, Aviram and Ratner suggested in their seminal
paper [8] that a single molecule with a donor-spacer-acceptor structure
would behave as a diode when placed between two electrodes.
Rectifying behavior was already observed in 1990 and 1993 by
two groups using a monolayer of hexadecylquinolinium tricyanoquin-
odimethanide sandwiched between dissimilar metal electrodes (magnesium
and platinum) [463, 464] and then confirmed later in 1997 and 2001 by
Metzger and coworkers, who used identical metals (first aluminum, then
gold) [14, 465, 466]. These papers use Langmuir-Blodgett monolayers (one
molecule thick), with maybe 1014 to 1015 molecules measured in parallel.
About nine similar rectifiers of vastly different structure have been found by
Metzger’s group between 1997 and 2006 [467]. Rectification has also been
studied at the level of single-molecule contacts, see for instance Ref. [468].
Let us see now what the resonant tunneling model can teach us about
the symmetry of the I-V characteristics. This model suggests that a possible
rectification mechanism is related to the voltage profile across the junction.
Let us consider an asymmetric situation, where the molecule is differently
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372 Molecular Electronics: An Introduction to Theory and Experiment

10

7.5 ΓL = ΓR
ΓL = 0.5ΓR
5 ΓL = 0.2ΓR
2.5 ΓL = 0.1ΓR
I (µA)

0
2
-2.5 1

-5 0
ΓL = 2 meV; ΓR = 20 meV
-1 ΓL = 20 meV; ΓR = 2 meV
-7.5
-2
-2 0 2
-10
-4 -2 0 2 4
V (V)

Fig. 13.9 Current-voltage characteristics in the resonant tunneling model for an asym-
metric situation for ǫ0 = 1 eV, ΓR = 20 meV and at room temperature (kB T = 25
meV). The different curves correspond to different values of the left scattering rate. The
inset shows very asymmetric situations where the scattering rates have bee interchanged.
Notice the that the I-V curves exhibit a clear rectification behavior.

coupled to the left and right electrodes. If the scattering rates ΓL and
ΓR are different, it is reasonable to assume that the voltage drops at the
interfaces accordingly to the ratio of the scattering rates. This can be simply
modeled by assuming that the voltage dependence of the level position is of
the form: ǫ0 (V ) = ǫ0 + (eV /2)(ΓL − ΓR )/Γ. This expression simply reflects
the fact that if one of the rates is much greater than the other, the level
follows the shift of the chemical potential of the electrode that is better
coupled.
With this simple model, we can now compute the I-V curves and an
example is shown in Fig. 13.9. Here, the different curves correspond to
different values of the ratio ΓL /ΓR . As we can see, when this ratio clearly
differs from one, the I-V curves become very asymmetric and the desired
rectification behavior becomes apparent. Notice that the polarity of the
curves can be controlled by exchanging the values of the scattering rates in
an asymmetric situation, as it is shown in the inset of Fig. 13.9.
It is easy to understand the shape of the I-V curves in Fig. 13.9. For
instance, if we focus on the situation where ΓL ≪ ΓR , the level is shifted
with the bias as ǫ0 (V ) = ǫ0 − eV /2, i.e. it follows the chemical potential
of the right electrode. Then, the resonant condition is reached for positive
voltages when the Fermi energy of the left electrode is aligned with the
level, i.e. when eV /2 = ǫ0 − eV /2, which implies ǫ0 = eV . For negative
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Coherent transport through molecular junctions I: Basic concepts 373

voltages, since the level follows the right electrode, the resonant condition
is never reached and then the current for this polarity is much lower than
for positive voltages. These arguments explain the curve in Fig. 13.9 for
ΓL = 0.1ΓR . Using similar arguments, one can easily explain the other
curves in this figure.
It is worth pointing out that the asymmetry in the coupling can be due
to extrinsic factors, like a different coupling between left and right due to
an asymmetric configuration of the molecular junction, or it can be due to
something intrinsic, like the geometry of the molecule under investigation.
Thus for instance, an asymmetric molecule has molecular orbitals with an
asymmetric charge distribution. This induces a different coupling with the
electrodes, which can lead in turn to an asymmetric voltage profile. In both
cases, the final result is the observation of asymmetric I-V curves. For an
illustrative experimental example, we refer to the reader to Ref. [469].

13.2.5 The resonant tunneling model at work

After the extensive discussion of the previous subsections about the trans-
port characteristics that can be deduced from the resonant tunneling model,
the reader may be wondering whether this model actually works. The pur-
pose of this subsection is to show that indeed it does.
The resonant tunneling model has been used by several authors to de-
scribe the experimental results in different types of molecular junctions.
Thus for instance, Grüter et al. [470] used this model to obtain information
about the tunneling rates in the transport through thiolated C60 molecules
in a liquid environment. As we shall discuss in detail in section 18.2, this
model was used by Poot et al. [471] to describe successfully the tempera-
ture dependence of I-V characteristics of three-terminal devices containing
individual tercyclohexylidene molecules.
More recently, Zotti et al. [472] have shown that the I-V curves of single
tolane molecules attached to gold electrodes via different anchoring groups
can be accurately fitted with the resonant tunneling model. In Fig. 13.10
we show typical examples of those I-Vs for three different molecules to-
gether with the corresponding fits to this model. The curves in this figure
correspond to symmetric I-Vs, but also asymmetric curves were fitted us-
ing the ideas of the previous subsection. It is worth mentioning that these
I-V curves could not be so accurately described with other models like the
Simmons one. On the other hand, as one can see in the figure caption,
the values of the scattering rate (Γ = ΓL = ΓR ) vary depending on the
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374 Molecular Electronics: An Introduction to Theory and Experiment

Fig. 13.10 I-V curves of tolane-based molecules junctions measured with the micro-
fabricated MCBJ technique at room temperature and under liquid environment [472].
The molecules investigated are shown in the upper part: 4,4′ -bisthiotolane (BTT), 4,4′ -
bisnitrotolane (BNT) and 4,4′ -biscyanotolane (BCT). The black lines in the different
panels correspond to the experimental results, while the lighter lines are the fits to the
resonant tunneling model. The parameters used in the fits of these symmetric curves
(Γ = ΓL = ΓR ) are: Γ = 42 meV and ǫ0 = 404 meV for BTT, Γ = 93 meV and ǫ0 = 271
meV for BNT and Γ = 1.8 meV and ǫ0 = 558 meV for BCT. Courtesy of Artur Erbe.

anchoring group used to bind the molecules. Let us also stress that Zotti et
al. showed by means of ab initio DFT-based calculations that the use of the
resonant tunneling model was justified. To be precise, they showed that the
transport in these molecules is indeed dominated by a single molecular or-
bital that gives rise to a Breit-Wigner resonance close to the Fermi energy.
In particular, the transport was found to be dominated by the HOMO in
the case of the thiolated molecule, while the LUMO was found to be re-
sponsible for the conduction in the other two cases with nitro and cyano
(or nitril) groups. The implications of this work for the role of anchoring
groups in the transport through molecular junctions will be discussed in
section 14.2.

13.3 A two-level model

In the previous section we have assumed that the coherent transport was
completely dominated by a single molecular level. Of course, this is not
always the case. For instance, the Fermi level may lie more or less in the
middle of the HOMO-LUMO gap and then both molecular orbitals would
contribute to the transport. In other situations, we can have other levels
very close to the HOMO or to the LUMO contributing significantly to the
transport. For these reasons, we want to refine the resonant tunneling
model to include a second level. Our goal is to learn how the conductance
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Coherent transport through molecular junctions I: Basic concepts 375

0110 11
00
00
11
10 00
11
1010 00
11
00
11
1010 00
11
00
11
10 t tH t11
00
1010 00
11
00
11
1010 ε0 ε0 00
11
00
11
10 00
11
1010 00
11
00
11
L 1010 00
11
00
11
R
10 00
11
1010 00
11
00
11
10 00
11

Fig. 13.11 Schematic representation of a two-level model where two sites with on-site
energy ǫ0 are coupled via a hopping tH . Each site is coupled to its closest electrode by
a hopping t (the same for both leads).

depends on the distance between the two levels and on the strength of the
coupling to the electrodes.
The model that we are about to describe is inspired by an important
example in molecular electronics, namely the transport through a hydrogen
molecule [569]. As we shall see in section 14.1.3, Smit and coworkers [127]
investigated the transport through hydrogen molecules with Pt contacts us-
ing the break junction technique. These authors concluded that a hydrogen
molecule can form a stable bridge between Pt electrodes and that such a
bridge has typically a conductance very close to the conductance quantum
G0 = 2e2 /h. Obviously, in this situation only two molecular levels can
participate in the transport, namely the bonding and antibonding state of
the hydrogen molecule.
With the hydrogen molecule in mind, we now proceed to analyze the
transmission in the model represented schematically in Fig. 13.11. In this
model we consider that the molecule is formed by two atoms with a single
relevant orbital per site. The on-site energy is denoted by ǫ0 and it is
assumed to be the same in both sites. The two sites are connected by a
hopping tH , while the symmetric coupling to the electrodes is described by
the hopping t. Notice that, for simplicity, we assume that the electrodes are
only coupled to its closest atom. The hopping tH is related to the splitting
between the bonding (ǫ+ ) and the antibonding state (ǫ− ) of the molecule,
namely ǫ± = ǫ0 ± tH . Thus, the HOMO-LUMO gap is simply 2tH in this
case. Obviously, within this model the conductance is made up of a single
channel because there is only one distinct path to cross the molecule.
The calculation of the zero-bias transmission is a simple exercise for
those who have followed the theoretical background (see Exercise 13.1).
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376 Molecular Electronics: An Introduction to Theory and Experiment

1 Γ = 1.0tH
(a) (b) Γ = 0.6tH
0.8 1.5

DOS(E)
Γ = 0.4tH
T(E)

0.6 Γ = 0.2tH
1 Γ = 0.05tH
0.4
0.2 0.5

0 0
-2 -1 0 1 2 -2 -1 0 1 2
(E - ε0)/|tH| (E - ε0)/|tH|
Fig. 13.12 (a) Transmission as a function of the energy for the two-level model. The
different curves correspond to different values of the scattering rate Γ. (b) Total density
of states (DOS) projected onto the molecule, i.e. the sum of the local DOS in both sites
vs. energy.

We just state here the final result that reads9

4Γ2 t2H
T (E) = . (13.7)
[(E − ǫ̃+ )2 + Γ2 ] [(E − ǫ̃− )2 + Γ2 ]
Here, ǫ̃± = ǫ0 ± tH + t2 Re{g a } are the renormalized molecular levels, g a (ǫ)
being the advanced Green function which describes the local electronic
structure of the leads. The scattering rate Γ, which determines the broad-
ening of the molecular levels, is given by Γ(ǫ) = t2 Im{g a } = πt2 ρ(ǫ), where
ρ(ǫ) is the LDOS of the metallic contacts. For the sake of simplicity, we
now assume that Γ is independent of the energy and that the levels are not
renormalized (ǫ̃± = ǫ± ). In Fig. 13.12(a) we show the transmission as a
function of energy for different values of Γ in units of tH . We also show
in Fig. 13.12(b) the corresponding total density of states (DOS) projected
onto the molecule.10 Let us recall that the linear conductance is finally
determined by the value of the transmission at the Fermi energy, which we
have not yet specified.
As one can see in Fig. 13.12(a), the energy dependence of the trans-
mission depends crucially on the ratio between the scattering rate and the
hopping tH . In a weak coupling situation, where Γ ≪ tH , the molecular
levels are clearly resolved and there is a pronounced pseudo-gap between
them. On the other hand, as Γ becomes of the order of tH , and therefore of
the order of the distance between the molecular levels, the gap is filled with
states, see Fig. 13.12(b). In this case one can reach a transmission close to
9 It is not important to understand the meaning of all the functions appearing in this

formula to appreciate the main conclusion that we want to draw.

10 This DOS is given by ρ + ρ , where πρ = Γ/{(ǫ − ǫ̃ )2 + Γ2 }.
+ − ± ±
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Coherent transport through molecular junctions I: Basic concepts 377

one even in the energy region between the two molecular states. The first
limit describes the typical situation in many organic molecules in which
the Fermi energy lies somewhere in the HOMO-LUMO gap and the broad-
ening of the levels (0.01-0.5 eV) is clearly smaller than the gap (3-8 eV).
This is the reason why most organic molecules, even those with delocalized
orbitals, are poorly conductive. The opposite limit describes the situation
that occurs in strongly coupled systems such as the hydrogen molecule [127]
and other short organic molecules coupled to transition metals [473, 474],
where the linear conductance can be as high as 1 G0 . In these cases the
strong hybridization between the molecules and the electrodes (made of
Pt) provides a broadening to the molecular levels of several electronvolts,
which is in some cases comparable to the gap of the molecules or it simply
facilitates the resonant condition for the relevant orbital for transport (see
sections 14.1.3 and 14.1.4). Thus, almost irrespective of the exact position
of the Fermi energy, the transmission reaches a value close to unity. This
is, in simple terms, the explanation for the high conductance observed in
those examples.
Another simple two-level model is that in which the transmission is
assumed to be the sum of two independent Lorentzian functions. We shall
make use such a model in section 19.3 in our discussion of the thermopower
of molecular junctions.

13.4 Length dependence of the conductance

One of the most studied issues in molecular electronics is the length de-
pendence of the conductance of molecular junctions. Typically the experi-
ments are restricted to low bias, but there are also studies of the influence
of a finite bias on this length dependence. Series of molecules like alkanes,
oligophenylenes, oligothiophenes, etc., have been extensively studied with
different techniques (see Ref. [41] for exhaustive list of references). The
most common finding is that conductance decays exponentially with the
length of the molecule, L, as
G(L) = Ae−βL , (13.8)
where the attenuation factor β depends on the particular type of molecule,
the presence of side groups, eventually on the bias voltage and not so much
on the anchoring group. Here, A is just a prefactor that determines the
order of magnitude of the conductance. Typical values of β range from 0.2-
0.4 Å−1 for conjugated molecules to 0.8-1.2 Å−1 for aromatic compounds.
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378 Molecular Electronics: An Introduction to Theory and Experiment

Fig. 13.13 Length dependence of the current through a self-assembled monolayer of

alkane thiols measured for different bias voltages with the nanopore technique. The
figure shows a log plot of the tunneling current densities multiplied by the molecular
length, which is denoted by d in this graph, at low bias and by d2 at high bias (symbols)
vs. molecular lengths. The lines through the data points are linear fittings. Reprinted
with permission from [130]. Copyright 2003 by the American Physical Society.

The exponential length dependence is expected in almost any tun-

neling model. Thus for instance, from the Simmons model (see sec-
tion 4.4) one expects at low voltages a length dependence of the type
G ∝ (1/L) exp(−βLV L), where βLV is a bias-independent decay coefficient
given by
√
2 2m √
βLV = α ϕB , (13.9)
~
where let us recall that ϕB is the barrier height, m is the electron mass and
α is a parameter that depends on the exact shape of the barrier. For higher
voltages (HV) (i.e. eV > ϕB ), the attenuation factor depends on the bias
as
µ ¶1/2
eV
βHV = βLV 1 − . (13.10)
2ϕB
We show a typical experimental example of this type length dependence
in Fig. 13.13 taken from Ref. [130]. Let us recall that in this experiment the
current through a self-assembled monolayer of alkanethiols was measured
for different bias voltages with the nanopore technique. The data corre-
spond to three different alkanethiols: CH3 (CH2 )n−1 SH with n = 8, 12, 16,
denoted as C8, C12 and C16. The current density has been normalized
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Coherent transport through molecular junctions I: Basic concepts 379

0110 0110
1010 1010
1010 1010
1010 t 2,3 t N−1,N 1010 t
1010 t L t1,2 1010 R
1010 ε2 εΝ−1 εΝ
1010
1010 ε1 1010
1010 1010
L 1010 1010 R
1010 1010
1010 1010

Fig. 13.14 Schematic representation of the bridge model to explain the exponential
length dependence of the conductance. For further explanations, see text.

following the expectation of Simmons model and as it can be seen, the fit
is satisfactory. On the other hand, in order to compare with other results
reported in the literature, the authors also performed a fit to Eq. (13.8).
They obtained a β value from 0.83 to 0.72 Å−1 in the bias range from 0.1
to 1.0 V, which is comparable to results reported previously with other
techniques [458, 475, 476].
From an atomistic point of view, the exponential length dependence
of the conductance can be understood using a simple tight-binding model,
often used in the field of electron transfer [38]. Let us briefly explain the
main idea. The model is schematically represented in Fig. 13.14. In this
model a molecular bridge formed by N sites (or segments) with on-site
energies ǫi (only one orbital per side) is coupled to two metallic leads via the
hoppings tL,R . In the bridge we only consider nearest-neighbor hoppings
denoted by ti,i+1 . Notice that this model is simply the inhomogeneous
version of the model that we have used to explain the even-odd effect in
gold atomic chains in section 11.8.
Let us briefly remind how the transmission through the molecular bridge
can be calculated.11 Using the result of the Exercise 7.5 or the general
formulas derived in section 8.1, the zero-bias transmission coefficient can
be written as

T (E) = 4ΓL (E)ΓR (E)|Ga1N (E)|2 , (13.11)

where the ΓL,R are the scattering rates determining the strength of the cou-
pling to the metallic electrodes. Usually they do not have a very significant
energy dependence and we assume here that they are constant. Moreover,
Ga1N is the (advanced) Green’s function connecting the first and last site in
11 Those readers not familiar with the Green’s function techniques described in the second
part of the book can skip this discussion and go directly to Eq. (13.14).
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380 Molecular Electronics: An Introduction to Theory and Experiment

the molecular bridge. In this sense, |Ga1N (E)|2 can be seen as the proba-
bility for an electron to propagate along the molecular wire. This function
can be calculated by taking the element (1, N ) of the following matrix (see
section 11.8)
−1
Ga (E) = [E a 1 − Hbridge − ΣaL − ΣaR ] , (13.12)
where E a = E − i0+ and Hbridge is the Hamiltonian of the molecular
bridge. Here, the only non-vanishing elements of the matrix self-energies
a
are (ΣaL )11 = t2L gL and (ΣaR )N N = t2R gR
a a
, where gL,R are the lead Green’s
functions (their exact expressions are irrelevant for our present discussion).
The scattering rates are giving by ΓL,R = t2L,R Im{gL,R a
}.
Rather than inverting exactly the previous N × N matrix, we compute
the first non-vanishing contribution to Ga1N . Obviously, this lowest-order
contribution corresponds to the sequential tunneling along the bridge with-
out any reflection. This is a good approximation to the exact expression in
the weak coupling regime, where max{ti,i+1 } ≪ min{|E − ǫi |}. Mathemat-
ically, this contribution can be written as
N −1
1 Y ti,i+1
Ga1N (E) ≈ . (13.13)
E a − ǫN i=1 E a − ǫi

For the sake of simplicity, we now assume that all bridge segments are
identical, i.e. ti,i+1 = t and ǫi = ǫ. Substituting the previous result into the
expression of the transmission, one obtains for the homogeneous bridge
¯ ¯2N
4ΓL ΓR ¯¯ t ¯¯
T (E) ≈ . (13.14)
|t|2 ¯ E − ǫ ¯
This result implies a simple form for the attenuation parameter of Eq. (13.8)
¯ ¯
2 ¯¯ E − ǫ ¯¯
β(E) = ln ¯ , (13.15)
a t ¯
where a measures the segment size, so that the bridge length is N a. Notice
that β is independent of the coupling to the leads and it is just determined
by intrinsic properties of the molecular bridge. The exponential dependence
on the bridge length is a manifestation of the tunneling character of this
process. Again, remember that the relevant energy for the linear conduc-
tance is the Fermi energy, EF . For typical values, e.g. |(EF − ǫ)/t| = 10
and a = 5 Å, Eq. (13.15) yields β = 0.92 Å−1 .
So in short, the general conclusion of our discussion is that the expo-
nential length dependence of the conductance is a signature of coherent
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Coherent transport through molecular junctions I: Basic concepts 381

tunneling in an off-resonant situation. Things may be different if the trans-

port occurs via a resonant molecular orbital. In this case, the conductance
could be length independent or at least a non-monotonic function (see Ex-
ercise 13.4). Indeed, it is easy to show that if an electron is injected within
the molecular bridge energy band, the conductance oscillates as a function
of both the injection energy and of wire length (see Refs. [255–259] and
Exercise 13.4). This is precisely the behavior found in the monoatomic
chains in section 11.8. However, this situation seems to occur very rarely
in the case of molecular junctions and it is reserved to “metallic” solid-like
molecules like the carbon nanotubes.

13.5 Role of conjugation in π-electron systems

We want to address in this section the role of the conjugation in a delo-

calized π-electron system.12 It is obvious that the “goodness” of the elec-
trical conduction in a molecular junction depends crucially on the degree
of delocalization of the molecular orbitals. After all, we have learned that
the conductance is governed, among other things, by the strength of the
metal-molecule coupling. In order to have a high current flowing through
a molecular orbital, it has to be strongly coupled to both electrodes, which
in turn implies that it has to be extended over the whole molecule. This is
the reason why conjugated molecules are believed to be good candidates for
molecular wires. A delocalized π-electron system in a conjugated molecule
can be interrupted by the introduction of adequate side groups that rotate
one part of the molecule with respect to the other. In this case the coupling
of the two subsystems, which is mainly determined by a matrix element (or
hopping) between two π-orbitals, decreases as the twist angle increases,
and eventually it vanishes when the two orbitals are exactly orthogonal at
an angle of π/2. This argument suggests a way of testing the role of the
conjugation in the conductance of a molecular junction.
A beautiful experimental illustration of this simple idea was reported by
Venkataraman and coworkers [477]. These authors investigated the trans-
port through different biphenyl molecules using an STM-based break junc-
tion technique. They studied in particular a series of biphenyl molecules
with different ring substitutions that alter the twist angle of the molecules.
They found that the conductance for this series decreases with increasing
12 This type of electron systems was discussed in section 9.5.1 using benzene as an ex-
ample.
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382 Molecular Electronics: An Introduction to Theory and Experiment

Fig. 13.15 (a) Structures of a subset of the biphenyl series studied, shown in order of
increasing twist angle or decreasing conjugation. (b) Conductance histograms of the
different molecules obtained with an STM at a bias voltage of 25 mV. (c) Position of
the peaks for all the molecules studied plotted against cos2 θ, where θ is the calculated
twist angle for each molecule, see Ref. [477] for more details. Reprinted by permission
from Macmillan Publishers Ltd: Nature [477], copyright 2006.

twist angle, consistent with a cosine-squared relation, which is expected in

transport through π-conjugated biphenyl systems, see Fig. 13.15.
Let us briefly explain how the cos2 θ-dependence comes about in these
biphenyl compounds. For this purpose, let us use the model of the previous
section. For simplicity, we assume that the bridge is composed of two
identical segments linked by a hopping t. In an off-resonant situation,
according to Eq. (13.14) the transmission is simply given by
¯ ¯4
4ΓL ΓR ¯¯ t ¯¯ 4ΓL ΓR 2
T (E) ≈ = |t| , (13.16)
|t|2 ¯ E − ǫ ¯ |E − ǫ|4
i.e. the transmission is proportional to |t|2 . Here, t is the hopping between
two π-orbitals that is simply proportional to cos θ, where θ is the angle
between them (see Exercise 13.3). Thus, we arrive at the result that the
transmission, and therefore the linear conductance, is expected to be pro-
portional to cos2 θ, as it was nicely observed in Ref. [477]. For a more
rigorous discussion of this cos2 θ-law, see Ref. [478].

13.6 Fano resonances

As we have discussed in sections 13.2 and 13.3, in most cases the coherent
transport through molecular junctions is determined by Breit-Wigner reso-
nances that originate from the different molecular orbitals. However, these
are not the only transmission line shapes that can be expected in molec-
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Coherent transport through molecular junctions I: Basic concepts 383

0110 11
00
00
11 1
1010 00
11
(a) 00
11 (b)
1010 00
11
00
11
1010 ε 00
11
00
11
0.1
1010 ΓL ΓR11
00
t 00
11

T(E)
1010 00
11
00
11 0.01
1010 ε0 00
11
00
11
t = 0.0 eV
1010 00
11 t = 0.1 eV
00
11 t = 0.3 eV
L 1010 00
11 R 0.001
00
11 t = 0.6 eV
1010 00
11
00
11
1010 00
11
00
11 0.0001
-1 0 1 2 3
E (eV)

Fig. 13.16 (a) Schematic representation of a simple that illustrates the physics of Fano
resonances. Here, the resonant tunneling model of section 13.2 is modified by introducing
an additional level (ǫ) that is coupled to the resonant level, but not to the leads. (b)
Zero-bias transmission as a function of the energy for the model of panel (a) for ǫ0 = 2.0
eV, ǫ = 0.0 eV, ΓL = ΓR = 0.1 eV and different values of the coupling t.

ular junctions. In the last years, different authors have discussed the role
of quantum interference [259, 479–482] and, in particular, Fano resonances
[483–487] in the transport through molecular contacts. It has been shown
that these phenomena can give rise of transmission line shapes that dif-
fer significantly from the standard Breit-Wigner resonances of section 13.2.
As an example, in this section we shall briefly discuss the physics of Fano
resonances in molecular wires.
In 1961 U. Fano showed that in the context of the excitation spectra of
atoms and molecules, the interference of a discrete autoionized state with a
continuum gives rise to characteristically asymmetric peaks [488], which are
nowadays referred to as Fano peaks or resonances. The appearance of this
type of resonances in transport experiments have been discussed in several
contexts in mesoscopic physics ranging from one-dimensional waveguides to
Kondo impurities. In the context of molecular junctions, a Fano resonance
can appear in the transmission, for instance, due to the interplay between
extended molecular orbitals and states that are localized in a side group of
the molecule which is decoupled from the electrodes [484].
Following Ref. [484], we shall use the toy model schematically repre-
sented in Fig. 13.16(a) to explain the origin of Fano resonances. This model
is based on the resonant tunneling model and the ingredient is the presence
of an additional site (or energy level) that represents a side group that is
not directly connected to the electrodes. The coupling to the resonant level
is given by the hopping t and the level position of this “side group” is de-
noted by ǫ. The calculation of the zero-bias transmission in this model is a
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384 Molecular Electronics: An Introduction to Theory and Experiment

simple exercise (see Exercise 13.5) and the final result reads
4ΓL ΓR
T (E) = 2 , (13.17)
[E − ǫ0 − t2 /(E − ǫ)] + Γ2
where Γ = ΓL + ΓR . This equation reduces to the Breit-Wigner formula
of Eq. (13.2) when the coupling element t vanishes. The main new feature
in this model is the appearance of an antiresonance at E = ǫ where the
transmission vanishes. This feature stems from a destructive quantum in-
terference between the direct path crossing the resonant level and a path in
which the electron “visits” the side group. Apart from this antiresonance,
the transmission exhibits two maxima at E = ǫ± , where ǫ± are given by
1n p o
ǫ± = (ǫ + ǫ0 ) ± (ǫ − ǫ0 )2 + 4t2 . (13.18)
2
In the limit t ≪ |ǫ − ǫ0 |, i.e. when the “side group” is weakly coupled to
the central backbone, the transmission exhibits a Breit-Wigner resonance
of width Γ in the vicinity of E = ǫ0 . Moreover, a Fano peak occurs near
the antiresonance (E = ǫ) separated from it by a distance of approximately
t2 /|ǫ−ǫ0 |. Thus, in this limit the hybridization with the weakly coupled side
group leads to the appearance of a peculiar asymmetric structure formed
by a peak followed by an antiresonance, which is the main fingerprint of
this phenomenon. Examples of those asymmetric line shapes are shown in
Fig. 4.73(b) in the limit of weak coupling (small t). Notice in particular
the dramatic change in the transmission that can go from 1 all the way
down to zero by changing slightly the energy. Obviously, in order to have
an impact in the transport properties, the Fano resonances needs to be
located close to the Fermi energy. If this is the case, they can give rise
to a pronounced structure in the I-V curves [480] or they can significantly
modify the thermoelectric properties of a molecular junction [487].
It is worth mentioning that an experimental situation that closely mim-
ics our simple model was reported in Ref. [489]. In this work, an artificial
quantum structure consisting of a single CO molecule adsorbed on a Au
chain was assembled by manipulating single Au atoms on NiAl(110) at 12
K with a STM. It was shown that the CO disrupts the delocalization of
electron density waves in the chain, as it suppresses the coupling between
neighboring chain atoms. In a subsequent paper, Calzoni et al. [490] showed
theoretically that the electronic properties of this system can be tuned by
the selective adsorption of small molecules. In particular, they showed that
a single CO group induces a quantum interference pattern that modulates
the electronic wave functions and modifies the coherent transport properties
of the system.
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Coherent transport through molecular junctions I: Basic concepts 385

13.7 Negative differential resistance

As explained in Chapter 1, one of the goals of molecular electronics is to

complement current Si technology. For this purpose, one must find molec-
ular systems that, at least, mimic some of today’s microelectronic compo-
nents. In this respect, one of the most studied issues in the last years is the
occurrence of negative differential resistance (NDR) in molecular junctions,
which indeed has already been observed in several systems [18, 491–499].
NDR is the key feature in the I-V characteristics of the semiconductor de-
vice known as resonant tunneling diode, which was pioneered by Esaki and
coworkers [500]. This device consists of two potential barriers in series,
the barrier being formed by thin layers of a wide-gap material like AlGaAs
sandwiched between layers of a material like GaAs having a smaller gap.
Both barriers are thin enough for electrons to tunnel through. The NDR
that occurs in this device forms the basis for practical applications as a
switching device and in high frequency oscillators [501–503].
In the context of molecular junctions, several mechanisms for NDR
have been suggested involving, for instance, charging and/or conformation
changes [18, 504–508] or polaron formation [509]. Following the philosophy
of this chapter, we are interested in the following question: Is it possible to
induce NDR simply by means of coherent tunneling processes? With our
analysis so far, based mainly on the resonant tunneling model, one might
get the impression that this is not possible. However, it is well-known that
the NDR in Esaki’s resonant diode is explained in terms of coherent trans-
port (for a didactic discussion of the essential physics of this device, see
Chapter 6 of Ref. [50]). The NDR in that device is originated from the
energy dependence of the electron injection rate (or scattering rate in our
usual language), which is due to the band structure of the semiconducting
leads. Thus, the take-home message is that coherent tunneling can lead
to NDR, but one needs to have a pronounced energy dependence of the
scattering rates.13 This is not easy to achieve with metallic leads because
they typically exhibit a rather flat density of states around the Fermi en-
ergy. An alternative is then to use of semiconductor electrodes (at least
one of them). Indeed, many experiments have demonstrated the feasibility
of attaching various organic molecules on Si substrates (see list of refer-
ences in Refs. [510, 511]). The first theoretical analysis of the transport
through metal-molecule-semiconductor junctions was carried out by Datta
13 Letus recall that so far we have always assumed that the scattering rates were energy
independent.
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386 Molecular Electronics: An Introduction to Theory and Experiment

00
11 01
00
11 10 15

(a) 00
11
00
11
00
11
1010
10 (b) 1
11
00 10
10
00
11 10 0
00
11
00
11 1010 5
00
11 ∆ 1010 Gap
-1
00
11

I (µA)
00
11 10 -2 -1 0 1 2 3
00
11 ΓR 1010
0
00
11 ΓL
00
11
00
11 ε0 1010 -5
ΓL = ΓR = 10 meV
L 00
11
00
11 1010 R ΓL = ΓR = 20 meV 2∆/e
00
11 1010 ΓL = ΓR = 30 meV
00
11
-10
00
11 1010
00
11 -15
-4 -2 0 2 4
V (V)

Fig. 13.17 (a) Schematic representation of a metal-molecule-semiconductor junction.

In equilibrium, the right electrode has a gap in the energy window E ∈ [EF , EF + ∆],
which simulates a heavily p-type doped semiconductor. The scattering rate ΓR van-
ishes inside the gap of the semiconductor. (b) I-V characteristics of the metal-molecule-
semiconductor junction of panel (a) for different values of the scattering rates and at
room temperature (300 K). The gap is ∆ = 1 eV and the level position is ǫ0 = −1
eV (measured with respect to EF ). The value of ΓR indicated in the legend refers to
value outside the gap of the semiconductor. We have assumed that the voltage drops
symmetrically at both interfaces. The vertical dotted lines indicate the voltage region
where the resonant level lies inside the gap of the right electrode. The inset shows a
blow up of the voltage region where the NDR occurs.

and coworkers [512]. These authors showed that indeed one can have NDR
in these systems by means of coherent resonant tunneling. The presence
of a semiconductor band-edge leads to NDR when the molecular levels are
driven by the external potential into the semiconducting band-gap. We
now proceed to illustrate this mechanism with a simple model.
Let us consider once more the resonant tunneling model of section 13.2.
In order to describe a metal-molecule-semiconductor junction, we now as-
sume that there is a gap in, let us say, the right electrode, see Fig. 13.17(a).
The size of this gap is denoted by ∆. In a heterojunction like this one, it is
important to describe correctly the band-bending in the semiconductor and
the overall level alignment. We shall ignore these important details, in order
to emphasize the basic conceptual issues. We assume that the equilibrium
band-alignment is as shown in Fig. 13.17(a). Here, the Fermi energy lies
near the semiconductor valence band-edge, i.e. we assume that the semicon-
ductor is heavily p-type doped. The presence of a gap in the right electrode
strongly modifies the scattering rate, which in particular vanishes inside the
gap. We model this situation by a rate, ΓR , that is constant outside the
gap region and equal to zero at energies E ∈ [EF − eV /2, EF − eV /2 + ∆].
Here, we have already taken into account the shift of the chemical potential
of the right electrode induced by the bias voltage. The energy dependence
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Coherent transport through molecular junctions I: Basic concepts 387

of ΓR is the only difference with respect to the standard resonant tunneling

model.
In Fig. 13.17(b) we show examples of the I-V characteristics obtained
with this simple model for a symmetric situation where the voltage drops
equally at both interfaces. Here, we have assumed that in equilibrium the
level lies 1 eV below the Fermi energy and the gap is ∆ = 1 eV. The most
prominent feature is the appearance of NDR (a decrease in the current) at
V = +2 V. This voltage corresponds to the bias at which the resonant level
reaches the semiconductor valence band-gap and therefore the transmis-
sion drops abruptly. Another important feature is the strong asymmetry
of the I-V’s with respect to voltage inversion. In particular, notice that
there is no NDR for negative bias. The reason is that for negative voltages
the resonant level is shifted down with respect to the chemical potential
of the right electrode and thus, it never “feels” (or reaches) the band-gap.
The I-V curves of Fig. 13.17(b) reproduce qualitatively the line shapes ob-
tained with a Hückel model and ab initio methods by Datta and coworkers
[512, 513]. These authors also pointed out that in order to see NDR at neg-
ative voltages, one would need to use n-type semiconductors (see Exercise
13.6).
The first observation of NDR through individual organic molecules on
silicon surfaces was reported by Hersam and coworkers [514]. This work re-
ported room temperature charge transport measurements performed on in-
dividual organic molecules mounted on degenerately doped Si(100) surfaces
using UHV STM. In particular, for 2,2,6,6-tetramethyl-1-piperidinyloxy
(TEMPO) molecules, NDR was observed only for negative sample bias on
n-type Si(100) and for positive sample bias on p-type Si(100). This unique
behavior is consistent with the resonant tunneling mechanism described
above. However, let us mention that the origin of the NDR in the n-type
junction is not so clear and it has been attributed to possible vibronic inter-
actions [513]. An example of the experimental results of Ref. [514] is shown
in Fig. 13.18. Since this first observation, the conditions that give rise to
electronic NDR on silicon within the coherent regime have been investigated
at length both experimentally and theoretically [510, 511, 513, 515–520].
We conclude this section by saying that the mechanism described above
is not the only possibility to obtain NDR in a situation where the transport
is mainly coherent. The electrostatic potential profile across a molecular
conductor is a key factor determining the shape of the I-V characteristics.
It has been suggested by Liang et al. [521] that a complex potential profile
might lead to NDR in a molecular junction.
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388 Molecular Electronics: An Introduction to Theory and Experiment

Fig. 13.18 Experimental observation of NDR in the transport through TEMPO

molecules on Si(100)-2 × 1 surfaces probed with STM. (A) Molecular mechanics opti-
mized structure of an individual TEMPO molecule on a truncated Si(100)-2 × 1 surface.
(B) STM topography image of isolated TEMPO molecules on a degenerately n-type
Si(100)-2 × 1 surface. (C) STM image of the isolated TEMPO molecule that is circled
in part (B). (D) I-V curves of an isolated TEMPO molecule bound to n-type Si(100).
At negative sample bias, three distinct NDR events are observed, while a shoulder is ob-
served at positive sample bias. (E) Current-voltage plot of an isolated TEMPO molecule
bound to p-type Si(100). At negative sample bias, a shoulder is observed, whereas two
NDR events are detected at positive sample bias. Adapted with permission from [514].
Copyright 2004 American Chemical Society.

13.8 Final remarks

The goal of this chapter has been to describe and illustrate some basic
concepts related to the coherent transport through molecular junctions. It
is often believed in the context of molecular electronics that the theory is
unable to reproduce the experimental observations. We hope to have shown
that this judgment is unfair. We have been able to explain qualitatively a
variety of effects by simply using toy models and handwaving arguments.
A different story is our quantitative understanding that, as we shall see in
the next chapter, is not yet that satisfactory.
We also want to stress that there are other basic issues related to the
coherent transport that we have not covered in this chapter. Probably the
most important one is the issue of the electrostatic potential profile. We
have learned in this chapter that the position of the energy levels plays
a crucial role determining the current through a molecular junction. At
finite bias the energy levels are shifted with the voltage in a way that de-
pends on the exact electrostatic profile across the junction. Therefore, the
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Coherent transport through molecular junctions I: Basic concepts 389

determination of such profile is crucial for the proper description of the

current-voltage characteristics. The theoretical analysis of the electrostatic
potential profile across atomic-scale junctions has been addressed by differ-
ent authors with a variety of methods such as simple tight-binding models
[246], model calculations based on a combination of the Schrödinger and
Poisson equations [522], a simple Thomas-Fermi-type screening model [523]
or ab initio approaches [293] mainly based on DFT (see section 10.8). For
a detailed discussion about the electrostatic potential profile in molecular
conductors we recommend Ref. [521] and references therein.

13.9 Exercises

13.1 Resonant tunneling model: Let us consider the resonant tunneling model
of section 13.2 for a symmetric situation (ΓL = ΓR = Γ/2). Calculate the current
at zero temperature up to third order in the bias voltage, i.e. determine the
relation I(V ) = AV + BV 3 at low bias and express the constants A and B as a
function of the two parameters of the model, namely Γ and ǫ0 .
13.2 Two-level model: Let us consider the two-level model of section 13.3.
(a) Use the general expressions derived in section 8.1, see Eq. (8.18) or (8.19),
to show that the zero-bias transmission is given by Eq. (13.7). Discuss also under
which conditions one recovers the expression of the transmission of the (single-
level) resonant tunneling model.
(b) Compute the I-V characteristics within the two-level model and discuss
the results. Hint: Assume that there is no voltage drop inside the molecular
bridge and that the scattering rates are independent of the energy.
13.3 The cos2 θ-law: Show that a matrix element (or hopping) between two
π-orbitals is proportional to cos θ, where θ is the angle formed by the axes of the
two orbitals. Hint: See discussion about two-center matrix elements in section
9.3.1.
13.4 Length and energy dependence of the transmission in molecu-
lar wires: In a series of papers Mujica and coworkers studied the conduction
through molecular wires using an effective tight-binding Hamiltonian (equivalent
to a Hückel model) [255–259]. They obtained the following interesting results for
the linear conductance of a molecular junction:

(1) The conductance achieves large (but bounded) values in the vicinity of any
of the wire energy eigenvalues.
(2) The conductance oscillates as a function of both injection energy and of wire
length when the electron is injected within the wire’s energy band.
(3) The conductance decreases exponentially with length when the electron is
injected outside the band of the wire.
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390 Molecular Electronics: An Introduction to Theory and Experiment

Use the model for a molecular bridge discussed in section 13.4, see Fig. 13.14,
to demonstrate the previous conclusions. For this purpose, solve the model ex-
actly inverting Eq. (13.12), rather than using perturbation theory as we did in
our discussion in section 13.4. On the other hand, model the leads as semi-infinite
linear chains and use Eq. (5.46) for the Green’s functions of the outermost atoms
of the chains that are coupled to the molecular bridge.
13.5 Fano resonances: Show that the transmission in the model introduced
in section 13.6 is given by Eq. (13.17). Hints: (i) Use the general expression of
Eq. (8.18) to show that the zero-bias transmission can be written as T (E) =
4ΓL ΓR |Ga00 (E)|2 , where Ga00 (E) is the advanced Green’s function in the resonant
level. (ii) Show that Ga00 (E) = 1/{E − ǫ0 − t2 /(E − ǫ) − iΓ}. This result leads
directly to Eq. (13.17).
Finally, investigate the impact of Fano resonances on the current through a
molecular junction by computing the I-V curves within this model. For simplicity,
assume that the system is symmetrically coupled to the leads and that the voltage
drops occur at the metal-molecule interfaces.
13.6 NDR in metal-molecule-semiconductor junctions: Using the model
of section 13.7 show that one can encounter NDR at negative voltages using
a heavily n-type doped semiconductor. For this purpose, (i) assume that in
equilibrium the Fermi level lies near the edge of the conduction band of the
semiconducting lead and (ii) assume that in equilibrium the resonant level lies
above the semiconductor gap.
13.7 Transmission of a benzene junction: The goal of this exercise is to com-
pute the transmission as a function of energy for a metal-benzene-metal junction.
Use for this purpose the Hückel approximation for the benzene molecule described
in section 9.5.1 with ǫ0 = 0 for the on-site energy of the π-orbital in each carbon
atom and t = −2.5 eV for the hopping between neighboring atoms. Assume that
the benzene molecule is coupled to the leads through a single carbon atom in each
side [e.g. atoms 1 and 4 in Fig. 9.4(a)] and describe the strength of the coupling
with a scalar and energy-independent scattering rate Γ (the same for both inter-
faces). Calculate the zero-bias transmission as a function of energy within this
model for different values of Γ. Determine also the linear conductance assuming
that the Fermi energy is EF = 0 (i.e. it lies in the middle of the HOMO-LUMO
gap of the benzene molecule) and estimate the value of Γ necessary to reach a
conductance larger than 0.1 G0 .
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Chapter 14

Coherent transport through molecular

junctions II: Test-bed molecules

In the previous chapter we have learned that the coherent transport through
molecular junctions is determined by the strength of the metal-molecule
coupling as well as by the intrinsic properties of the molecules, including
their length, conformation, the HOMO-LUMO gap and the alignment of
this gap to the metal Fermi level. Moreover, we have shown that in many
cases the experimental observations can be explained by means of very
simple qualitative arguments. In this chapter we shall go on discussing the
coherent transport in single-molecule junctions, but from a more quanti-
tative point of view. Our goal is twofold. On the one hand, we want to
calibrate our present level of understanding and for this purpose, we shall
compare different experimental and theoretical results for various test sys-
tems. On the other hand, we shall illustrate some of the basic concepts
discussed in the previous chapter in more quantitative terms.
Bearing these goals in mind, we shall discuss in the next section the
results obtained so far for some test-bed molecules of special interest in
molecular electronics. Then, we shall review recent advances in the un-
derstanding of the role of the metal-molecule interface and the efforts to
chemically tune the conductance with the use of side-groups. Moreover,
we shall briefly describe a set of controlled experiments performed with
the STM, in which the junctions are fully characterized providing thus im-
portant test systems. We shall finish this chapter with a summary of the
main conclusions and some comments about the future challenges and open
problems.
Before getting started, let us say that the current status of the under-
standing of the electronic transport through molecular junctions has been
reviewed several times in this decade. In particular, we recommend the
following articles by Lindsay and collaborators [524–529].

391
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392 Molecular Electronics: An Introduction to Theory and Experiment

Fig. 14.1 I-V characteristics and differential conductance of Au-benzenedithiol-Au junc-

tions measured with the MCBJ technique at room temperature. (A) Typical I-V curves,
which illustrate a gap of 0.7 V; and the differential conductance G(V ) = dI/dV , which
shows a steplike structure. (B) Three independent G(V ) measurements, offset for clar-
ity, illustrating the reproducibility of the conductance values. From [16]. Reprinted with
permission from AAAS.

14.1 Coherent transport through some test-bed molecules

In order to establish to what extend we understand the electronic transport

through single-molecule junctions, we shall review in this section several
representative examples related to small molecules, in which the electrical
conduction is believed to be dominated by coherent tunneling. These ex-
amples will also serve to illustrate in more detail some of the basic concepts
discussed in the previous chapter.

14.1.1 Benzenedithiol: how everything started

As we discussed in our brief review of the history of molecular electronics
in section 1.2, the experiment of Reed and coworkers in 1997 [16] is often
considered as the beginning of the field of single-molecule conduction. This
experiment was performed using a mechanically controllable break junction
(MCBJ) device working at room temperature, with the junction immersed
in a solution of the organic compound of interest. The compound that they
selected was 1,4-benzenedithiol (BDT), which has become a workhorse in
this field. In the experiment the broken gold wire was allowed to interact
with the molecules for a number of hours so that a self-assembled monolayer
covered the surface. Next, the junction was closed and re-opened a number
of times and I-V curves were recorded at the position just before contact
was lost completely. The I-V curves showed some degree of reproducibility
with a large energy gap feature of about 0.7 V, see Fig. 14.1, which was
attributed to a metal-molecule-metal junction.
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Coherent transport through molecular junctions II: Test-bed molecules 393

Transport through BDT molecules have been studied by several experi-

mental groups with different setups [530–536]. The reported values for the
linear conductance vary between 5 × 10−5 G0 and 0.1 G0 , i.e. they are scat-
tered over more than 3 orders of magnitude. From the theory side, many
authors have calculated the linear conductance of Au-BDT-Au junctions
[537–546]. The typical values lie in the range of (0.05-0.4) G0 , which in
general overestimate the observed linear conductance.
A certain level of disagreement between different experiments and differ-
ent theories might be understandable because the transport depends on the
microscopic details of the junction. Indeed, the conductance of BDT con-
tacts has been found theoretically to be strongly dependent on the bonding
site of the S atom [540, 543], while variations in the Au-S bond length only
affects the transmission function weakly [541]. However, it is difficult to
understand the differences found in the conductance histograms, where a
statistical analysis is supposed to average out the microscopic details. Thus
for instance, Xiao et al. [530] using a STM-based break-junction setup found
that the most probable value for the room temperature linear conductance
is ∼ 0.01 G0 , while Lörtscher et al. [534] reported a value of ∼ 5 × 10−5 G0
using microfabricated break junctions. Also using this latter technique,
Martin et al. [535, 536] found no distinct peaks in the histograms. More-
over, in Refs. [534, 535] I-V characteristics were reported that clearly differ
from those of the original work of Reed and coworkers [16]. Both Lörtscher
et al. [534] and Martin et al. [535] found non-linear I-Vs that were sensitive
to temperature and, although some features like the gap at low bias were
similar, there were significant differences in the magnitude of the measured
current in these experiments. The origin of these discrepancies still remains
unclear.
With respect to the theory, what is the origin of the differences be-
tween different theoretical results and why does the theory seem to over-
estimate the value of the linear conductance? There is no definite answer
to these questions, but let us try to give some ideas. Most of the calcu-
lations mentioned above are based on the combination of nonequilibrium
Green’s function techniques (NEGF) and density functional theory (DFT),
which was explained in detail in section 10.8. At this stage the techni-
cal details related to the implementation of this combination still matter
and in some cases the deficiencies in the implementation of this approach
lead to artificial results, which has nothing to do with the limitation of
the NEGF-DFT approach (for a discussion of this issue, see Ref. [545]).
On the other hand, the discrepancy between experiment and theory might
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394 Molecular Electronics: An Introduction to Theory and Experiment

be due to the intrinsic approximations made in the NEGF-DFT method.

For instance, Delaney and Greer have suggested that the problem might
lie in the insufficient description of the electronic correlation [547]. These
authors claim that by reformulating the transport problem using boundary
conditions suitable for correlated many-electron systems, one can obtain
I-V curves for BDT that are close to experimental observations. Although,
this method is not generally accepted (see Ref. [548] for severe objections
to this approach), it is quite reasonable to believe that correlations beyond
the scope of the NEGF-DFT approach play a fundamental role in molecular
junctions. The development of those theoretical methods is presently one
of the major challenges in the field.
Bearing in mind the limitations of the existent theories, let us try to give
a simple picture of the expected transport mechanism in benzenedithiol.
First of all, since this molecule is rather small and the Au-S bond is suf-
ficiently strong, one does not expect the transport in Au-BDT-Au junc-
tions to be dominated by vibronic degrees of freedom or correlation effects
like Coulomb blockade. In other words, it is reasonable to assume that
the transport in this case is coherent and therefore, it is probably deter-
mined by the electronic structure of the contact.1 With respect to the
molecule itself, it possesses an electronic structure that closely resembles
that of benzene (see section 14.1.4). In Fig. 14.2(a) we show the frontier
orbitals of this molecule, as obtained from a DFT calculation of the isolated
molecule.2 With respect to the vacuum level, the HOMO and the LUMO of
the molecule lie at -4.95 eV and -1.42 eV, respectively. It is worth stressing
that when the molecule is coupled to the electrodes, its levels are shifted
and broadened depending on the strength of the interaction with the metal.
Anyway, since the gold Fermi energy lies at approximately -5 eV and the
Au-S bond is rather strong, one naively expects a rather high conductance
dominated by the HOMO level. This picture is indeed confirmed by calcu-
lations based on the DFT-NEGF combination. Apart from the numerical
discrepancies mentioned above, these calculations show that the transport
proceeds through the tail of the HOMO of the molecule that lies at around
1 eV below the Fermi energy. An example of the zero-bias transmission as
a function of energy of an Au-BDT-Au junction taken from Ref. [545] is

1 The temperature dependence of the I-V curves in Refs. [534, 535] cannot be easily

explained within a coherent transport picture, unless such changes are related to the
thermal stability of the contacts.
2 These DFT results were obtained with the code TURBOMOLE v5.7 [575] using a split

valence polarization basis set and the BP86 exchange-correlation functional.

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Coherent transport through molecular junctions II: Test-bed molecules 395

(a)
(a) (b)
−1.42 eV LUMO
1.6 (b)
(c) WFs (0.28)
DZP (0.24)
SZP (0.21)

Transmission
1.2

0.8

−4.95 eV
0.4
HOMO
0
-6 -5 -4 -3 -2 -1 0 1 2
ε−εF (eV)

Fig. 14.2 (a) Frontier orbitals of a benzenedithiol (BDT) molecule as obtained from a
DFT calculation (see footnote 2). (b) Supercell used to model the central region of a
Au(111)-BDT-Au(111) junction with S at the fcc hollow site. (c) The calculated trans-
mission functions with two different methods and different basis sets. The transmission
at the Fermi level is indicated in the parentheses following the legends. Reprinted with
permission from [545]. Copyright 2008, American Institute of Physics.

shown in Fig. 14.2(c). In this case the leads are ideal Au(111) surfaces and
the S atoms were place at the minimum energy positions in the fcc hollow
sites. The linear conductance obtained in this case is ∼ 0.28 G0 in line with
the naive expectation and clearly higher that in the experiments.

14.1.2 Conductance of alkanedithiol molecular junctions: a

reference system
From our discussion about benzenedithiol in the previous section, one may
infer that the level of agreement between experiments, theories, and exper-
iment and theory is certainly disappointing. We shall see in this section
that the situation is definitively improving and for this purpose we shall
discuss the transport through alkanedithiols.
Alkanes3 (Cn H2n+2 ) are simple saturated chains of carbon atoms that
constitute the most popular test-bed molecules for studies of the electrical
conductance of molecular junctions in the last few years. Their chemical
stability and large HOMO-LUMO gap (see below) make them ideal for
investigating the contribution of the metal-molecule coupling to the con-
ductance. In most cases thiol groups (SH) have been attached to the ends
3 There are different types of alkanes such as branched alkanes or cyclic alkanes. Here,

we restrict ourselves to the linear alkanes. A brief discussion of the properties of these
molecules can be found in section 3.2.
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396 Molecular Electronics: An Introduction to Theory and Experiment

of alkane molecules to investigate the transport with gold electrodes, mak-

ing use of the well-known chemistry of the covalent Au-S bond.4 Trans-
port through thiolated alkanes has been studied extensively both at the
level of single molecules [125, 549–556, 535] and self-assembled monolo-
yares (SAMs), indexself-assembled monoloyares (SAM) see Refs. [130, 41]
and references therein. Furthermore, as we shall discuss later in this chap-
ter, alkanes have also been used as a platform for testing the anchoring
efficiency of different chemical groups [557, 126, 559].
It is presently acknowledged that a reliable measurement of the trans-
port properties of single-molecules junctions requires a detailed statistical
analysis. In this sense, the method introduced by Xu and Tao in 2003 [549]
has been adopted by many authors, especially in the context of STM and
break junctions. In this statistical analysis, the conductance of a molecular
contact is measured by repeatedly forming thousands of junctions. Of-
ten the corresponding conductance histograms reveal well-defined peaks at
integer multiples of a fundamental conductance value, which is typically
interpreted as the conductance of a single molecule. Xu and Tao presented
in their seminal paper [549] the first statistical results on the conductance
of alkanedithiols. In particular, they reported zero-bias resistances of 10.5
± 0.5, 51 ± 5, and 630 ± 50 MΩ for hexanedithiol, octanedithiol, and
decanedithiol. Moreover, the attenuation factor (βN ) for N-alkanedithiols
was 1.0 ± 0.1 per carbon atom and was weakly dependent on the applied
bias, which is in qualitative agreement with the values reported in SAMs
by various authors (see Refs. [130, 460] and references therein).
Despite using the same statistical analysis and comparable experimental
techniques, Xu and Tao [549] and Haiss et al. [550] obtained qualitatively
different results for both the average conductance of an N-alkanedithiol
and the length dependence. Further studies showed that the analysis of the
conductance histogram would not yield a unique trace for the conductance,
but rather several traces or peaks [551, 553]. The initial puzzling situation
is by now resolved to a large extent. Presently, different groups agree that
molecular junctions based on alkanedithiols are typically characterized by
three conductance values. These can be labeled Gl (low), Gm (medium)
and Gh (high). The authors explain that each G value corresponds to a
single molecular junction of a different type, which is characterized by the
atomic configuration at the molecule-electrode bond [551, 553, 555, 561].
Changes in the internal alkane conformation (from trans to gauche) can also
4 When a thiolated molecule is adsorbed on gold surfaces the H of the thiol terminations
desorbs and the sulfur atoms at each end bond strongly to the Au surfaces [560].
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Coherent transport through molecular junctions II: Test-bed molecules 397

Fig. 14.3 Conductance histograms of Au-octanedithiol-Au junctions measured with a

microfabricated MCBJ setup. (a) Log G-histograms built from sets of 100 G(z) traces
(figure 1(c)). A peak structure is observed when C8 is in solution in contrast to the flat,
pure solvent histogram. The broad peak in contact mode can be fitted to a Gaussian
curve (dotted line). A finer, superimposed structure is observed (black arrows). In
the non-contact mode, new peaks appear between - 4 and -3. (b) and (c) Linear G-
histograms for C8 in two different G ranges. A peak centered at 2.2 × 10−4 G0 appears
only in the non-contact mode. Reprinted with permission from [556]. Copyright 2008
IOP Publishing Ltd.

result in different conductance values [553, 555, 561]. These assumptions

are supported by several ab initio calculations that predict a significant
conductance variation upon atomic rearrangement [562, 563, 555].
The present situation has been summarized by González et al. [556].
There is good agreement among the values assigned to Gl , Gm and Gh
by different groups [551, 553, 555, 561]. Also, individual G values re-
ported in initial experiments (where only a restricted conductance range
was explored) [549, 550] are in good agreement with one of the three con-
ductance values. The important exception was until recently Gh . While
this value is reported by several groups working with STM break junctions
[549, 555, 561], no molecular signature was initially observed in that con-
ductance range in MCBJ experiments [552]. In Ref. [556] González et al.
studied the conductance of octanedithiol using a MCBJ setup and found for
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398 Molecular Electronics: An Introduction to Theory and Experiment

Table 14.1 Values of the three main peaks (low, medium and high)
of the conductance histograms of Au-alkanedithiol-Au junctions. The
integer N indicates the number of C atoms in the molecule.∗

alkanedithiol Gl (G0 ) Gm (G0 ) Gh (G0 )

N =5 2.45×10−5 8.26×10−4 -
N =6 3.16×10−5 2.58×10−4 1.22×10−3
N =8 1.14×10−5 5.68×10−5 2.71×10−4
N =9 6.06 ×10−6 2.58×10−5 1.27×10−4
N = 10 2.84 ×10−6 5.81×10−6 2.17×10−5

∗ Taken from Ref. [555].

the three peaks. The first one was found at Gl = 1.2 × 10−5 G0 ,5 which was
attributed to the conductance of a single-molecule junction. The other two
peaks appear at Gm = 4.5×10−5 G0 , and Gh = 2.3×10−4 G0 , see Fig. 14.3.
They found that the Gm has the strongest statistical weight, whereas Gh
is only observed in a non-contact mode, in which the electrodes do not get
into contact before each new molecular junction formation. They proposed
that these two values reflect the formation of several molecular junctions
in parallel between the electrodes.
Then, what is the linear conductance of Au-alkanedithiol-Au? In Table
14.1 we have reproduced the experimental results of Wandlowski’s group
obtained with STM break junctions for the three main peaks in the conduc-
tance histograms of alkanedithiol molecules of different length [555]. These
values show an exponential decay of the linear conductance as a function of
the number of C atoms (or length) for both the medium and the high peaks
with exponents of 0.94 and 0.96 per carbon atom (βN ), respectively. How-
ever, the low peak does not exhibit such an exponential decay, see Ref. [555]
for further details.
Let us discuss now how the transport takes place through alkane
molecules. As we explained in the previous chapter, the analysis of the I-
V characteristics in experiments involving alkane SAMs have shown clearly
that the transport mechanism is coherent tunneling [41, 130]. This has also
been confirmed in single-molecule experiments [557]. This is indeed what
is naively expected from the electronic structure of these carbon chains. In
Fig. 14.4 we have summarized some of the main features of such electronic
structure, as obtained from DFT-based calculations (see footnote 2). As
one can see, these molecules exhibit a very large HOMO-LUMO gap of
5 This peak was followed by several ones at multiples of Gl .
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Coherent transport through molecular junctions II: Test-bed molecules 399

2
C8
0
LUMO
+1.09 eV
LUMO 9.2

Gap (eV)
Energy (eV)
-2
8.8

-4 Au Fermi energy 8.4

8
−7.28 eV 5 10 15
-6 N
HOMO
HOMO
-8

2 4 6 8 10 12 14 16
N

Fig. 14.4 Electronic structure of alkane molecules as computed from DFT (see text).
(a) Frontier orbitals (HOMO and LUMO) of octane (C8). (b) HOMO and LUMO levels
for alkanes of different length (N is the number of carbon atoms). The dashed line
indicates the approximate position of the Fermi energy of gold. The inset shows the
HOMO-LUMO gas vs. N.

more than 8 eV. The HOMO lies around 2-3 eV below the Fermi energy (or
negative work function) of gold.6 Thus, it is reasonable to assume that the
transport in Au-alkanedithiols-Au junctions takes place through the tails of
the HOMO of these molecules. This simple picture is basically confirmed
by the existent DFT-based calculations of the linear conductance of these
junctions [562, 563, 555]. However, there are still significant discrepancies
between the different theoretical studies, as we now proceed to explain.
The DFT-based study of Ref. [555] indicates that the conductance of
these junctions strongly depends on the binding geometry. These authors
proposed values of 0.83 and 0.88 for the attenuation factor per C atom
(βN ) for the medium and high conductance peaks, respectively, which is in
fair agreement with the experimental results reported in that work. They
also indicated that these exponents are sensitive to the functional used in
the DFT calculations and differences up to 20% between functional can be
expected. On the other hand, the estimates based on a complex band struc-
ture analysis, performed by Tomfohr and Sankey [299] and by Picaud et al.
[564], suggested βN ≈ 1.0 and 0.9, respectively. However, their estimates
for the tunneling barrier (distance between the HOMO of the molecule and
the gold Fermi energy) of 3.5-5.0 eV exceed the values of Ref. [555] by
a factor of 2. Another study by Müller [563] reported a comprehensive
6 The inclusion of thiol groups at the end of the carbon chains introduces states close
to the gold Fermi energy. These states are mainly localized in the sulfur atoms and
therefore, they are not expected to play a role in the conduction, at least for long
molecules. The situation may be different in the case of short alkanes.
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400 Molecular Electronics: An Introduction to Theory and Experiment

dL 1.0 3.0 5.0 7.0 9.0

Bonds
2
15.6 Ang

Angles
60
-60
1

Transmission
-2
10

-4
10

-6
10 0 5 10 15 20
11.0 13.0 15.0 17.0 19.0 Distance [Ang]

Fig. 14.5 (left) Snapshots of the formation of an octanedithiol molecular junction sim-
ulated using DFT-based molecular dynamics. As the junction is being stretched, the
molecule migrates into the junction and pulls out a short gold chain before finally break-
ing. (right) Calculated electron transmission probability as a function of stretching
distance. The number of Au-S bonds (defined by rAu−S < 3.3 Å) and dihedral angles
(0o ∼ straight molecule; 60o ∼ gauche defect) for the S-C8 -S chain are also shown.
Reprinted with permission from [566]. Copyright 2009 American Chemical Society.

transport calculation using the TRANSIESTA package and also showed

a strong dependence of the conductance on the contact geometry. How-
ever, the obtained exponent of βN = 1.25 is in clear disagreement with the
other theoretical results. Moreover, the HOMO-LUMO gaps reported in
that study were unrealistically large (17 eV). An attempt to go beyond the
DFT approach for the conductance through alkanes was made by Fagas
et al. [565] using a configuration interaction method. Unfortunately, the
discrepancy between the obtained value, βN = 0.5, and the experimental
observation is even larger than the DFT-related uncertainty.
One of the main problems of ab initio theories in molecular electronics
is the fact that the numerical calculations are so time-consuming that at
the moment it is practically impossible to do a proper statistical analysis
of the transport properties of a molecular junction. In this sense, most the-
oretical studies are restricted to the analysis of a few idealized geometries
and the comparison of these results with the experiment should be taken
with caution. In some cases, it has been possible to perform some small
molecular dynamic simulations to get an insight into the most probable con-
tact geometries, see our discussion about hydrogen and benzene in the next
subsections. In the case of alkanes, Paulsson et al. [566] have recently re-
ported a study of the formation and conductance of alkanedithiol junctions
using DFT-based molecular dynamics. This study provides a very valu-
able insight into the formation mechanism of junctions based on thiolated
molecules and gold electrodes. This work also shows that the conductance
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Coherent transport through molecular junctions II: Test-bed molecules 401

along the last “plateau” is very sensitive to the contact geometry, and one
can observe upon stretching large variations in the conductance of an or-
der of magnitude when gauche defects are present. We show an example
of these simulations in Fig. 14.5 for octanedithiol (C8), where one can see
both the evolution of the contact geometry upon stretching and the cor-
responding transmission probability. From these simulations, the authors
constructed rudimentary conductance histograms from which they deduced
a value of βN = 1.19 and values of the conductance peaks of 2.2 ×10−3 ,
3.1 ×10−4 , and 2.0 ×10−5 in units of G0 for C6, C8, and C10, respectively.
Notice that these values tend to overestimate the experimental results of
Table 14.1 (see Ref. [566] for further details).

14.1.3 The smallest molecular junction: hydrogen bridges

Since the goal of this section is to discuss the coherent transport through
certain reference systems, it seems natural to include here the analysis
of probably the simplest molecular junction that one can think of. Smit
et al. [127] obtained molecular junctions of a hydrogen molecule between
platinum leads using the MCBJ technique. In Fig. 14.6 we reproduce some
of the results of this experiment. The inset shows a conductance curve
for clean Pt (black) at 4.2 K, before admitting H2 gas into the system.
About 10000 similar curves were used to build the conductance histogram
shown in the main panel (black, normalized by the area). After introducing
hydrogen gas the conductance curves were observed to change qualitatively
as illustrated by the gray curve in the inset. The dramatic change is most
clearly brought out by the conductance histogram (gray, hatched). Clean
Pt contacts show a typical conductance of 1.5 ± 0.2 G0 for a single-atom
contact, as it can be inferred from the position and width of the first peak
in the Pt conductance histogram. Below 1 G0 very few data points are
recorded, since Pt contacts tend to show an abrupt jump from the one-
atom contact value into the tunneling regime towards tunnel conductance
values well below 0.1 G0 . In contrast, after admitting hydrogen gas a lot of
structure is found in the entire range below 1.5 G0 , including a pronounced
peak in the histogram near 1 G0 .
Apart from the simplicity of the hydrogen molecule, what makes this
system so special is the thorough characterization of these junctions that
was carried out both in the original work and in subsequent papers. The
information gathered in the different works can be summarized as follows:
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402 Molecular Electronics: An Introduction to Theory and Experiment

Fig. 14.6 Conductance curves and histograms for clean Pt, and Pt in a H2 atmosphere.
The inset shows a conductance curve for clean Pt (black) at 4.2 K recorded with a bias
voltage of 10 mV, before admitting H2 gas into the system. About 10000 similar curves
are used to build the conductance histogram shown in the main panel (black). After
introducing hydrogen gas the conductance curves change qualitatively as illustrated by
the grey curve in the inset, recorded at 100 mV. This is most clearly brought out by
the conductance histogram (grey; recorded with 140 mV bias). Reprinted by permission
from Macmillan Publishers Ltd: Nature [127], copyright 2002.

• The presence of the molecules was confirmed with the signatures of

vibration modes at energies between 40-70 meV in I-V characteristics
[127]. Such signatures cannot be attributed to Pt, which has a Debye
energy of around 20 meV.
• The shift in the vibrational energies upon isotope substitution of H2
by D2 and HD confirmed that the modes were indeed associated to a
hydrogen molecule.
• Upon stretching of the contacts, the energy of the lowest modes in-
creased, which indicates that these modes are transverse ones [567].
• The analysis of the conductance fluctuations [127] and especially shot
noise measurements showed that the conductance in the range of 1 G0
is largely dominated by a single channel [568].

The physics behind the signatures of vibration modes and the impor-
tance of transport properties like the shot noise will be discussed in detail
in subsequent chapters, where we shall come back to this example.
All the observations detailed above offer a very stringent test to the the-
ory, which should explain consistently all the experimental results. Before
reviewing the existent work in the literature, it is interesting to discuss the
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Coherent transport through molecular junctions II: Test-bed molecules 403

1.4 8 k-points
1.2 Γ-point
Transmission

1
0.8

Fermi level
0.6
0.4
0.2
0
-8 -6 -4 -2 0 2 4
Energy (eV)

Fig. 14.7 Calculated transmission for the molecular hydrogen contact shown in the in-
set. For comparison both the k-point sampled transmission and the Γ-point transmission
are shown. The wide plateau with T ≈ 1 extending across the Fermi level indicates a
single, robust conductance channel with nearly perfect transparency. Reprinted with
permission from [571]. Copyright 2005 by the American Physical Society.

naive expectation for the conductance of a hydrogen molecule. Obviously,

in the transport through this molecule, only the bonding and antibonding
states (formed by the hybridization of the 1s orbitals of the H atoms) can
contribute. A typical DFT calculation yields a HOMO-LUMO gap of about
10.5-11 eV, depending on the functional used, and the HOMO turns out
to be located at ∼ −10 eV with respect to the vacuum level. These results
are for the equilibrium geometry, where the H-H distance is equal to 0.74
Å. This means that Fermi energy of Pt lies more or less in the middle of
the gap. Thus, one might naively think that in view of the huge gap, H2
should be poorly conductive, in clear contrast to the experiments. We shall
see below that this naive picture fails because the molecule is significantly
distorted in the Pt junction.
Different authors have studied theoretically the conductance of Pt-H2 -
Pt junctions [127, 569–572]. The most satisfactory explanation so far has
been proposed by Thygesen and Jacobsen [571], who presented conductance
calculations based on density functional theory (DFT) showing that a hy-
drogen molecule bridging a pair of Pt contacts can have a conductance close
to 1 G0 . In Fig. 14.7 we reproduce the main result of Ref. [571], where one
can see the transmission as a function of energy for the junction shown in
the inset. Notice in particular the presence of a plateau with a transmission
close to 1 in an energy window of 4 eV around the Fermi level, suggesting
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404 Molecular Electronics: An Introduction to Theory and Experiment

the existence of a single conduction channel with nearly perfect transmis-

sion. The geometry of the inset was fully optimized and it is characterized
by the bond lengths dH−H = 1.0 Å and dPt−H = 1.76 Å. This means that
in this stable configuration the molecule has been largely deformed and,
in particular, the HOMO-LUMO gap is significantly reduced with respect
to its value in vacuum. On the other hand, the authors showed that the
vibration modes of the hydrogen molecule in this configuration are in fair
agreement with the experimental results [567].
With respect to the physical mechanism, by performing a Wannier func-
tion analysis, they could establish that the transport is dominated by the
antibonding state of the molecule. In particular, the transmission plateau
in Fig. 14.7 is a result of a strong hybridization between the H2 antibonding
state and a combination of d- and s-like orbitals located on the neighboring
Pt atoms. The antibonding orbital was found to be 0.1 eV above the Fermi
energy (EF ), while the bonding orbitals lied 6.4 eV below EF . Moreover,
a coupling matrix element of 1.9 eV between the antibonding state and
the leads was obtained. Therefore, the conclusion is that one has resonant
transport through the antibonding orbital that has been largely broadened
due to the strong hybridization with the Pt electrodes.
Other DFT calculations have been performed [569, 570, 572]. Using a
slightly different approach Garcı́a et al. [570] obtained a conductance well
below 1 G0 . They propose an alternative atomic arrangement to explain
the high conductance for the Pt-H bridge, consisting of a Pt-Pt-bridge
with two H atoms bonded to the sides in a perpendicular arrangement.
However, this configuration gives rise to three conduction channels, which is
excluded based on the analysis of shot noise and conductance fluctuations as
discussed above. The origin of this discrepancy is still unclear (see Ref. [545]
for some ideas).
Let us conclude this discussion by saying that conductance histograms
recorded using Fe, Co and Ni electrodes in the presence of hydrogen also
show a pronounced peak near 1 G0 [403], indicating that many transition
metals may form similar single-molecule junctions. Also Pd seemed a good
candidate, but Csonka et al. [573] find an additional peak at 0.5 G0 in the
conductance histogram, and it was argued that hydrogen is incorporated
into the bulk of the Pd metal electrodes.
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Coherent transport through molecular junctions II: Test-bed molecules 405

14.1.4 Highly conductive benzene junctions

As we shall discuss in detail in the next section, the most common approach
in fabrication of molecular junctions utilizes functional side groups attached
to the main molecule structure as anchoring “arms” that chemically bind to
metallic leads (e.g. thiol [16], amine [126] and carboxylic [557] groups). In
many cases anchoring groups act as resistive spacers between the electrodes
and the molecule. This leads to low conductance and sensitivity to differ-
ent environmental effects such as neighbor adsorbed species [574]. In order
to overcome these problems, Ruitenbeek and coworkers [473] have recently
reported on a highly conductive molecular junction achieved by direct bind-
ing of a π-conjugated organic molecule (benzene) to metallic electrodes (Pt)
without the use of anchoring groups. Again, the thorough analysis of the
transport properties through these junctions makes Pt-benzene-Pt contacts
a nice test system. In this sense, the goal of this section is to briefly describe
the work of Ref. [473].
The measurements were performed using the MCBJ technique and they
were conducted at 4 K. Following the formation of the Pt junction, the
benzene was admitted using a leak valve via a heated capillary to the Pt
junction while the latter is broken and formed repeatedly. During the
benzene introduction, the typical Pt peak is observed to be suppressed, and
a single peak appears near 1 G0 accompanied with a low conductance tail
(Fig. 14.8, filled curve). In some cases, the histogram exhibits a peak near
0.2 G0 on top of the tail. These findings imply that after the introduction
of benzene, the formation of pure Pt junctions is suppressed while new
junctions with preferred conductance of 1 G0 and sometimes 0.2 G0 are
formed while stretching the contact.
Following the spirit of the experiments on hydrogen, the presence of
the molecule was identified by vibrational spectroscopy that revealed a
well-defined mode at around 42 meV in the zero-bias-conductance region
of 0.05-0.4 G0 , which was rather insensitive to stretching of the contact.
On the other hand, shot noise measurements showed that the number of
channels is eventually reduced to one when the conductance is reduced to
0.2 G0 , while at higher conductance (also well below 1 G0 ) multiple channels
make up the transport across the junction.
What is the naive expectation for the conductance of a benzene
molecule? As we discussed in section 9.5.1, the electronic structure of ben-
zene is determined by a delocalized π-orbital system formed by 6 π-orbitals
(the p-orbitals pointing out of the benzene plane), one per C atom. This
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406 Molecular Electronics: An Introduction to Theory and Experiment

Fig. 14.8 Conductance histograms for a Pt junction (black), and for Pt after introducing
benzene (filled) measured with the MCBJ technique. Each conductance histogram is
constructed from more than 3000 conductance traces recorded with a bias of 0.1 V during
repeated breaking of the contact. Reprinted with permission from [473]. Copyright 2008
by the American Physical Society.

simple picture is confirmed by DFT calculations (see footnote 2), which

also predict a HOMO-LUMO gap of 5.14 eV with the two-fold degenerate
HOMO lying at -6.26 eV (measured with respect to vacuum), see Fig. 9.4.
Taking into account that the Fermi energy (or negative work function) of Pt
is around -5.4 eV, one naively expects that if there is no substantial charge
transfer, the transport must be dominated by the HOMO. With respect to
the value of the conductance, it will depend crucially on the strength of the
metal-molecule coupling (see Exercise 13.7).
The authors of Ref. [473] performed DFT structural simulations to de-
termine the contact geometry and conductance calculations based on the
method detailed in Ref. [576]. Their main conclusions are: (i) benzene can
indeed form a stable bridge between Pt contacts with a conductance as high
as 1 G0 and (ii) stretching of the junction leads to tilting of the molecule
which reduces both the conductance and the number of transmission chan-
nels across the junction as a consequence of sequential breaking of the Pt-C
bonds. The main take-home message from the theoretical analysis is that
the high conductance can be attributed to the strong hybridization of the
benzene molecule with the Pt contacts. This can be seen in Fig. 14.9, where
we show both the transmission and density of states (DOS) projected onto
the frontier molecular orbitals as a function of energy for two geometries at
different stages of the stretching process. Notice that when the conductance
is close to 1 G0 (see left panels), the transport is dominated by the HOMOs
of the molecule, which are no longer degenerate because of the different
coupling to the Pt electrodes. In this geometry, the conductance is due
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Coherent transport through molecular junctions II: Test-bed molecules 407

HOMO1 HOMO2 LUMO1 LUMO2

o Ttot o
4.9 A 6.5 A
2 T1 1.2
T2 1
Transmission

1.5
T3 0.8
1 0.6
0.4
0.5
0.2
0 0
LDOS (1/eV)

HOMO1
0.4 HOMO2 0.4
LUMO1
LUMO2
0.2 0.2

0 0
-10 -8 -6 -4 -2 0 -10 -8 -6 -4 -2 0
E (eV) E (eV)
Fig. 14.9 Transmission and density of states (DOS) as a function of energy in a Pt-
benzene-Pt junction as calculated in Ref. [473]. The left panels show the results for
a geometry where the outermost Pt atoms were separated a distance of 4.9 Å, while
the right ones show the corresponding results for a separation of 6.5 Å. The contact
geometries are shown on top of these panels. The transmission plots show both the total
transmission and its decomposition into individual transmission coefficients, Ti . The
local DOS has been projected onto the four benzene frontier orbitals, which are shown
in the upper part of the figure. The vertical dashed lines indicate the position of the
Fermi energy (-5.4 eV). Courtesy of Sören Wohlthat.

to two channels and the frontier orbitals of the benzene acquired a large
broadening due to the strong interaction with the metallic leads. When
the elongation of the contact proceeds, the reduction of the metal-molecule
coupling becomes apparent in the transmission curve with the appearance
of a pseudo-gap around the Fermi energy (see right panels in Fig. 14.9).
In this case, the transport is dominated by a single conduction channel.
The reason for this is not really obvious from the information of the local
DOS. Then, what determines the number of channels in this case? As a
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408 Molecular Electronics: An Introduction to Theory and Experiment

rule of thumb, an upper limit to the number of conduction channels when

the molecular junction is formed, is simply given by the number of C atoms
bonded to the Pt tip atoms. This can be understood as follows. Since each
C atom has only one orbital taking part in the π-system of the benzene
ring, each C atom can build at most one π-channel. This is nothing else
than another example of the simple rule that we discussed in the context
of the conductance of a single-atom contact (see section 11.5).

14.2 Metal-molecule contact: The role of anchoring groups

As we discussed in the previous chapter, one of the fundamental ingredients

that determines the coherent transport through molecular junctions is the
strength of the metal-molecule coupling. This strength can be tuned chem-
ically, at least up to certain degree, by using appropriate anchoring groups
to bind a molecule to metallic electrodes. How to choose the linker group?
The choice depends primarily on the type of metal-molecule combination
used to build the junctions and usually only a few anchoring groups are
possible. On the other hand, the choice also depends on the functionality
that one wants to implement in the system. If the goal is to achieve a high
conductance, then the anchoring group is chosen to maximize the strength
of the metal-molecule coupling.7 Other important factors to bear in mind
are the stability of the contact and the variability of the bonding between
the terminal group and the metal, which can play a fundamental role in
the reproducibility of the experimental results.
The majority of candidates for end-group/metal pairings for molecular
electronics come from studies of self-assembled monolayers (SAMs) [577],
such as thiolated molecules on gold surfaces. The combination of thiol as
an end group and gold electrodes is by far the most studied metal-molecule
binding motif in molecular electronics so far. In the previous section we have
given several examples of this combination. Lately, it has been argued that
the variability in the bonding between thiol groups and gold may be harm-
ful for the reliability of electrical measurements on single molecules [126].
For this reason, different alternatives are currently being explored in many
laboratories. An interesting possibility was put forward by Venkataraman
and coworkers in Ref. [126], where the authors suggested the use of amine
7 A strong coupling is not always the goal. In some cases, one may want to partially

decouple the molecule from the leads, like in the case of the molecular transistors (see
next chapter).
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Coherent transport through molecular junctions II: Test-bed molecules 409

no molecules 1,4−benzenediamine 1,4−benzenedithiol 1,4−benzenediisonitrile

{
{
{
{

Fig. 14.10 (a) Sample conductance traces measured with STM Au break junc-
tion without molecules and with 1,4-benzenediamine, 1,4-benzenedithiol, and 1,4-
benzenediisonitrile shown on a semilog plot. (b) Conductance histograms constructed
from over 3000 traces measured in the presence of the three molecules shown on a log-log
plot. The control histogram of Au without molecules is also shown. Inset: same data on
a linear plot showing a Gaussian fit to the peak (black curve). Adapted with permission
from [126]. Copyright 2006 American Chemical Society.

(NH2 ) groups to obtain well-defined values of the conductance of molecular

junctions. In this work, the conductance of amine-terminated molecules was
measured by breaking Au atomic contacts in a molecular solution at room
temperature. It was found that the variability of the observed conductance
for the diamine molecule-Au junctions is much less than the variability for
diisonitrile- and dithiol-Au junctions. This narrow distribution enabled the
authors to unambiguously determine the conductance of single molecules.
The conductance histograms obtained in Ref. [126] for three differently
substituted aromatics, 1,4-benzenedithiol, 1,4-benzenediisonitrile, and 1,4-
benzenediamine, are shown in Fig. 14.10(b). Notice that in comparison
to the data for the dithiol or the diisonitrile, the conductance histogram
for 1,4-benzenediamine is particularly well-defined. From this histogram, a
conductance value for this molecule of 0.0064 ± 0.0004 G0 was deduced.
With the help of DFT-based calculations, it was suggested in Ref. [126]
that the reproducible electrical characteristics result from the selective bind-
ing between the gold electrodes and amine link groups through a donor-
acceptor bond to under-coordinated gold atoms. The amine end groups
have been used by Venkataraman and coworkers to study the transport
through alkanes [558], to analyze the role of the conjugation in the trans-
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410 Molecular Electronics: An Introduction to Theory and Experiment

port through biphenyl molecules [477] and to establish a detailed com-

parison with theory [578]. For more information, see Ref. [579], which
provides a comprehensive review of single-molecule junction conductance
measurements across families of molecules measured while breaking gold
point contacts in a solution of molecules with amine end groups.
Tao and coworkers have systematically studied and compared the
single-molecule conductance of alkanes terminated with dicarboxylic-acid
(COOH), diamine, and dithiol anchoring groups [557]. The conductance
values of these molecules were found to be independent of temperature,
indicating coherent tunneling. For each anchoring group, the authors re-
ported an exponential decay of the conductance with the molecular length,
given by G = A exp(−βN N ), which also suggests the tunneling mecha-
nism. The prefactor of the exponential function, A, a measure of contact
resistance, turned out to be highly sensitive to the type of the anchoring
group, which varies in the order Au-S > Au-NH2 > Au-COOH. This de-
pendence was attributed to the different coupling strengths provided by
the different anchoring groups between the alkane and the electrodes. On
the other hand, with respect to the spread of the peaks in the conduc-
tance histograms, there were no significant differences between thiols and
amines. Something similar has also been reported by Martin et al. [535].
Using microfabricated gold break junctions, these authors measured the
conductance histogram for benzenediamine. In contrast to Ref. [477], they
did not find a pronounced peak structure. According to these authors, the
difference may be due to the absence of a solvent in their experiment and
also to the fast rupture of the metal-molecule bond that must have reduced
the probability of forming stable molecular junctions.
From the previous discussion it is obvious that the conductance values
are not necessarily correlated with the selectivity of the binding that leads
to narrow peaks in the conductance histograms. It would be desirable to
find linker groups with the properties of amines, but providing a stronger
coupling. With this goal in mind, Park et al. [559] have compared the
low bias conductance of a series of alkanes terminated on their ends with
dimethyl phosphines, methyl sulfides, and amines and found that junctions
formed with dimethyl phosphine terminated alkanes have the highest con-
ductance. Furthermore, they observed a clear conductance signature with
these linker groups, indicating that the binding is well-defined and electron-
ically selective.
As we discussed in a previous section devoted to the transport through
benzene molecules, an interesting possibility is the use of other metals than
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Coherent transport through molecular junctions II: Test-bed molecules 411

0
10 (a) (b)
120

100
-1
10
/h)

counts (ms/trace)
2
ctance (2e

10
-2 80
BDC60

60
10
-3 BDA
40
condu

-4
10
20 BDT

10
-5
0 clean solv ent
0 5 10 15 20 25 30 35 10
-5
10
-4
10
-3
10
-2
10
-1
10
0

ctance (2e
2
time (s) condu /h)

Fig. 14.11 (a) Conductance traces of 1,4-bis(fullero[c]pyrrolidin-1-yl)benzene (BDC60),

1,4-benzenediamine (BDA) and 1,4-benzenedithiol (BDT) measured using lithographic
gold MCBJs. (b) Conductance histograms on a semilog scale, constructed from 400 con-
secutive traces. The arrow marks the typical junction conductance of BDC60. All curves
are offset for clarity. Colors in panel (a) correspond to those in panel (b). Reprinted
with permission from [536]. Copyright 2008 American Chemical Society.

gold. As we showed in that section, the use of a transition metal like Pt

allows exploring the chemistry of unsaturated carbon bonds. In the case
of benzene, this led to a very high conductance, of the order of 1 G0 , to
be compared with the conductance of 0.0064 ± 0.0004 G0 reported for ben-
zenediamine in Ref. [477]. This illustrates the fact that in many cases the
anchoring groups are acting as spacers or potential barriers that diminish
the conductance of the junctions. Of course, the use of other metals is
often hindered by the oxidation of those metals, which can only be avoided
working under UHV conditions.
The direct binding of carbon structures, like C60 , to gold electrodes
has also been explored in the literature. C60 is known to hybridize
strongly with gold surfaces [580], and in single-molecule junctions it can
exhibit conductances on the order of one tenth of G0 [128]. These re-
sults suggest that one could also use C60 as an anchoring group. In-
deed, this possibility has been recently investigated by Martin et al. [536].
These authors have designed and synthesized a linear and rigid C60 -capped
molecule, 1,4-bis(fullero[c]pyrrolidin-1-yl)benzene (BDC60), and compared
the electrical characteristics to those of 1,4-benzenediamine (BDA) and
1,4-benzenedithiol (BDT) using lithographic MCBJs. The main conclu-
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412 Molecular Electronics: An Introduction to Theory and Experiment

sion of this work is the suitability of fullerene-anchoring for single-molecule

transport measurements. In particular, compared to thiols the fullerene-
anchoring leads to a considerably lower spread in low-bias conductance due
to the higher junction stability that minimizes fluctuations due to atomic
details at the anchoring site.
More recently, Zotti et al. [472] have studied both experimentally and
theoretically the transport through tolane molecules attached to gold con-
tacts via different anchoring groups. From the experimental side, they
showed that the molecules with thiol and nitro groups can sustain a much
higher current (see Fig. 13.10). From the theory side, and with the help
of DFT-based calculations, they showed that the anchoring not only deter-
mine the strength of the metal-molecule coupling (i.e. width of the molecu-
lar resonances), but they also control the position of the molecular energy
levels. In particular, they showed that in the case of thiol and amine groups,
their electron-donating character is reflected in the fact that the HOMO of
the molecules dominates the transport. On the contrary, nitro and cyano
groups have an electron-withdrawing character, which means in practice
that the LUMO is pushed closer to the gold Fermi energy and it dominates
the electrical conduction. Moreover, these authors showed that there is
no direct relation between the metal-molecule binding energies for different
anchoring groups and the corresponding junction conductances. This is ob-
vious in the case of molecules where the LUMO dominates the transport,
since this orbital plays practically no role in the binding energy.
As a last comment, let us say that not only the type of anchoring group
matters, but also its exact position. In a nice work, Mayor et al. [581]
showed that the conductance of a thiol-terminated indexanchoring groups!
thiol rod-like conjugated molecule depends crucially on the position of the
thiol group. They showed that by placing the thiol group in the meta
position of the last phenyl ring, the conjugation is partially interrupted
and the current decreases significantly as compared with the case in which
the thiol group is in the para position.

14.3 Tuning chemically the conductance: The role of side-

groups

As it is clear from our discussions in the previous chapter, the coherent

transport through a molecular junction depends crucially on the position of
the relevant orbitals of the molecule with respect to the metal Fermi energy
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Coherent transport through molecular junctions II: Test-bed molecules 413

and also on their character (degree of delocalization). Thus, the internal

electronic structure of a molecule plays a fundamental role and it can be
chemically tuned to certain extend with the inclusion of appropriate side-
groups or substituents . In principle, side-groups can have two main effects:
(i) they can control the structure of a molecule which in turn determines
the degree of conjugation (delocalization of the molecular orbitals) and (ii)
they can tune the position of the frontier orbitals. Both effects can have
an impact in the conductance of a junction. These effects are well-known
in the field of electron transfer [582, 583], but so far they have been quite
difficult to test systematically in molecular junctions.
The fact that the conformation must have a major impact on the con-
duction through a molecular junction has been predicted long ago [584, 585]
and it is very easy to explain, as we saw in section 13.5. Such impact have
been illustrated in different experiments [586–588], but probably the most
illustrative example have been reported by Venkataraman and coworkers
in Ref. [477]. As we explained in section 13.5, these authors carried out
a detailed study of the conductance of a series of biphenyl molecules with
different twist angles, θ, that were coupled to gold electrodes via amino
linking groups. They showed that the conductance follows a cos2 θ depen-
dence, as expected for transport through π-conjugated biphenyl systems
(see Refs. [584, 260] and section 13.5).
As it was shown by Pauly et al. [478], ab initio calculations based on
DFT show that the low-temperature conductance of biphenyl derivatives
follows closely the cos2 θ law consistent with an effective π-orbital coupling
model. A comparison between theory and the results of Ref. [477] has been
reported by Finch et al. [589]. These authors studied the conductance of
the series of 8 molecules shown in Fig. 14.12(a), with both thiol and amine
anchoring groups. They showed that if the Fermi energy EF lies within the
HOMO-LUMO gap, then the experimental results are reproduced. More
generally, however, if EF is located within either the LUMO or HOMO
states,8 the presence of resonances destroys the linear dependence of the
conductance on cos2 θ and gives rise to non-monotonic behavior associated
with the level structure of the different molecules. These results are illus-
trated in Fig. 14.12(b).
It is worth mentioning that the conduction in the experiment of
Ref. [477] is not completely suppressed when θ = π/2. In this limit also
σ-orbitals contribute to the effective coupling that allows a finite current
8 Inan experiment, EF may differ from the computed value for a number of reasons,
including the presence of a dielectric environment, such as air or water.
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414 Molecular Electronics: An Introduction to Theory and Experiment

(a) (b) 1

0.8

G(θ)/G(θ=0)
0.6

S
0.4 N
N Exp.

0.2

0
0 0.2 0.4 0.6 0.8 1
2
cos θ
Fig. 14.12 Theoretical results on the conductance of biphenyl derivatives. (a) Molecules
studied capped with NH2 . The dark vertex in the backbone of molecule 3 corresponds
to N and the side groups of molecules 6 and 7 (other than H) correspond to F and
Cl atoms, respectively. (b) Zero-bias conductance in a hollow configuration, for sulfur
contact (circles), nitrogen contact (squares) and values from Ref. [477] (triangles). All
cases have been normalized to the θ = 0 value. Adapted with permission from [589].
Copyright 2008 IOP Publishing Ltd.

to flow through the system [478]. In this sense, Pauly et al. [546] have
shown theoretically that the conductance of oligophenylenes of different
length remains finite when the molecules are modified with methyl side-
groups, although these substituents induce a rotation of the neighboring
phenyl rings of about 90o . The typical reduction of the conductance, in
comparison with the conjugated molecules, is about two order of magni-
tude. Recently, Lörtscher et al. [590] have shown experimentally that such
non-conjugated molecules are still conductive.
The role of the conjugation in the conduction through molecular systems
can also be illustrated without resorting to side-groups. Thus for instance,
the comparison of the conductance through alkanes to that through proto-
typical molecular wires with extended π-electron states, like oligophenyle-
neethynylene (OPE) or oligophenylenevinylene (OPV), shows substantially
higher conductance through the conjugated molecules and a rational de-
pendence on the HOMO-LUMO gap [475, 591–593].
As mentioned above, the second main effect of side-groups is to shift
the frontier orbitals of a molecule. In this sense, side-groups can be used
to improve the usually bad alignment between the molecular levels and
the Fermi energy of the metallic electrodes. In other words, and using
terminology of semiconductor physics, one can use side-groups to “dope”
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Coherent transport through molecular junctions II: Test-bed molecules 415

(a) (b) 1.1

Molecules HOMO LUMO
1

1 −6.25 +0.06 0.9

G (nS)
0.8
−6.62 −0.59 3
2
0.7
4
0.6 1
3 −5.23 +0.09
0.5

0.4 2
4 −5.97 +0.12
-7 -6.5 -6 -5.5 -5
HOMO (eV)

Fig. 14.13 Chemical control of double barrier tunneling in α, ω-dithiaalkane molecular

wires. (a) Molecules used and their frontier orbital energies (in eV) as obtained from a
DFT calculation. R = HS(CH2 )6 - in all cases. (b) Plot of conductances determined by
I(t) method (with standard deviations) against HOMO energy for molecules 1-4. From
[594]. Reproduced by permission of The Royal Society of Chemistry.

molecular junctions. This effect has been studied experimentally by several

groups [558, 594, 102], but it seems that it is still rather difficult to show
this basic effect in a systematic manner. Let us briefly describe the work
of Leary et al. [594], where the authors studied the low-bias conductance
of 1,4-bis-(6-thiahexyl)-benzene derivatives using the STM-based I(t) and
I(s) methods that will be discussed in the next section. In particular, they
investigated the four benzene derivatives shown in Fig. 14.13(a). In order
to achieve a contact with gold electrodes, these molecules contain radicals,
which act a linking groups, consisting of thiolated alkyl chains [HS(CH2 )6 ].
The central idea of this work was to study the correlation between the low-
bias conductance and the position of the frontier orbitals. For this purpose,
the authors determined theoretically the position of these orbitals by means
of DFT calculations [see Fig. 14.13(a)]. In panel (b) of the same figure one
can see the experimental results for the conductance plotted as a function of
the theoretical position of the HOMO of the isolated molecules. This graph
shows that the more electron-rich benzene rings (with a higher HOMO) give
higher conductances, which is consistent with hole conduction (i.e. via the
benzene HOMO). These results constitute a beautiful illustration of the
doping effect, although the change in the conductance is still rather small
(smaller than a factor 2). Anyway, let us stress that what really determines
the conductance is the actual position of the frontier orbitals of the molecule
in the junction, which in principle may differ from the corresponding ones in
gas phase. In that sense, it would be highly desirable to obtain information
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416 Molecular Electronics: An Introduction to Theory and Experiment

in-situ about the level alignment with the electrode Fermi energy, along
the lines of Ref. [461]. This is of course extremely challenging in the case
of single-molecule junctions, although not impossible as we shall see in
Chapter 20.
Another example of this doping effect was presented by Venkataraman
et al. [558]. In this case, the authors studied the single molecule con-
ductances of a series of very short conjugated molecules (substituted 1,4-
diaminobenzenes) using an STM-based break junction technique. They
found that electron donating substituents resulted in higher molecular con-
ductances, and there was an approximate correlation between the conduc-
tance and the Hammett σp parameter,9 consistent with hole transport (i.e.
transport dominated by the HOMO of the molecules). Another interest-
ing example related to the influence of side-groups has been reported by
Baheti et al. [102]. In this work the thermopower of molecular junctions
based on several 1,4-benzenedithiol (BDT) derivatives was investigated.
The BDT molecule was modified by the addition of electron-withdrawing
or -donating groups such as fluorine, chlorine, and methyl on the benzene
ring. It was found that the substituents on BDT generated small and pre-
dictable changes in conductance depending on their character. Moreover,
the authors showed that by replacing the thiol end groups by cyanide end
groups the transport changes radically and it turns out to be dominated by
the LUMO of the molecule. These results will be discussed in more detail
in Chapter 19 in the context of thermoelectricity in molecular junctions.

14.4 Controlled STM-based single-molecule experiments

One of the major problems in most of the experiments that we have dis-
cussed so far is the fact that it is not easy to prove that one is dealing with
a single molecule. In principle, the STM constitutes an ideal tool to resolve
this issue.10 The STM can be utilized to perform controlled transport
experiments through individual molecules that have been deposited onto
metal surfaces by bringing the metallic tip into contact to the molecule. The
obvious advantage of the STM is that the structure under investigation–a
molecule along with its substrate–can be imaged with submolecular preci-
sion prior to and after taking conductance data. In this way, parameters
9 Roughly speaking, the Hammett parameter (or constant) describes the change in re-
action rates upon introduction of substituents. For a precise definition, see Ref. [595].
10 The STM as a tool to fabricate molecular junctions has extensively described in section

3.4.4.
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Coherent transport through molecular junctions II: Test-bed molecules 417

Fig. 14.14 Conductance of a C60 molecule deposited on a Cu(100) surface measured

with a STM as a function of the tip displacement ∆z. Data are an average of 500
measurements. Zero displacement corresponds to the tip position before freezing the
feedback loop at V = 300 mV and I = 3 nA. The solid line correspond to the exper-
imental data, while the square symbols correspond to calculations performed with the
TRANSIESTA package. The inset shows a single conductance curve revealing a discon-
tinuity at ∆z = 3.3 Å. Reprinted with permission from [116]. Copyright 2007 by the
American Physical Society.

such as molecular orientation or binding site can be monitored. Another

advantage of STM is the possibility to characterize to some extent the sta-
tus of the second electrode, the microscope tip, by recording conductance
data on clean metal areas. Maybe the main disadvantage of the STM is
its mechanical stability, which does not reach level of the break-junction
methods. On the other hand, it is sometimes said that in this approach
there is an inherent asymmetry in the contact and the strong substrate-
molecule interaction that may distort some of the intrinsic properties of
the molecule. In any case, the high degree of control in these experiments
is extremely valuable at this stage in order to establish the basic transport
mechanisms at the molecular scale. In this sense, it is somewhat surprising
that STM data for molecular contacts are so scarce. In this section we
shall briefly describe some illustrative examples of this type of controlled
single-molecule experiments.
One of the first experiments of this kind was performed by Joachim et
al. [115] who used a STM at room temperature to study the contact con-
ductance of a C60 on Au(110). This experiment has been revisited more
recently by Néel et al. [116], but at 8 K and under UHV conditions. In
this experiment the molecules were deposited by sublimation onto a clean
Cu(100) surface and were probed by a Cu-covered tip. The orientation
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418 Molecular Electronics: An Introduction to Theory and Experiment

of the molecules on top of the Cu(100) surface could be resolved, and

only those molecules were selected that exposed a C-C bond between a
hexagon and a pentagon at the top. When approaching the tip towards the
molecule they observed a very reproducible jump into contact from about
G = 2.5 × 10−2 G0 to G = 0.25 G0 (see Fig. 14.14). When approaching the
tip further towards the molecule a jump up to G ≈ G0 was observed. The
detailed information provided in this experiment makes it ideal to compare
with the theory. Indeed, in the same work a theoretical analysis of the
linear conductance based on DFT yielded a satisfactory agreement, as one
can see in Fig. 14.14. From the modeling the authors inferred that the con-
trolled contact to a C60 molecule does not significantly deform its spherical
shape and they also showed that the conductance around the tip-molecule
contact formation is affected by a fluctuation between different microscopic
configurations.
In the context of the STM, two important methods have been intro-
duced by the Nichols’ group for the measurement of single-molecule con-
ductance [550, 596–599]. These methods are referred to as the I(s) and
I(t) methods. In these methods the starting point for the measurements
is the adsorption of a low coverage of the molecules under investigation on
a Au surface. This condition typically results in flat-lying molecules and
enables the formation of single-molecule wires with high probability. To
attach a molecule to the STM tip, usually made also of Au, the tip is low-
ered onto the surface by fixing the tunneling current I0 at relatively high
values and then lifted, while keeping a constant position in the x-y plane.
This procedure is illustrated in Fig. 14.15. The current decay shows distinc-
tive current plateaus when molecular wires bridge the gap between the tip
and substrate, whereas in the absence of wire formation the current simply
decreases nearly exponentially with tip-sample separation, see Fig. 14.15.
The current plateaus obtained with this method have been related to
electron tunneling through molecular wires bridging the STM tip and the
substrate [550, 596]. Statistical analysis of the data using histogram plots
has shown that the current-plateau values group themselves into discrete
values, which are integer multiples of a lowest value. The lowest current
peak in the histogram corresponds to a single molecule, whereas the next
discrete conductance step has been assigned to conduction through two
wires and so on.
An alternative method is the so-called I(t) method [550]. This involves
holding the Au STM tip at a given distance above the substrate while moni-
toring current jumps as molecular wires bridging the tip and substrate form
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Coherent transport through molecular junctions II: Test-bed molecules 419

Fig. 14.15 Schematic illustration of the I(s) STM method of forming molecular wires.
(A) A low coverage of the studied molecule is formed on the Au(111) surface, and the
set-point current is increased. (B) Attachment of the molecule at one end to the Au STM
tip is achieved, and then (C-D) the tip is retracted from the surface while recording the
current. The graph shows the conductance decay with distance for a clean Au substrate
(lower curve) and for a molecule on Au (upper curve). In the latter curve the different
stages of the contact formation are indicated (B, C and D). Notice the presence of a
plateau before rupture. Courtesy of Edmund Leary.

and subsequently break. It has been shown that both the I(s) and the I(t)
method result in the same single-molecule conductance for alkanedithiols
[550]. A nice application of the I(t) method can be found in Ref. [599],
where the authors showed, in combination with ab initio transport calcu-
lations, that the tilt-angle dependence of the electrical conductance is a
sensitive spectroscopic probe, providing information about the position of
the Fermi energy.
The use of methods in the spirit of the I(s) and the I(t) ones are opening
new ways of looking at molecular conductance. Thus for instance, Temirov
et al. [119] have reported beautiful results on a complex system, PTCDA
(4,9,10-perylenetetracarboxylic-dianhydrid), on a Ag(111) surface. They
demonstrated that one can controllably contact the molecule to the STM
tip at one of the four oxygen corner groups and peel the molecule gradually
from the surface. The conductance clearly varies in the process of peeling,
but when pulled to an upright position the conductance is approximately
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420 Molecular Electronics: An Introduction to Theory and Experiment

0.15 G0 . During the process an interesting Kondo-like resonance develops

that can be tuned by the electrode position. Another spectacular exam-
ple has been recently reported by Lafferentz et al. [600]. These authors
have measured the conductance and mechanical characteristics of a single
polyfluorene wire by pulling it up from a Au(111) surface with a STM tip,
thus continuously changing its length up to more than 20 nm. They showed
that the conductance not only decays exponentially but also exhibits char-
acteristics oscillations as one molecular unit after another is detached from
the surface during stretching.

14.5 Conclusions and open problems

Although in this chapter we have only talked about some concrete aspects
of single-molecule conduction, we can already draw a few general conclu-
sions and point out some of the main challenges for the near future. It
is clear that in the last years a significant progress has been made in the
experimental approaches to study single-molecule junctions as well as in
the qualitative understanding of their transport properties. From the ex-
perimental side, the introduction of statistical methods to determine the
conductance has partially eliminated the discrepancies between different
experimental results which appear when only individual traces are com-
pared. The use of new techniques to measure other transport properties
such as shot noise or thermopower provides very valuable additional infor-
mation that is not contained in the standard conductance measurements
(see Chapter 19). The use of low temperatures and the improvement in
the stability of the devices allow now making use of the inelastic tunneling
spectroscopy (see Chapter 16), which gives an essential information about
the presence of the molecules and the geometry of the junctions.
From the theory side, the development of ab initio methods makes now
possible to study both the mechanical and the electrical properties in a
much more reliable way. In particular, DFT-based calculations provide, for
instance, a detailed information about the possible structure of the con-
tacts, the relevant vibration modes and the conductance of the junctions.
These theoretical methods are now able to describe the general experimen-
tal trends, see e.g. Ref. [579], although they still fail in general to describe
quantitatively the transport results.
So in short, there are good reasons to be optimistic about the develop-
ment of this field. However, it has to be acknowledged that there are still
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Coherent transport through molecular junctions II: Test-bed molecules 421

basic issues to be resolved. The challenges for the experiments concern,

in the first place, the reproducibility of the results. We have seen that
the statistical methods are not the panacea and the interpretation of the
conductance histograms is not always straightforward. One of the main
goals should be to find strategies to rigidly bind molecules to electrodes
via selective anchoring groups or by means of new trapping techniques. It
would also be desirable to improve the stability of the contacts to be able
to extend the statistical analysis also to the I-V curves [534], which contain
much more information. Most of these requirements are indeed met by the
controlled STM experiments that we discussed in the previous section and,
in this sense, there is no doubt that they will play an important role in the
near future.
The theory has also to face several basic problems. One of the main
things to do is to understand the origin of the discrepancies between the
DFT-based methods which, in principle, are supposed to deliver the same
results. Thus, systematic comparisons between different implementations of
the DFT-NEGF approach are necessary, as proposed in Ref. [545]. On the
other hand, as we explained in section 10.8, DFT as it is used in molecular
electronics has clear limitations. Thus, the biggest challenge for the theory
is the introduction of new methods to describe properly the role of the
electronic correlations in the transport through these systems. DFT does
not describe correctly the energy spectrum of a system and, in particular, it
tends to give small values for the HOMO-LUMO gap, as compared with the
experiment. This is the main reason behind its systematic overestimation
of the low-bias conductance of molecular junctions. Moreover, DFT is not
well-founded in an out-of-equilibrium situation and its use to describe the
transport at finite bias is then doubtful.11 Finally, so far most theoretical
methods used to describe the I-V curves are not able to take into account
the possible conformational changes that may appear when a molecule is
subjected to a rather high electric field. This issue is certainly playing an
important role in many experiments and it is presently out of the scope of
most theories in molecular electronics.
11 The reader has surely noticed that we have not presented or discussed any comparison
of the I-V curves and we have focused our attention on the low-bias conductance. There
are several reasons for that. First, there are very few statistical analyses of the I-V
characteristics and there are no yet test systems in which different experiments agree
on the shape of the I-Vs. Second, most of the existent theoretical methods fail to
describe quantitatively the level spectrum of a molecule (or a molecular junction), and
as consequence no quantitative agreement between theory and experiment has been
obtained yet.
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422 Molecular Electronics: An Introduction to Theory and Experiment

January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Chapter 15

Single-molecule transistors: Coulomb

blockade and Kondo physics

15.1 Introduction

In the previous two chapters we have considered the coherent transport

regime in which the electrons proceed elastically (without exchanging en-
ergy) through the junctions. What is the range of validity of the coherent
picture? Intuitively, the coherent mechanism will be the dominant one as
long as the time that an electron needs to cross the molecular bridge is
smaller than the time that it takes to interact with other electrons or to ex-
cite vibronic degrees of freedom, i.e. the time that is needed for an electron
to undergo an inelastic scattering event. A problem here is that the time
that an electron spends in a junction, sometimes referred to as tunneling
traversal time, is not easy to define unambiguously. Close to a resonant sit-
uation, i.e. when a molecular level is close to the Fermi energy of the leads,
a measure of this time scale is ~/Γ, where Γ is the width of the molecu-
lar resonance due to the coupling to the electrodes. The scale ~/Γ can be
viewed as the lifetime of an electron for escaping into the leads. Away from
the resonant condition, the traversal time, τ , is mainly determined by the
injection gap, ∆E, which is the energy difference between the leads’ Fermi
energy and the relevant molecular orbital (HOMO or LUMO).1 Büttiker
and Landauer have shown that the traversal time obtained in the deep
tunneling
p limit for a square barrier of energy height ∆E and width D is
τ = D m/2∆E, where m is the electron mass [601]. If, instead, the bridge
is described in terms of a one-dimensional lattice of N equivalent sites, this
time is given by τ = ~N/∆E [602].
√ Then, for practical purposes we can use
the unified expression τ = ~/( ∆E 2 + Γ2 ) as an estimate of the traversal

1 In
the resonant tunneling model, the injection gap is simply the energy ǫ0 of the level,
measured with respect to the Fermi energy.

423
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424 Molecular Electronics: An Introduction to Theory and Experiment

time, which covers the different situations.2 Thus, if ~/τ is larger than the
energy scales associated with inelastic interactions like electron-electron,
U , or electron-vibration, λ, then the transport is mainly coherent. At the
contrary, if a molecule is weakly coupled to the electrodes (Γ < max{U, λ})
and the system is brought close to resonance (∆E ≈ 0), the transport will
very likely be dominated by the Coulomb interaction in the molecule or by
the excitation of internal degrees of freedom like vibrational modes.
There are by now many examples in nanophysics in which the elec-
tronic transport through a small object which is weakly coupled to metal-
lic electrodes has been explored. Let us mention, for instance, the cases
of semiconductor quantum dots, carbon nanotubes or metallic nanopar-
ticles. In all these systems, the transport in the weak-coupling regime is
governed by single-electron tunneling processes that lead to phenomena like
the Coulomb blockade effect. Moreover, if the coupling is not so weak, other
interesting many-body phenomena, like the Kondo effect, can show up at
show temperatures. We shall see in this chapter that these phenomena also
appear in single-molecule junctions. These effects have been understood
in great detail in different devices with the help of a gate electrode. This
third terminal is only capacitively coupled to the small object and it allows
to tune its energy level spectrum and to explore different charge (or redox)
states. The gate electrode allows us in turn to control the current that flows
through the system with an external field, very much like in the case of field-
effect transistors in microelectronics. Due to this analogy and also to the
fact the transport is usually dominated by single-electron processes, these
weakly coupled systems are known as single-electron transistors (SETs). In
the last decade it has become possible to incorporate a gate electrode into
single-molecule devices. We shall refer to these three-terminal molecular
devices as single-molecule transistors (SMTs).
The goal of this chapter is to discuss the electronic transport through
SMTs with special emphasis in the role of the Coulomb interaction in the
molecules. The role of the vibrational modes in these systems will be dis-
cussed in the next chapter. With this idea in mind, we shall first review
briefly the general conditions necessary to observe charging effects and we
shall recall the basic signatures of these effects in the transport characteris-
tics. Then, in section 15.3 we shall recall the main experimental techniques
that have been used so far to fabricate SMTs. The experimental results in
SMTs are often analyzed in the light of the “orthodox” theory of Coulomb
2 See Ref. [38] and references therein for a detailed discussion about the tunneling traver-
sal time.
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Single-molecule transistors: Coulomb blockade and Kondo physics 425

blockade. For this reason, we have included a detailed description of this

theory in section 15.4. The attempts to generalize the standard theory to
the specific problem of SMTs are discussed in section 15.5. Section 15.6
is devoted to the intermediate transport regime (Γ not too small) and we
shall pay special attention to the Kondo effect. In the last section we shall
review some of the most representative experimental results obtained in the
context of SMTs.
The physics, results and challenges related to STMs have been discussed
in the reviews of Refs. [39, 603–607].

15.2 Charging effects in transport through nanoscale de-

vices

In this section we examine the circumstances under which Coulomb charg-

ing effects are important in the transport through small devices and we
briefly recall the main main signatures of these effects in the transport
characteristics.
Following Ref. [608], we want to address first the following question:
How small and how cold should a conductor be so that adding or subtracting
a single electron has a measurable effect? To answer this question, let
us consider the electronic properties of the generic conductor depicted in
Fig. 15.1, which is coupled to three terminals. Particle exchange can occur
with only two of the terminals. These source and drain terminals connect
the small conductor to macroscopic current and voltage meters. The third
terminal provides an electrostatic or capacitive coupling and can be used as
a gate electrode. If we first assume that there is no coupling to the source
and drain contacts, then the small conductor acts as an island for electrons.
The number of electrons on this island is an integer N , i.e. the charge on
the island is quantized and equal to N e. If we now allow tunneling to the
source and drain electrodes, then the number of electrons N adjusts itself
until the energy of the whole circuit is minimized.
When tunneling occurs, the charge on the island suddenly changes by
the quantized amount e. The associated change in the Coulomb energy
is conveniently expressed in terms of the capacitance C of the island. An
extra charge e changes the electrostatic potential by the charging energy
EC = e2 /C. This charging energy becomes important when it exceeds
the thermal energy kB T . A second requirement is that the barriers are
sufficiently opaque such that the electrons are located either in the source,
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426 Molecular Electronics: An Introduction to Theory and Experiment

Source Drain

Gate

V VG

Fig. 15.1 Schematic representation of a generic three-terminal device. The sphere rep-
resents the dot (or island), which is weakly coupled to the source and drain electrodes
by tunnel junctions. Finally, a third electrode (the gate) is capacitively coupled to the
island.

in the drain, or on the island. This means that quantum fluctuations in

the number N due to tunneling through the barriers are much less than
one over the time scale of the measurement. (This time scale is roughly
the electron charge divided by the current.) This requirement translates
to a lower bound for the tunnel resistances Rt of the barriers. To see this,
consider the typical time to charge or discharge the island ∆t = Rt C. The
Heisenberg uncertainty relation: ∆E∆t = (e2 /C)Rt C > h implies that
Rt should be much larger than the resistance quantum h/e2 = 25.813 kΩ
in order for the energy uncertainty to be much smaller than the charging
energy. To summarize, the two conditions for observing effects due to the
discrete nature of charge are
Rt ≫ h/e2 and e2 /C ≫ kB T. (15.1)
The first criterion can be met by weakly coupling the small object (or dot) to
the source and drain leads. The second criterion can be met by making the
dot small or by lowering the temperature. Let us recall that the capacitance
of an object scales with its radius R and for a sphere, C = 4πǫ0 R. Thus for
instance, the charging energy of a C60 molecule, which has a radius of ∼ 4
Å, can be estimated to be e2 /4πǫ0 R ∼ 3.6 eV. This indicates that charging
effects can in principle be readily observed in single-molecule junctions even
at room temperature, as along as the molecules are weakly coupled to the
electrodes.
The conditions summarized in Eq. (15.1) are met by many different
nanoscale systems and for this reason charging effect have been observed,
among other systems, in metallic islands [609, 610], semiconducting quan-
tum dots [608, 611], nanoparticles [612], carbon nanotubes [613, 614], and
semiconducting nanowire quantum dots [615, 616]. While the behavior of
these type of quantum dots is fairly well understood, the properties of SMTs
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Single-molecule transistors: Coulomb blockade and Kondo physics 427

are much less established mainly because it is difficult to fabricate them in

a reliable way (see next section).
When discussing charging effects, an important energy scale is the en-
ergy level spacing ∆E in the dot, i.e. the separation between the discrete
energy states of the small conductor. To be able to resolve these levels,
the spacing must be much larger than kB T . The level spacing at the Fermi
energy EF for a box of size L depends on the dimensionality. Including
spin degeneracy, we have

N/4 (1D)
~2 π 2 

∆E = × 1/π (2D) , (15.2)
mL2   ¡1/3π 2 N ¢1/3 (3D)

where m is the electron mass and N the number of electrons. The charac-
teristic energy scale is thus ~2 π 2 /(mL2 ). For a 1D box, the level spacing
grows for increasing N , in 2D it is constant, while in 3D it decreases as N
increases. The level spacing of a 100 nm 2D dot is ∼ 0.03 meV, which is
large enough to be observable at dilution refrigerator temperatures of ∼ 100
mK. Thus, dots made in semiconductor heterostructures are true artificial
atoms, with both observable quantized charge states and quantized energy
levels. Using 3D metals to form a dot, one needs to make nanoparticles as
small as ∼ 5 nm in order to observe atom-like properties. In the case of
molecular junctions, the spacing ∆E, which is basically the HOMO-LUMO
gap, is typically of the order of several electronvolts. Therefore, level quan-
tization should be easily observable in SMTs even at room temperature.
Now that we have identified the relevant scales for the occurrence of
charging effects, let us now see how they are revealed in the transport
characteristics. The tunneling of a single charge changes the electrostatic
energy of the island by a discrete value, a voltage VG applied to the gate
(with capacitance CG ) can change the island’s electrostatic energy in a con-
tinuous manner. In terms of charge, tunneling changes the island’s charge
by an integer while the gate voltage induces an effective continuous charge
q = CG VG that represents, in some sense, the charge that the dot would
like to have. This charge is continuous even on the scale of the elementary
charge e. If one sweeps VG , the build up of the induced charge will be com-
pensated in periodic intervals by tunneling of discrete charges onto the dot.
This competition between continuously induced charge and discrete com-
pensation leads to the so-called Coulomb oscillations in a measurement
of the current (or conductance) as a function of gate voltage at a fixed
source-drain voltage.
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428 Molecular Electronics: An Introduction to Theory and Experiment

(a) (b)

(c) (d)

Fig. 15.2 Coulomb blockade in a single-wall carbon nanotube. (a) AFM image of a
carbon nanotube on top of a Si/SiO2 substrate with two 15-nm-thick Pt electrodes, and
a corresponding circuit diagram. The total length of the tube is 3 µm, with a section
of 140 nm between the contacts to which a bias (source-drain) voltage is applied. A
gate voltage Vgate applied to the third electrode in the upper-left corner of the image is
used to vary the electrostatic potential of the tube. (b) Current versus gate voltage at
Vbias = 30 µV. Two traces are shown that were performed under the same conditions.
(c) Current-voltage curves of the tube at a gate voltage of 88.2 mV (trace A), 104.1
mV (trace B) and 120.0 mV (trace C). (d) Conductance G = I/Vbias versus ∆Vgate
at low bias voltage Vbias = 10 µV and different temperatures. Solids lines are fits of
G ∝ cosh−2 (e∆Vgate /α2kB T ), corresponding to the model of a single molecular level
that is weakly coupled to two electrodes. The factor α is the gate coupling parameter (see
text) and for this peak equals 16. Reprinted by permission from Macmillan Publishers
Ltd: Nature [613], copyright 1997.

An example of these oscillations in a single-wall carbon nanotube weakly

coupled to two metallic electrodes is shown in Fig. 15.2(b). As one can see,
there appear a series of peaks or spikes in the current versus the gate voltage
at very low bias (source-drain) voltage in a quasi-periodic fashion.3 In the
valley of the oscillations, the number of electrons in the nanotube is fixed
and necessarily equal to an integer N . In the next valley to the right the
number of electrons is increased to N + 1. At the crossover between two
stable configurations N and N + 1, a charge degeneracy exists where the
number can alternate between N and N + 1. This allowed fluctuation in
3 In this case the bias voltage was quite low (linear regime) and the conductance exhib-
ited the same peak structure as the current.
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Single-molecule transistors: Coulomb blockade and Kondo physics 429

the number (i.e. according to the sequence N → N + 1 → N → · · · ) leads

to a current flow and results in the observed peaks.
An alternative measurement is performed by fixing the gate voltage, but
varying the source-drain voltage VSD . As shown in Fig. 15.2(c) for a carbon
nanotube junction, one observes in this case a non-linear current-voltage
characteristic exhibiting a series of steps. This characteristic structure is
known as Coulomb staircase. A new current step occurs at a threshold volt-
age (∼ e2 /C) at which an extra electron is energetically allowed to enter the
nanotube. It is seen in Fig. 15.2(c) that the threshold voltage can be mod-
ulated with the gate voltage until the low bias gap completely disappears,
in accordance with the Coulomb oscillations. Finally, as one can see in
Fig. 15.2(d), the conductance versus the gate voltage has a very character-
istic temperature dependence close to a resonance, where the conductance
maximum decreases as the temperature increases.
The origin of these peculiar transport characteristics will be analyzed
in detail in detail in the next sections. Moreover, we shall show that one
can obtain very valuable spectroscopic information about the charge state
and energy levels of the dot by analyzing the precise shape of the Coulomb
oscillations and the Coulomb staircase.

15.3 Single-molecule three-terminal devices

Most of the experiments described in the previous two chapters in our dis-
cussion of the coherent transport have been performed with two-terminal
devices fabricated with the break-junction technique and the STM. These
techniques have several advantages, but it is however very difficult to in-
corporate a gate electrode in their set-ups. This is an important drawback
since, as discussed in the previous section, a gate electrode allows us to
extract much more information about the junctions. Thus for instance, the
gate makes possible to study the conduction through molecules in differ-
ent transport regimes by bringing the energy levels into and out of reso-
nance with the Fermi energy. This way, one can also probe excited states
and different charge states can be accessed. Excited states can either be
vibrational [22, 617, 678], electronic [618], or related to spin transitions
[619, 620]. These excitations serve as a fingerprint of the molecule under
study.
An important parameter in three-terminal devices is the gate coupling
parameter, α. This parameter quantifies the shift of the orbital levels that
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430 Molecular Electronics: An Introduction to Theory and Experiment

can be induced with a gate electrode potential, VG . In an experiment, the

gate coupling should be as large as possible in order to access as many
charge states as possible. The geometry plays an important role in the gate
coupling and one should take care that the electrodes themselves do not
screen the gate potential as this would decrease α. The electrode separation
(and therefore the length of the molecule) and the breakthrough voltage of
the gate oxide are other important parameters. Currently, two gate mate-
rials are frequently used: heavily doped silicon substrates with thermally
grown SiO2 on top and aluminum strips with a native Al2 O3 oxide of only
a few nanometers. For aluminum gates with an oxide thickness of 3 nm, the
gate coupling is about 0.1 so that, with a typical breakthrough voltage of 4
V at low temperatures, the potential of the molecular levels can be shifted
by ±0.4 eV. On the other hand, in silicon devices with an SiO2 thickness
of 250 nm, the gate coupling is about 10−3 ; with a typical breakthrough
voltage of 100 V, the range over which the potential on the molecule can
be varied equals ± 0.1 eV.
As we have seen in Chapter 3, three-terminal devices have been fab-
ricated using different techniques. We follow here Ref. [606] and we now
proceed to briefly describe the most successful approaches so far. They no-
tably differ in the way the nanogap or the molecular junction is created. The
most popular technique is electromigration, in which a large current den-
sity breaks a narrow and thin metal wire to form two physically separated
electrodes [21]. Electromigration-induced nanogap formation has been im-
aged in situ by transmission (and scanning) electron microscopy [82, 621].
Several of these electrode pairs can be fabricated on top of a conducting
substrate (coated by an insulating layer) which can then serve as a gate elec-
trode. Although some control has been obtained over the electromigration
process by using a feedback mechanism, the resulting nanogap geometry or
size remains uncontrollable. The advantage of electromigrated devices on
a Al/Al2 O3 gate electrode is their large gate coupling (αmax ∼ 0.27). The
planar geometry [see Fig. 15.3(a)] offers a large stability for systematic stud-
ies as a function of gate voltage, temperature and magnetic field. Molecules
are deposited from solution either prior to gap formation or afterward.
A second technique involves the fabrication on top of a gate electrode
of two gold electrodes using a shadow mask technique as illustrated in
Fig. 15.3(b). If the tilt angle of evaporation is high there is no overlap
between the source and drain shadows. Reducing the tilt angle decreases
the source-drain gap. In situ measurements of the conductance allow for
fine tuning of the gap distance when performed at low temperatures (∼ 4.2
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Single-molecule transistors: Coulomb blockade and Kondo physics 431

a b Gold deposition
angle
Mask
Molecule

Gold
SOURCE DRAIN
Al2O3 electrodes
Al Gate
Al2O3
Al Gate
c d
Molecule Gold
Countersupport
electrodes Gold
particle Molecule

SiO2
SOURCE DRAIN
Doped Si Pushing rod
SiO2
Doped Si Gate

Fig. 15.3 Schematic diagrams of different three-terminal device techniques. (a) Elec-
tromigrated thin metal wire on top of a Al/Al2 O3 gate electrode. (b) Angle evaporation
technique to fabricate planar electrodes with nanometer separation on top of a Al/Al2 O3
gate electrode. (c) Gated mechanical break junction. (d) The dimer contacting scheme
(see text). Reprinted with permission from [606]. Copyright 2008 IOP Publishing Ltd.

K). Molecules are deposited by quench condensation without disruption of

the vacuum [622, 623]. The advantages of this evaporation technique in-
clude all the ones from the electromigrated devices, plus the control over
the gap distance and the ability for molecule deposition inside a clean en-
vironment. Typical gate coupling values are of the same order as the ones
for electromigrated junctions.
Only recently it has been possible to integrate a gate electrode in me-
chanical controllable break junctions (MCBJ) [86], see Fig. 15.3(c). So far
it has been possible to place the gate electrode from the gap at a distance of
40 nm [86]. Although the gate coupling remains low as compared to other
techniques with a planar geometry (α ∼ 0.006 in Ref. [86]), MCBJs have
the clear advantage of precise control over the gap distance; the reported
breakthrough voltage [86] was 12 V. Molecule deposition is carried out from
solution.
Another three-terminal approach was reported by Dadosh et al. [114].
Their method is based on synthesizing in solution a dimer structure con-
sisting of two colloidal gold particles connected by a dithiolated molecule.
The dimer is then electrostatically trapped between two gold electrodes de-
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432 Molecular Electronics: An Introduction to Theory and Experiment

fined on top of a gate electrode [see Fig. 15.3(d)]. According to the authors,
this dimer-based contacting scheme provides several advantages such as the
ability to fabricate single-molecule devices with high certainty in which the
contacts to the molecule are well defined. The gold particles in this set-up,
however, efficiently screen the gate potential. Moreover, at low tempera-
tures spectroscopic features of the gold particles were sometimes observed
to be superimposed on the characteristics of the molecule conduction.

15.4 Coulomb blockade theory: Constant interaction model

Most of the results obtained so far in single-molecule transistors (SMTs),

i.e. in weakly coupled three-terminal molecular devices, have been analyzed
with the help of the “orthodox” theory of Coulomb blockade [609], which
has been very successful explaining the basic transport properties of semi-
conductor quantum dots. For this reason, and before describing some of
the main experiments reported to date, it is important to discuss in cer-
tain detail this theory, which is often referred to as the constant interaction
model.
In the next subsections we present an introductory description of the
standard theory of Coulomb blockade paying special attention to the rel-
evant regime for molecular devices. In a later section we shall present an
alternative formulation of this theory which is better adapted to SMTs, but
it is much more involved. The next subsections are based on Refs. [624, 625]
and on the didactic review of Ref. [605]. We also recommend the review on
single-molecule junctions of Ref. [39].

15.4.1 Formulation of the problem

We consider a quantum dot or molecule,4 which is weakly coupled via
tunnel barriers to two metallic electrodes and it is also capacitively coupled
to a gate electrode. The quantum dot has single-particle energy levels at
Ep (p = 1, 2, · · · ), labeled in ascending order and measured relative to
the equilibrium chemical potential of the electrodes, which we set to zero
(EF = 0). Each level contains either one or zero electrons. Spin degeneracy
can be included by counting each level twice, and other degeneracies can be
included similarly. Each electrode is considered to be in thermal equilibrium
4 Although we have in mind molecular junctions, we shall use throughout this section

the name quantum dot to refer generically to a small island weakly coupled to the source
and drain.
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Single-molecule transistors: Coulomb blockade and Kondo physics 433

11
00
00
11 11
00
00
11
L 00
11
00
11 00
11
00
11 R
00
11
00
11 00
11
00
11
00
11 00
11
EF 00
11
00
11 00
11
00
11
00
11
00
11 Ep 00
11
00
11
00
11 00
11
00
11
00
11 00
11
00
11
00
11 00
11
00
11
00
11 00
11
00
11
EF
00
11
00
11 00
11
00
11
eV 00
11 00
11
00
11 00
11
ηeV−eVG

Fig. 15.4 Schematic drawing of the energy level diagram and electrostatic potential
profile of a generic quantum dot. The dot possesses a single-particle spectrum with
discrete levels, Ep . The Fermi levels in the left and right reservoirs are indicated. We
measure the levels Ep with respect to EF , which from now on we set to zero. The single
particle spectrum may be shifted by the external potential. Here, η is the portion of the
bias voltage that drops at the right interface and VG corresponds to the gate voltage.

at temperature T and the continuum of states in the reservoirs is occupied

according to the Fermi-Dirac distribution
· µ ¶¸−1
E
f (E) = 1 + exp . (15.3)
kB T
In Fig. 15.4(a) we show schematically the energy level diagram of the quan-
tum dot as well as the profile of the electrostatic potential.
Because in the weak coupling regime the number N of electrons localized
in the dot can take integer values only, a charge imbalance, and hence a
potential difference Vdot (Q) can arise between the dot and reservoirs in
equilibrium (Q = −N e is the charge on the dot). Following the orthodox
model of the Coulomb blockade [609], one can express Vdot in terms of an
effective N independent capacitance C between dot and the outside world,
Vdot (Q) = Q/C + Vext , (15.4)
where Vext is a contribution from external charges (in particular
R −N e those on a
nearby gate electrode). The electrostatic energy U (N ) = 0 Vdot (Q)dQ
then takes the form
U (N ) = (N e)2 /2C − N eVext . (15.5)
Thus, the result for the total energy of a dot that contains N electrons,
including the quantum energy due to the orbital energies is
N
X
Edot (N ) = U (N ) + Ep . (15.6)
p=1
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434 Molecular Electronics: An Introduction to Theory and Experiment

This expression for the total energy summarizes the constant interaction
model, in which the capacitance C does not vary with N .
It is important to clarify the meaning of the external potential in
Eq. (15.5). This can be done with the help of the equivalent circuit shown
in Fig. 15.5. Elementary electrostatics gives the following relation between
the different potentials and the charge Q on the island:
CVdot − CS VS − CD VD − CG VG = Q, (15.7)
where C = CS + CD + CG . Comparing this expression with Eq. (15.4) we
arrive at the following result for the external potential
Vext = (CS VS + CD VD + CG VG )/C. (15.8)
Thus, we see that the potential on the dot depends on the induced potential
Vext of the source, drain and gate. Notice that the change in the external
potential due to a change in the gate voltage carries a factor α = CG /C,
which is the gate coupling parameter that was mentioned in the previous
section. On the other hand, assuming that the drain is grounded as in
Fig. 15.5, the factor η introduced in Fig. 15.4 can now be simply expressed
as the capacitance ratio η = CS /C.
A key quantity determining whether the current can flow through the
dot is its chemical potential, which is the minimum energy required to add
an extra electron to the dot. From Eq. (15.6) it is easy to see that this
chemical potential is given by
1 e2
µ ¶
µdot (N ) = Edot (N ) − Edot (N − 1) = N − − eVext + EN . (15.9)
2 C
Before discussing the main predictions of this theory, it is important to
be aware of the conditions for which the constant interaction model gives

CS CD
Source Vdot Drain

CG
Gate

VS VG

Fig. 15.5 Schematic representation of the capacitance model of a quantum dot. The dot
is connected to source and drain electrodes with tunnel junctions and the gate electrode
shifts the electrostatic potential of the dot. Here, we assume that drain electrode is
grounded (VD = 0).
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Single-molecule transistors: Coulomb blockade and Kondo physics 435

a reliable description of the device. This is first of all weak coupling to the
leads. A second condition is that the size of the device should be sufficiently
large to make a description with single values for the capacitances possible.
Finally, the single-particle spectrum Ep should not vary with the charge
N residing on the dot. The constant interaction model works well for
weakly coupled quantum dots for which it is very often used. However,
the previous conditions are not fulfilled in general in molecular devices.
Neither the charging energy nor the energy level spectrum are expected
to be independent of the number of electrons in the molecule, specially
for small ones. Thus, the constant interaction model should be used with
caution in this case.

15.4.2 Periodicity of the Coulomb blockade oscillations

Let us now discuss the expected periodicity of the Coulomb oscillations,
i.e. the distance between the peaks in the conductance as a function of the
gate voltage in the limit of small (linear regime) source-drain voltage.
Electrons can flow from left to right when µdot is between the potentials,
µL and µR , of the leads (with eVSD = µL − µR ), i.e. µL > µdot > µR , see
Fig. 15.6. For small bias voltages, VSD ≈ 0, the N -th Coulomb peak is a
direct measure of the lowest possible energy state of an N -electron dot, i.e.
the ground state electrochemical potential µdot (N ). From Eqs. (15.9) and
(15.8) we obtain

µdot (N ) = (N − 1/2)e2 /C − eαVG + EN , (15.10)

where α = CG /C is the gate coupling. The addition energy is given by

e2 e2
∆µ(N ) = µdot (N + 1) − µdot (N ) = + EN +1 − EN = + ∆E, (15.11)
C C
where ∆E = EN +1 − EN is the level spacing mentioned in section 15.2.
In the absence of charging effects, the addition energy, ∆µ(N ), is deter-
mined by the irregular spacing ∆E of the single-electron levels in the quan-
tum dot. The charging energy e2 /C regulates the spacing, once e2 /C & ∆E.
If there is spin degeneracy of the levels, it is lifted by the charging energy.
In the limit (e2 /C)/∆E → 0, Eq. (15.11) is the usual condition for reso-
nant tunneling. In the limit (e2 /C)/∆E → ∞, Eq. (15.11) describes the
periodicity of the classical Coulomb-blockade oscillations in metallic islands
where the level spacing is negligible [609]. In molecular physics the related
energies are defined as A = Edot (N ) − Edot (N + 1) for the electron affinity
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436 Molecular Electronics: An Introduction to Theory and Experiment

Coulomb Blockade N−−−>N+1−−−>N−−−>N+1−−−>...

11
00 00
11 1
0 1
0 11
00 1
0
(a) 00
11 00
11 (b) 0
1 0
1 (c) 00
11 0
1
00
11
00
11 00
11
00
11 0
1
0
1 0
1
0
1 00
11
00
11 0
1
0
1
L 00
11 00 R
11 L 0
1 0
1 R L 00
11 0
1 R
00 µ(N+1) 11
11 00 0
1 0
1 00
11 0
1
00 dot 11
11 00 0
1 0
1 00
11 0
1
00
11
00 µ (N) 11
11 00
00 e2/C + ∆E
11 0
1
0
1 0
1
0
1 00
11
00
11 0
1
0
1
00 dot 11
11 00 0
1 0
1 00
11 0
1
00
11 11
00 0
1 0
1 00
11 0
1
00
11 00
11 0
1 0
1 00
11 0
1
00
11 00
11 0
1 0
1 00
11 0
1
00
11
00
11 00
11
00
11 0
1
0
1 0
1
0
1 00
11
00
11 0
1
0
1
00
11 00
11 0
1 0
1 00
11 0
1
00
11 00
11 0
1 0
1 00
11 0
1
00
11
00
11 00
11
00
11 0
1
0
1 0
1
0
1 00
11
00
11 0
1
0
1
00
11 00
11 0
1 0
1 00
11 0
1
00
11 00
11 0
1 0
1 00
11 0
1
eVSD

Fig. 15.6 Potential landscape through a quantum dot. The states in the contacts are
filled up to the electrochemical potentials µL and µR , which are related by the external
voltage VSD = (µL − µR )/e. The discrete single-particle states in the dot are filled
with N electrons up to µdot (N ). The addition of one electron to the dot raises µdot (N )
(i.e. the highest solid curve) to µdot (N + 1) (i.e. the lowest dashed curve). In (a) this
addition is blocked at low temperatures. In (b) and (c) the addition is allowed since here
µdot (N + 1) is aligned with the reservoir potentials by means of the gate voltage. (b)
and (c) show two parts of the sequential tunneling process at the same gate voltage. (b)
shows the situation with N and (c) with N + 1 electrons on the dot.

and I = Edot (N − 1) − Edot (N ) for the ionization energy. Their relation to

the addition energy is ∆µ(N ) = I − A.
From an experimental point of view, the Coulomb oscillations are mea-
sured as a function of the gate voltage. The peak spacing in terms of the
gate voltage is given by
∆VG = ∆µ(N )/eα = (e2 /C + ∆E)/eα, (15.12)
while the condition eαVGN = (N − 1/2)e2 /C + EN gives the gate voltage of
the N -th Coulomb peak.

15.4.3 Qualitative discussion of the transport characteris-

tics
Before discussing how to compute the transport properties in the Coulomb
blockade regime, it is convenient to describe qualitatively the main expected
features. Here, we shall follow Ref. [605] closely. As said in the previous
subsection, in the weak coupling regime and at low temperature, the cur-
rent is suppressed when all chemical potential levels lie outside of the bias
window. As we can tune the location of these levels using the gate voltage,
it is interesting to study the current and differential conductance of the
device as a function of the bias and of the gate voltage. A two-dimensional
plot of the current or conductance as a function of the two voltages is often
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Single-molecule transistors: Coulomb blockade and Kondo physics 437

referred to as stability diagram.

Let us now determine the line in the stability diagram (VSD -VG plane)
that separates a region of suppressed current from a region with finite
current. This line is given by the condition that the chemical potential
of the source (or drain) is aligned with that of a level on the dot. We
again assume the drain to be grounded as in Fig. 15.5. From the expression
of Eq. (15.9) for the chemical potential and using the definition for Vext ,
Eq. (15.8), we find the following condition for the dot chemical potential to
be aligned with the source one (µL = eV ) keeping the dot’s charge constant
VSD = β(VG − VC ), (15.13)
where β = CG /(CG + CD ) and VC = (N − 1/2)e/CG + CEN /(eCG ), which
can be seen as the voltage corresponding to the chemical potential on the
dot in the absence of an external potential. If the chemical potential is
aligned with the drain (µR = 0), we have
VSD = γ(VC − VG ), (15.14)
with γ = CG /CS . The expressions given here are specific for a grounded
drain electrode.5
Each dot resonance generates two straight lines in the VSD -VG plane,
separating regions of suppressed current from those with finite current.
For a sequence of resonances, one obtains generically the picture shown in
Fig. 15.7(a). The diamond-shaped regions are traditionally called Coulomb
diamonds, as they are often studied in the context of metallic dots, where
the chemical potential difference of the levels is mainly made up of the
Coulomb energy. The name is also used in molecular transport, although
this is strictly speaking not justified since in this case the level spacing can
be of the same order as the Coulomb interaction.
From the Coulomb diamond picture we can infer the values of some im-
portant quantities. Thus for instance, the addition energy, see Eq. (15.11),
can be read off from the height of the diamond, see Fig. 15.7(a), or from
the distance of the degeneracy points, although in this case one needs to
know the gate coupling parameter, α. If the addition energy is dominated
by the charging energy, we can find the total capacitance. Combining this
with the slopes of the diamond sides, which give us the relative values of
CG , CS and CD , we can find all these capacitances explicitly.
5 It is easily verified that, irrespective of the grounding, it holds that
C 1 1 1
= = + .
CG α β γ
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438 Molecular Electronics: An Introduction to Theory and Experiment

(a)
current

)
−V
C
VG
Eadd /e

β(
∆E
VSD

0
α Eadd/e

γ ( VC
N N+1

−VG
)
(b)
VG

Fig. 15.7 (a) Generic two-dimensional plot of the current as a function of bias and
gate voltage (stability diagram) for a quantum dot in the Coulomb blockade regime. For
small bias current only flows in the three degeneracy points indicated with circles. Upper
shadow region: positive currents. Lower shadowed region: negative currents. White:
blockade, no current. The dotted lines indicate the presence of excitations (see text).
(b) Measured stability diagram of a metallic single-walled carbon nanotube showing the
expected fourfold shell filling. Blockade regime is white. Reprinted with permission from
[626]. Copyright 2005 by the American Physical Society.

An interesting consequence of the previous analysis is that, if the ca-

pacitances do not depend on the particular state we are looking at, the
height of successive Coulomb diamonds is constant. If, in addition to the
Coulomb energy, the level splitting is significant, this homogeneity will be
destroyed, as can be seen in Fig. 15.7(b), which shows the diamonds for
a carbon nanotube [626]. The alternation of a large diamond with three
smaller ones can be explained in terms of the electronic structure of the
nanotubes [627]. In the case of transport through molecules there is no
obvious underlying structure in the diamonds.
A stability diagram cannot only be used for finding addition energies,
but it can also constitute a spectroscopic tool for revealing subtle excita-
tions that arise on top of the ground state configurations of a dot with a
particular number of electrons on it. This fact has been exploited in dif-
ferent contexts to study the level spectroscopy of a variety of systems such
as metallic nanoparticles [612] or few-electron quantum dots [628]. The
excitations appear as lines running parallel to the Coulomb diamond edges,
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Single-molecule transistors: Coulomb blockade and Kondo physics 439

see dotted lines in the upper panel of Fig. 15.7. At such a line, a new
(electronically or vibrationally) excited state enters the bias window, cre-
ating an additional transport channel. The result is a step-wise increase of
the current and a corresponding peak in the differential conductance. The
energy of an excitation, ∆E in Fig. 15.7(a), can be determined by reading
off the bias voltage of the intersection point between the excitation line and
the Coulomb diamond edge through the same argument that we used for
finding addition energies. The excitations correspond to the charge state of
the Coulomb diamond they end up in [see Fig. 15.7(a)]. The width of the
lines in the dI/dVSD plot (or, equivalently, the voltage range over which the
step-wise increase in the current occurs) is determined by the larger one
of the energies kB T and Γ, which in the Coulomb blockade regime must
be the first one. In practice, this means that sharp lines and thus accurate
information on spectroscopic features are obtained at low temperatures and
for weak coupling to the leads.
There are other important issues like the role of the asymmetry in the
coupling that can be discussed at a qualitative level. For more details,
recommend the review of Ref. [605].

15.4.4 Amplitudes and line shapes: Rate equations

We now want to put in more quantitative terms the statements of the
previous subsection. For this purpose, we shall introduce here the so-called
rate or master equations that allow us to compute, among other things,
the amplitudes of the Coulomb blockade oscillations, the shape of the I-
V curves (Coulomb staircase) and the stability diagrams. This section is
based on Ref. [624].
In the weak coupling regime that we are interested in, the transport is
determined by the tunnel rates from a given level p to the left and right
(p) (p)
reservoirs, see Fig. 15.4, which we shall denote by ΓL and ΓR , respec-
tively.6 We assume here that kB T, ∆E ≫ (ΓL + ΓR ) (for all levels partic-
ipating in the conduction), so that the finite width Γ = (ΓL + ΓR ) of the
transmission resonance through the quantum dot can be disregarded. This
assumption allows us to characterize the state of the quantum dot by a set
6 To (p)
be consistent with our notation in previous chapters, the rates ΓL,R have dimen-
(p)
sions of energy. Thus, ΓL,R /~ gives
the probability per unit of time of having a tunneling
event that connects the level p of dot with the leads. The definition the rates in this
chapter are a factor two larger than those used, for instance, in our discussion of the
resonant tunneling model in Chapter 13.
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440 Molecular Electronics: An Introduction to Theory and Experiment

of occupation numbers, one for each energy level. Notice that the restric-
tion kB T, ∆E ≫ Γ results in the conductance being much smaller than
e2 /h. We also assume conservation of energy in the tunneliing processes,
thus neglecting contributions of higher order in Γ from tunneling via virtual
intermediate states in the quantum dot. We finally assume that inelastic
scattering takes place exclusively in the reservoirs — not in the quantum
dot. The effect of inelastic scattering in the quantum dot is considered in
Ref. [624] (see also Exercise 15.4).
Energy conservation upon tunneling from an initial state p in the quan-
tum dot (containing N electrons) to a final state in the left reservoir at
energy Epf,l (in excess of the local electrostatic potential energy), requires
that7
Epf,l (N ) = Ep + U (N ) − U (N − 1) − (1 − η)eV. (15.15)
Here η is the fraction of the applied voltage V which drops over the right
barrier,8 see Fig. 15.4. The energy conservation condition for tunneling
from an initial state Epi,l in the left reservoir to a final state p in the quantum
dot is
Epi,l (N ) = Ep + U (N + 1) − U (N ) − (1 − η)eV, (15.16)
where N is the number of electrons in the dot before the tunneling event.
Similarly, for tunneling between the quantum dot and the right reservoir
one has the conditions
Epf,r (N ) = Ep + U (N ) − U (N − 1) + ηeV, (15.17)
Epi,r (N ) = Ep + U (N + 1) − U (N ) + ηeV, (15.18)
where Epi,r
and Epf,r
are the energies of the initial and final states in the
right reservoir.
The stationary current through the left barrier equals that through the
right barrier, and is given by
∞
e X X (p)
ΓL P ({ni }) δnp ,0 f (Epi,l (N )) − δnp ,1 f¯(Epf,l (N )) . (15.19)
£ ¤
I=
~ p=1
{ni }

where we have used the shorthand notation f¯(E) ≡ 1 − f (E). The second
summation is over all realizations of occupation numbers {n1 , n2 , . . .} ≡
7 Let us remind that the energies E are measured with respect to the Fermi energy of
p
the leads.
8 Notice that this definition differs from the one of Ref. [624]. This change has been

introduced to preserve the convention our convention about the current direction and
the sign of the bias voltage.
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Single-molecule transistors: Coulomb blockade and Kondo physics 441

{ni } of the energy levels in the quantum dot, each with stationary proba-
bility P ({ni }). here, the numbers ni can take only the values 0 and 1. In
equilibrium, this probability distribution is the Gibbs distribution in the
grand canonical ensemble " ̰ !#
1 1 X
Peq ({ni }) = exp − Ei ni + U (N ) − N EF , (15.20)
Z kB T i=1
P
where N ≡ i ni , and " Z is the
̰ partition function given!#by
X 1 X
Z= exp − Ei ni + U (N ) − N EF . (15.21)
kB T i=1
{ni }
The non-equilibrium probability distribution P is a stationary solution
of the kinetic equation
∂
~ P ({ni }) = 0
∂t h i
(p) (p)
X
=− P ({ni })δnp ,0 ΓL f (Epi,l (N )) + ΓR f (Epi,r (N ))
p
h i
(p) (p)
P ({ni })δnp ,1 ΓL f¯(Epf,l (N )) + ΓR f¯(Epf,r (N ))
X
−
p
X
+ P (n1 , . . . np−1 , 1, np+1 , . . .)δnp ,0
p
h i
(p) (p)
× ΓL f¯(Epf,l (N + 1)) + ΓR f¯(Epf,r (N + 1))
X
+ P (n1 , . . . np−1 , 0, np+1 , . . .)δnp ,1
p
h i
(p) (p)
× ΓL f (Epi,l (N − 1)) + ΓR f (Epi,r (N − 1)) , (15.22)
The kinetic equation, Eq. (15.22), for the stationary distribution function is
equivalent to the set of detailed balance equations (one for each p = 1, 2, . . .)
(p) (p)
P (n1 , . . . np−1 , 1, np+1 , . . .)[ΓL f¯(Epf,l (Ñ + 1)) + ΓR f¯(Epf,r (Ñ + 1))]
(p) (p)
= P (n1 , . . . np−1 , 0, np+1 , . . .)[ΓL f (Epi,l (Ñ )) + ΓR f (Epi,r (Ñ ))], (15.23)
P
with the notation Ñ ≡ i6=p ni . A similar set of equations formed the basis
for the work of Averin, Korotkov, and Likharev on the Coulomb staircase
in the non-linear I-V characteristics of a quantum dot [629–631].
Eq. (15.22), together withX the normalization condition
P ({ni }) = 1, (15.24)
{ni }
form a set of linear algebraic equations that can be easily solved numerically.
For those readers not familiar with rate or master equations, we recommend
Exercise 15.1, in which a single-level dot is considered.
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442 Molecular Electronics: An Introduction to Theory and Experiment

15.4.4.1 Linear response

As shown by Beenakker in Ref. [624], in the linear response regime,
the conductance can be calculated analytically, see also Exercise 15.2.
The result can be expressed in terms of the equilibrium joint probabil-
ity Peq (N, np = 1) that the quantum dot contains N electrons and that
level p is occupied is
X
Peq (N, np = 1) = Peq ({ni })δN,Pi ni δnp ,1 . (15.25)
{ni }
In terms of this probability distribution, the conductance is given by
∞ ∞ (p) (p)
e2 X X ΓL ΓR
G= (p)
P (N, np = 1)f¯(Epf,l (N )),
(p) eq
(15.26)
~kB T p=1
N =1 ΓL + ΓR

where in this case Epf,l (N ) = Ep + U (N ) − U (N − 1) since the bias volt-

age is vanishingly small. This particular product of distribution functions
expresses the fact that tunneling of an electron from an initial state p in
the dot to a final state in the reservoir requires an occupied initial state
and empty final state. The same formula was obtained independently by
Meir, Wingreen, and Lee [632] by solving an Anderson model in the limit
kB T ≫ Γ (see Exercise 8.9).
Eq. (15.26) is valid irrespective of the relative values of the temperature,
charging energy and level splitting. The most relevant limit for molecular
devices is kB T ≪ e2 /C, ∆E. In this case Eq. (15.26) can be written in a
simplified form. Now, the single term with p = N = N0 gives the dominant
contribution to the sum over p and N . If we consider that V0 is the gate
voltage at which the resonance associated with N0 is reached, the depen-
dence of linear conductance on the gate voltage, VG , around V0 is given
by
µ 2¶ (N ) (N ) µ ¶
e π ΓL 0 ΓR 0 −2 eα(VG − V0 )
G(VG , T ) = cosh . (15.27)
h 2kB T Γ(N0 ) + Γ(N0 ) 2kB T
L R
This line shape is characterized by a maximum value of Gmax =
(N ) (N ) (N ) (N )
(e2 /2hkB T )ΓL 0 ΓR 0 /(ΓL 0 + ΓR 0 ) attained when the gate voltage
reaches the resonance V0 ; this is the so-called Coulomb peak. The full-
width at half maximum (FWHM) of this peak is 3.525kB T /(eα), and the
peak height decreases with temperature as 1/T . Notice that Eq. (15.27) is
nothing but the result that we obtained in section 13.2.3 in the context of
the coherent tunneling through a single resonant level, see Eq. (13.6).9 It is
9 Let us remind that in Eq. (15.27) we do not consider spin degeneracy and the tunneling
rates are a factor 2 larger than in Chapter 13.
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Single-molecule transistors: Coulomb blockade and Kondo physics 443

natural to recover this result, since in the Coulomb blockade theory detailed
in this section we have only considered elastic tunneling processes. Indeed,
it is sometimes difficult to distinguish experimentally between Coulomb
blockade effect and coherent transport through a weakly coupled system.
From the theory side, there is an obvious difference. While in the coherent
case the electrons tunnel through the single-particle levels of the dot, in
the Coulomb blockade regime the resonances are also determined by the
charging energy.

15.4.4.2 Non-linear transport: A simple example

In order to illustrate the main features of the non-linear transport char-
acteristics of a quantum dot in the Coulomb blockade regime within the
constant interaction model, we discuss now in detail an example of a two-
level system, see Fig. 15.8(a). This system has two non-degenerate single-
particle levels with energies E1 = 50 meV and E2 = 80 meV, which are
measured with respect to the equilibrium chemical potential of the leads
(set to zero). Level 2 plays the role of an excited state and we shall show
how it is revealed in the different transport characteristics. We assume that
the charging energy is e2 /C = 100 meV, which is larger than the excita-
tion energy ∆E = E2 − E1 = 30 meV. For simplicity, we assume that all
(p)
tunneling rates are identical and equal to ΓL,R = 1 meV (p = 1, 2). The
temperature is kB T = 2.5 meV (i.e. T ≈ 30 K) and we take η = 0.6, where η
is the parameter that describes the portion of the voltage that drops at the
right barrier. Finally, we assume an arbitrary value for the gate coupling
parameter, α, and in the different plots the gate voltage, VG , will carry the
factor α.
In this model there are four possible configurations for the dot: {ni } =
(n1 , n2 ), with (0, 0), (1, 0), (0, 1) and (1, 1). The first configuration has
zero electrons in the dot, the second and the third ones correspond to one
electron in the dot, and the fourth one to two electrons. To determine the
different transport properties we have solved the stationary kinetic equa-
tion, see Eq. (15.22), to obtain the probabilities of the four configurations
and we have then computed the current using Eq (15.19). The details of
this simple calculation can be found in Exercise 15.3. Let us now proceed
to describe the results of this model:
(i) Coulomb oscillations: As we explained in section 15.4.2, in the lin-
ear response regime (VSD ≈ 0), the current can flow when the chemi-
cal potential of the dot equals the equilibrium chemical potential of the
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444 Molecular Electronics: An Introduction to Theory and Experiment

15
(b)
e/C + ∆E/e

dI/dV (µS)
10
0 1 2
(a) 5
E2
∆E
L E1 R 0
0 50 100 150 200 250 300
αVG (mV)
0.3
0.2 (c)
0.1
I (µA)

0
αVG = 0 mV
-0.1 αVG = 50 mV
-0.2 αVG = 100 mV

-0.3
-400 -200 0 200 400 600
VSD (mV)
Fig. 15.8 (a) Two-level model to illustrate the transport characteristics of a quantum
dot in the Coulomb blockade regime within the constant interaction model. The single-
particle energies are E1 = 50 meV and E2 = 80 meV (measured with respect to the
equilibrium chemical potential of the leads). The excitation energy is thus ∆E = E2 −
E1 = 30 meV. The charging energy is chosen to be e2 /C = 100 meV and all tunneling
rates are assumed to be equal to Γ = 1 meV. The temperature is kB T = 2.5 meV (i.e.
T ≈ 30 K) and η = 0.6. (b) Differential conductance vs. the gate voltage (including the
gate coupling constant α) corresponding to the model of panel (a). The source-drain (or
bias) voltage is 20 µV (linear regime). The numbers 0, 1 and 2 indicate the number of
electrons in the different regions separated by the Coulomb peaks. (c) Corresponding
non-linear current-voltage characteristics for several gate voltages.

leads (which we have set to zero): µdot (N, VSD ≈ 0) = (N − 1/2)e2 /C +

EN − eαVG = 0. This implies that conductance peaks will appear at
eαVGN = (N − 1/2)e2 /C + EN , which in this example correspond to
αVG1 = 100 mV and αVG2 = 230 mV. This is illustrated in Fig. 15.8(b).
Notice that the distance between the Coulomb peaks times the electron
charge is equal to the addition energy e2 /C + ∆E = 130 meV.
(ii) Coulomb staircase: In Fig. 15.8(c) we show the corresponding results
for the current as a function of the source-drain or bias voltage (VSD ) for
several values of the gate voltage. Notice that the I-V curves exhibit a
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Single-molecule transistors: Coulomb blockade and Kondo physics 445

series of steps, which correspond to the opening of new channels when the
reservoir chemical potentials cross the different resonances in the dot. Thus
for instance, at VG = 0 and positive bias voltage, the current is blocked
until µdot (N = 1) equals the chemical potential of the left reservoir, i.e.
µdot (N = 1) = e2 /2C + E1 + ηeVSD = eVSD . This occurs at VSD = 250
mV. Then, the next step corresponds to the crossing of the excited state
without changing the net charge in the dot. This requires an additional bias
voltage equal to ∆E/e(1 − η) = 75 mV, which explains the appearance of
a step at VSD = 325 mV. Following this line of reasoning, one can explain
the position of all the steps in the I-V curves.
Two additional features in Fig. 15.8(c) are worth mentioning. First,
notice that the gap in the low-bias voltage region can be completely closed
by increasing the gate voltage, in accordance with the Coulomb oscillations.
Second, notice that the I-V curves are not symmetric with respect to the
inversion of the bias voltage. This is simply due to the fact that we have
chosen an asymmetric electrostatic profile with η 6= 0.5.
(iii) Stability diagrams: As we discussed in section 15.4.3, the different
energy scales and capacitances of the problem can be extracted from the
so-called stability diagrams, where either the current or the differential
conductance are plotted as a function of both the gate voltage and the
bias voltage. In Fig. 15.9 we show the stability diagrams for our two-level
example, which nicely illustrate the main conclusions of our qualitative
discussion in section 15.4.3. In particular, notice that the addition energy
can be extracted from the height of the middle diamond or from its width
(distance between two consecutive degeneracy points). On the other hand,
the energy of the excitation, ∆E = 30 meV, can be read off from the
bias voltage of the intersection point between the excitation line and the
Coulomb diamond edge. The excitation line is particularly visible in the
diagram of the differential conductance. Finally, notice that the diamonds
are “inclined” due to the asymmetric potential profile (η = 0.6).

15.5 Towards a theory of Coulomb blockade in molecular

transistors

There are two assumptions of the constant interaction model that are not
met in general in SMTs. Neither the charging energy nor the level spectrum
are expected to be independent of the number of electrons in a molecule. In
this sense, the orthodox theory of Coulomb blockade may not be adequate
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446 Molecular Electronics: An Introduction to Theory and Experiment

Fig. 15.9 Stability diagrams corresponding to the example of Fig. 15.8. (a) Current
vs. gate voltage and source-drain (or bias) voltage. (b) Differential conductance vs. gate
voltage and source-drain voltage.

for SMTs. The natural question is now how to generalize the standard
theory to deal with molecular devices. It is worth mentioning that this
question has also emerged in the other contexts like few-electron quantum
dots [628] and ultrasmall metallic grains [633].
Part of the answer to this question is rather simple, at least conceptually
speaking. Any theory of Coulomb blockade in SMTs should include an
appropriate description of the molecular many-body spectrum as a function
of the number of electrons in the molecule. In other words, we need as a
starting point the ground state and excited states of the molecules not
only for the neutral species, but also for the stable cations and anions.
In principle, this requires the use of ab-initio (post Hartree-Fock) quantum
chemistry methods like, for instance, the configuration interaction approach
[176]. In practice, both model Hamiltonians and approximate methods like
density functional theory10 have been used for this purpose. Once the
many-body spectrum for different number of electrons is known, one needs
to solve a master equation to determine the occupation of the different
states and finally their contribution to the current. By now many authors
10 Density functional theory is not designed to give the level spectrum of a system and,

although it may give reasonable results for neutral molecules, it has severe problems with
charged species (anions and cations) [273].
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Single-molecule transistors: Coulomb blockade and Kondo physics 447

have implemented such a procedure at different levels of sophistication, and

we just mention here a few works from which it should be easy to trace back
the entire literature [634–645].
Another important aspect that a theory of Coulomb blockade in SMTs
should account for is the possible renormalization of the molecular levels due
to the surrounding electrodes. As we shall discuss in the section 15.7, the
addition energies found experimentally in SMTs are clearly much smaller
than what is expected from the known ionization potential and the electron
affinity of the molecules in gas phase. It has been suggested that this
reduction of the addition energies is caused by image charges in the metallic
electrodes [622], giving rise to a localization of the charges near the leads.
This issue is a crucial one and surprisingly it has received little attention
so far. Fortunately, some groups are starting to tackle this problem, see
Ref. [646] and references therein, but this issue is by no means yet settled.
In the rest of this section, we shall sketch how more realistic single-
tunneling theories for SMTs are formulated, and we shall present a simple
example to illustrate those theories. The rest of this subsection is rather
technical and it can be skipped in a first reading.

15.5.1 Many-body master equations

We describe in this subsection how to compute the current through a SMT
in the weak-coupling regime within a model where the molecular part is
described with a truly microscopic many-body approach. This discussion
is based on Ref. [635], which we strongly recommend.
The starting point of our description of a molecular junction is the
P
following model Hamiltonian: H = HM + r Hr + HT , which incorporates
the molecule (M ), reservoirs r = L, R and the tunneling (T ). The last two
terms are given by
ǫkσ a†kσr akσr
X
Hr = (15.28)
kσ

tri a†kσr ciσ + h.c.

X
HT = (15.29)
kiσr
The molecular part HM is chosen to contain the various strong interac-
tions on the molecule which require an exact treatment. Generally, it is
expressed in a basis of single-electron operators ciσ for orbitals on the
molecule labeled by i and spin projection σ. After diagonalization one can
P
write HM = s Es |sihs|, where the discrete molecular many-body states
have a summary label s which includes the total charge N , spin S, and other
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448 Molecular Electronics: An Introduction to Theory and Experiment

possible quantum numbers. Eq. (15.28) models the electrodes r = L, R as

non-interacting quasi-particle reservoirs which are fixed at electro-chemical
potential µr = µ ± eV /2 and temperature T . Here, akσr are electron oper-
ators of electrode r labeled by k and spin σ. For simplicity, the density of
states in the electrodes ρe is assumed to be flat around the Fermi-energy
in order to focus on effects of the molecular part. The tunneling term,
Eq. (15.29), describes charge transfer between electrode and molecule on
a very small time scale. The level-dependent coupling strength is charac-
terized by the intrinsic line width Γri = 2π|tri |2 ρe , where ρe is the density
of states of the electrodes at the Fermi energy. Γ = max {Γri } denotes the
overall coupling strength (in units of energy) between electrodes and the
molecule and serves to define the scale of the current. The molecular states
may additionally be coupled to the electromagnetic field (photons) and/or
a mechanical environment which dissipate the energy accumulated on the
molecule due to the tunneling. Whereas this does not change the charge
on the molecule, it does have an effect on the non-equilibrium distribution
of the molecular states and may thereby strongly influence the current, see
Ref. [635] for more details. We do not explicitly discuss here the coupling
to such bosonic reservoirs.11
Since we are particularly interested in the Coulomb blockade regime, it
is important to treat the strong intramolecular interactions exactly, while
the tunneling to the reservoirs is treated in a systematic perturbation theory
in the electrode-molecule tunneling, Eq. (15.29). Assuming that Γ ≪ kB T ,
one only needs to compute the current to the lowest order in Γ. As in the
constant interaction model, such a lowest order perturbation theory leads
to master equations for the occupation probabilities ps of molecular many-
body states s. In principle, this approach can be improved systematically
by going to higher orders in Γ. We summarize now the equations necessary
to compute the current in lowest order perturbation theory in the coupling
strengths Γ. Their systematic derivation using a diagrammatic technique
has been discussed in [647, 648]. For time-independent applied bias, the
time derivative of the probabilities ps vanishes in the stationary state. The
stationary nonequilibrium probabilities pst s are uniquely determined by the
transition rates Wss′ from state s′ to s (forming a matrix Ŵ ) through the
stationary master equation Ŵ p~ st = 0 together with the normalization of

11 This will be done in section 17.3.1 in the context of the study of the influence of the

electron-phonon interaction in the transport properties of a molecular junction in the

Coulomb blockade regime.
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Single-molecule transistors: Coulomb blockade and Kondo physics 449

the distribution ps for arbitrary times. We can then write

ˆ )−1~v ,
p~ st = (W̃ (15.30)

where the matrix W̃ ˆ is identical to Ŵ but with one (arbitrarily chosen) row
s0 replaced with (Γ, · · · , Γ) and ~v is a vector, vs = Γδs,s0 . The transition
rates Wss′ , with s 6= s′ (in the absence of bosonic coupling) are the sum
r
P
Wss′ = r Wss ′ of the Golden rule rates for the tunneling of an electron

to/from electrode r = L, R:
X fr (Es − Es′ )¯ P tr hs|c† |s′ i¯2 Ns′ < Ns
( ¯ ¯
r i i iσ
Wss′ = 2πρe 2 , (15.31)
f¯r (Es′ − Es )¯ i tri hs|ciσ |s′ i¯ Ns′ > Ns
¯ P ¯
σ

where f (x) = 1/(exp(x/kB T ) + 1) is the Fermi function, fr (x) = f (x − µr )

and f¯r (x) = 1 − f (x − µr ). The decay rates are Wss = − s′ 6=s Ws′ s .
P

The stationary current I = hIi is related to the current operator

I = (IR − IL )/2. We can use the symmetrized combination of currents
Ir = −i(e/~) ikσ (tri a†kσr ciσ − h.c.) into electrode r = L, R since in the
P

stationary limit hIR i = −hIL i. The current can be explicitly calculated

from the expression
e T I st
I= ~e Ŵ p~ . (15.32)
2~
The vector ~e is given by es = 1 for all s. The rates entering the current are
I R L
Wss ′ = ±(Wss′ − Wss′ ), Ns ≶ Ns′ . (15.33)
The inclusion of dissipative environments (photons, phonons) modifies only
the rates Ŵ and thereby p~ st , but the rates Ŵ I are not affected. For an
alternative, but equivalent, formulation of these master equations, see for
instance Ref. [637].

15.5.2 A simple example: The Anderson model

In order to illustrate the formalism discussed above, let us now apply it to
a simple example. We first need a Hamiltonian for the molecular part. Let
us assume that its electronic structure can be described by a single-level
Anderson model (see section 5.4.3 and Appendix A)
X
HM = ǫ0 nσ + U n↑ n↓ . (15.34)
σ

Here, ǫ0 is the energy of a (spin-degenerate) resonant level, nσ = c†σ cσ is

the number operator which describes the occupation of that level for spin σ
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450 Molecular Electronics: An Introduction to Theory and Experiment

20 0.3
(a) U/e 0.2 (b)
dI/dV (µS)

15
0.1

I (µA)
10 0 1 2 0
αVG = 0 mV
-0.1 αVG = 25 mV
5
-0.2 αVG = 50 mV
0 -0.3
0 50 100 150 200 -400 -200 0 200 400
αVG (mV) V (mV)
Fig. 15.10 (a) Differential conductance vs. the gate voltage (including the gate coupling
constant α) for a molecular transistor described with the Anderson model. Here, ΓL =
ΓR = 1 meV, ǫ0 = 50 meV and U = 100 meV. The temperature is kB T = 2.5 meV (i.e.
T ≈ 30 K) and the bias voltage is 20 µV (linear regime). The numbers 0, 1 and 2 indicate
the number of electrons in the different regions separated by the Coulomb peaks. (b)
Corresponding non-linear current-voltage characteristics for several gate voltages. The
voltage is assumed to drop symmetrically at the interfaces.

and U is the Coulomb repulsion energy on the molecule. This Hamiltonian

has four eigenstates: |1i ≡ |00i with N = 0, |2i ≡ |10i and |3i ≡ |01i with
N = 1 and |4i ≡ |11i with N = 2, where N is the number of electrons
on the molecule. Here, we have used the notation |n↑ n↓ i, where nσ = 0, 1
is the occupation number of the single-particle level with spin σ. The
corresponding eigenenergies are: E1 = 0, E2 = E3 = ǫ0 and E4 = 2ǫ0 +
U . In this case, it is straightforward to compute the transition rates of
Eq. (15.31) (see Exercise 15.5). They are determined by the scattering rates
Γr ≡ 2πρe |tr |2 (r = L, R). Once the transition rates have been computed,
one can obtain the stationary probabilities ps (s = 1, ..., 4) for the four
states by solving numerically the 4 × 4 master equation, see Eq. (15.30).
Finally, the current can be calculated from Eq. (15.32).
In Fig. 15.10 we present the results of this model for ΓL = ΓR = 1 meV,
ǫ0 = 50 meV and U = 100 meV. Panel (a) of this figure shows the linear
conductance versus the gate voltage. Here, this voltage has been introduced
by shifting rigidly the level position, i.e. ǫ → ǫ − αVG , where α is the gate
coupling parameter. As one can see, the conductance exhibits Coulomb
peaks at the degeneracy points. Notice that these peaks are separated by
a distance U/e, which indicates that U plays here the role of the charging
energy (the level splitting is zero in this case).
Fig. 15.10(b) shows the current as a function of the bias (source-drain)
voltage for different values of the gate voltage. Here, we have assumed that
the bias voltage drops symmetrically at both interfaces, i.e. η = 0.5. Due
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Single-molecule transistors: Coulomb blockade and Kondo physics 451

to the spin degeneracy, the I-V curves only exhibit two plateaus. Notice
that the two steps (for a given voltage polarity) are separated by a distance
U/(eη).
In this example the transport characteristics are very similar to those
obtained with the constant interaction model (orthodox theory). The differ-
ences become more pronounced when there are more charge states involved
or additional quantum numbers play a fundamental role.

15.6 Intermediate coupling: cotunneling and Kondo effect

So far in this chapter, we have focused on the limit where the tunnel cou-
pling, Γ, is much smaller than any other energy scale in the problem. If
this coupling strength is increased, higher-order tunneling processes begin
to give a significant contribution to the transport properties [649]. In the
opposite limit (strong coupling regime) where Γ ≫ e2 /C, ∆E, kB T , the
electronic states in the molecule and electrodes are strongly hybridized. In
that case, as we have discussed in previous chapters, the elastic coherent
tunneling dominates transport and signatures of the Coulomb blockade are
washed out by quantum fluctuations of the molecular charge. Between the
weak coupling and strong coupling regime one can identify a third regime
which we shall refer to as the intermediate coupling regime. In this regime
it is still possible to observe Coulomb diamonds, but higher-order processes
lead to a non-negligible current inside the blockade regions. In this section
we shall discuss three different types of higher-order tunneling processes:
elastic and inelastic cotunneling and spin-flip cotunneling. This latter pro-
cess is behind the appearance of the Kondo effect.

15.6.1 Elastic and inelastic cotunneling

The first process that we want to describe is the so-called elastic cotunnel-
ing process. This second-order process is illustrated in Fig. 15.11. In the
situation depicted in this figure, energy conservation forbids the number of
electrons to change as this would cost an energy ∆E, which is not avail-
able at the bias voltage considered in Fig. 15.11. Nevertheless, an electron
can tunnel off the molecule, leaving it temporarily in a classically forbid-
den virtual state (middle diagram in Fig. 15.11). By virtue of Heisenberg’s
energy-time uncertainty principle this is allowed as long as another electron
tunnels into the molecule in the same quantum process in order not to vi-
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452 Molecular Electronics: An Introduction to Theory and Experiment

initial state virtual state final state

01 00
11 1
0 00
11 11
00 1
0
10 µ(N+1) 11 00 0
1 00
11 00 µ(N+1)
11 0
1
1010 dot 11 00
00
11 0
1
0 dot 00
1 µ(N+1) 11
00
11 00
11
00 dot
11 0
1
0
1
10 00
11 0
1 00
11 00
11 0
1
10 00
11 0
1 00
11 00
11 0
1
10 00
11 0
1 00
11 00
11 0
1
1010 00
11
00 e2/C + ∆E
11 0
1
0
1 00
11
00
11 00
11
00
11 0
1
0
1
10 00
11 0
1 00
11 00
11 0
1
1010 00
11 0
1 00
11 00
11 0
1
00
11 0
1 00
11 00
11 0
1
1010 00
11 0
1 00
11 00
11 0
1
L 1010 µdot (N) 11
00
00
11 L 0
1
0
1 µ (N) 11
00 L 00
11 µ (N) 0
1
dot 11
00 00
11 0
1 R
11
00 R 0
1 11
00 R 00
11 dot 0
1
1010 00
11
00
11 0
1
0
1 00
11
00
11 00
11
00
11 0
1
0
1
1010 00
11 0
1 00
11 00
11 0
1
00
11 0
1 00
11 00
11 0
1
10 00
11 0
1 00
11 00
11 0
1
eVSD

Fig. 15.11 Elastic cotunneling process. The N th electron on the dot jumps to the drain
(virtual state) to be immediately replaced (final state) by an electron from the source
(black arrow sequence). A similar process involves the unoccupied state (light arrow
sequence). In both examples, an electron is effectively transported from source to drain.

olate energy conservation. The final state then has the same energy as the
initial one, but one electron has been transported through the molecule.
This elastic cotunneling process is analogous to the superexchange mecha-
nism in chemical electron transfer theory [582]. It occurs at arbitrarily low
bias as the energy of the tunneling electron and the molecule are unchanged
and leads to a nonzero background conductance in the blockade regions.
A cotunneling event that leaves the molecule in an excited state is called
inelastic. An example of such a process is depicted in Fig. 15.12. As one can
see, the onset of the cotunneling event occurs at eVSD = ∆E, the condition
dictated by the energy conservation principle. In transport measurements,
inelastic cotunneling appears in the stability diagram inside the Coulomb
diamonds as two symmetric lines running parallel to the gate axis as repre-
sented by grey lines in Fig. 15.14(a). Their energy, ∆E, is the distance of
the excitation to the zero-bias axis as illustrated in Fig. 15.14(a). Further-
more, the inelastic cotunneling line is expected to intersect, at the diamond
boundary, the corresponding excitation line inside the single-electron tun-
neling region [650].
It is worth stressing that that higher-order coherent processes appear as
sharp spectroscopic features as the conductance of the i-th order process is
proportional to Γi , while for first-order incoherent single-electron tunneling,
the current is proportional to Γ.
If the electron spin is taken into account, one can encounter another
elastic cotunneling process connected to the Kondo effect. This will be
analyzed in detail in the next subsection.
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Single-molecule transistors: Coulomb blockade and Kondo physics 453

initial state virtual state final state

11
00 00
11 0110 00
11 11
00 00
11
00
11 µ(N+1) 11
00 1010 µ(N+1) 11 00 00
11 µ(N+1) 11
00
00
11 00
11 00
11 00
11 00
11
00
11 dot 00
11 00
11 00
11 dot 00
11
00
11 00
11 1010 dot
00
11 00
11 00
11
00
11 00
11 00
11 00
11 00
11
00
11 00
11 1010 00
11 00
11 00
11
00
11 00
11 00
11 00
11 00
11
00
11 00
11 1010 00
11 00
11 00
11
00
11 00
11 00
11 00
11 00
11
00
11 00
11 1010 00
11 00
11 00
11
00
11 ∆E 1100 00
11 00
11 00
11
00
11 00
11 1010 00
11 00
11 00
11
L 00
11 00
11 00
11 00
11 00
11
00
11 µ (N) 00
11 L 1010 00
11 L 00
11 µ (N) 11
00 R
00
11 dot 11
00 1010 dot 00 (N) 11
R µ R 00
11 dot 11
00
00
11 00
11 00
11 00
11 00
11
00
11 00
11 00
11 00
11 00
11
00
11 00
11 1010 00
11 00
11 00
11
00
11 00
11 00
11 00
11 00
11
eVSD 00
11 00
11 10 00
11 00
11 00
11

Fig. 15.12 Inelastic cotunneling process. For eVSD ≥ ∆E, the N th electron on the dot
may jump from the ground state to the drain (virtual state) to be immediately replaced
by an electron from the source (final state), which enters the excited state.

15.6.2 Kondo effect

The Kondo effect is a many-body phenomenon that occurs when a local-
ized spin interacts with surrounding conduction electrons [651]. This effect
is known to be the origin of the resistance increase at low temperatures
in metals with magnetic impurities [652]. In recent years, there has been
a renewed interest in the Kondo effect thanks to its observation in a va-
riety of nanodevices [653]. Thus for instance, Kondo physics has been
reported in the last decade in semiconductor quantum dots [654–656], in
magnetic impurities on the surface of metals [657–659], and carbon nan-
otubes [660, 661]. More importantly for our discussion, this phenomenon
has also been observed in single-molecule transistors [23, 24, 662–664] and
for this reason, we shall briefly review in this section the basics of this ef-
fect. For a detailed discussion of the Kondo physics in different mesoscopic
systems, see Refs. [653, 665–667].
In molecular transistors, and quantum dots in general, the Kondo effect
can arise when the molecule has a net spin (magnetic moment). This,
for example, occurs for an odd occupancy in the molecule (one electron
is unpaired, S = 1/2). We shall mainly consider this simple case. The
conduction mechanism for the Kondo effect involves spin-flip events such
as the one illustrated in Fig. 15.13(a). The Heisenberg uncertainty principle
allows the electron to tunnel out for only a short time of about ~/|ǫ0 |, where
ǫ0 is the energy of the electron relative to the Fermi energy. During this
time, another electron from the Fermi level at the opposite lead can tunnel
onto the dot, keeping the total energy of the system conserved (elastic
cotunneling). The exchange interaction causing the majority spin in the
leads to be opposite to the original spin of the dot causes the probability
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454 Molecular Electronics: An Introduction to Theory and Experiment

(a) initial state virtual state final state (b)

01 01 01 0
1 1
0 1
0 density of states
10 µ(N+1) 10 10 0
1 0 µ(N+1)
1 0
1
1100 1100 1100 µ(N+1) 1 0
0
1 0
1
0 dot
1 0
1
0
1
10 dot 10 10 dot 0
1 0
1 0
1
10 10 10 0
1 0
1 0
1
10 10 10 0
1 0
1 0
1
1010 1010 1010 0
1
0
1 0
1
0
1 0
1
0
1 TK
10 10 10 0
1 0
1 0
1

energy
10 ε0 10 10 0
1 0
1 0
1
10 10 10 0
1 0
1 0
1
10 10 10 0
1 0
1 0
1
1010 1010 1010 0
1 0
1 0
1
L 1010 µdot (N) 1010 R L 1010 µdot (N) 1
0
0
1 R L 0
1
0
1 µdot
(N)
0
1
0
1 R Γ
1
0 0
1 0
1
1010 1010 1010 0
1 0
1 0
1
1010 1010 1010 0
1
0
1 0
1
0
1 0
1
0
1
0
1 0
1 0
1

Fig. 15.13 (a) Spin-flip cotunneling. A spin-up electron jumps out of the dot (virtual
state) to be immediately replaced by a spin-down electron (final state). (b) Kondo
resonance in the density of states that appears as a consequence of the spin-flip tunneling
processes.

for the new electron to have spin opposite to the first to be very high.
This spin exchange qualitatively changes the energy spectrum of the
system. When many such processes are taken together, one finds that a
new state, known as Kondo resonance, is generated exactly at the Fermi
level, see Fig. 15.13(b).12 In this situation, the localized spin is completely
screened and the many-boy ground state turns out to be a singlet state
(S = 0). It is important to note that the Kondo state is always “on reso-
nance” since it is fixed to the Fermi energy. Even though the system may
start with an energy, ǫ0 , which is very far away from the Fermi energy, the
Kondo effect alters the energy of the system so that it is always on reso-
nance. For this reason, these many-body correlations can lead to a great
enhancement of the conductance. The only requirement for this effect to
occur is that the system is cooled to sufficiently low temperatures below
the Kondo temperature TK (see next paragraph).
The width of the Kondo resonance is proportional to the characteristic
energy scale for Kondo physics, the so-called Kondo temperature TK . For
ǫ0 ≫ Γ, TK is given by [668]
√ · ¸
ΓU πǫ0 (ǫ0 + U )
kB TK = exp . (15.35)
2 ΓU
Here, Γ is the coupling strength and U can be seen as the charging en-
ergy, e2 /C. Typical Kondo temperatures are TK ∼ 1 K for semiconductor
quantum dots [655], TK ∼ 1 − 10 K for carbon nanotubes [660, 661] and
TK ∼ 20 − 50 K for molecular devices [23, 24, 662, 664]. This increase of
TK with decreasing dot size can be understood from the prefactor, which
12 Insection 6.9 we presented a discussion of the origin and description of the Kondo
resonance in the framework of the Anderson model.
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Single-molecule transistors: Coulomb blockade and Kondo physics 455

Fig. 15.14 Schematic representation of the main characteristics of the Kondo effect in
electron transport through a molecular quantum dot. (a) In the stability diagram, the
Kondo effect results in a zero-bias resonance (white line) for an odd number of electrons
in the dot. Inelastic cotunneling excitations appear as lines running parallel to the gate
axis at finite bias. (b) For T ≪ TK , the full width at half-maximum (FWHM) of the
Kondo resonance is ∼ kB TK . (c) Temperature dependence of the Kondo-peak height in
the middle of the Coulomb diamond. Reprinted with permission from [606]. Copyright
2008 IOP Publishing Ltd.

contains the charging energy (U = e2 /C). Notice that the Kondo temper-
ature depends on the position of the level and therefore it can be tuned in
three-terminal devices by means of the gate voltage.
The theoretical description of the Kondo effect is very challenging. The
reason is that below TK , high order spin-flip processes contribute signifi-
cantly to both the electronic structure and the transport properties. This
implies that one needs to employ non-perturbative methods to describe
properly this phenomenon. Different many-body methods have been used
to account for the Kondo correlations in quantum dots and related struc-
tures. The description of such techniques is out of the scope of this book
and in the rest of this subsection we shall concentrate ourselves on the dis-
cussion of its main transport characteristics and refer the interested reader
to Refs. [651, 652, 669, 670, 665, 666] for more details about the theory.
The Kondo effect is manifested in the stability diagrams as a zero-
bias resonance in the differential conductance, dI/dVSD , versus VSD , in-
side the Coulomb diamond connecting both degeneracy points as shown
in Fig. 15.14(a). For an even number of electrons with all spins paired,
S = 0 and there is no Kondo resonance. This even-odd asymmetry is very
helpful in assigning the parity of the charge state which can then add extra
information to the understanding of the spectroscopic features observed in
the stability plots. In the low temperature limit (T ≪ TK ), the full width
at half-maximum (FWHM) of the Kondo resonance, as observed in a plot
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456 Molecular Electronics: An Introduction to Theory and Experiment

of the differential conductance versus VSD , is of the order of kB TK , see

Fig. 15.14(b). In the middle of the Coulomb diamond, the linear conduc-
tance exhibits a characteristic temperature dependence given by [670, 655]
G0
G(T ) = £ ¤s , (15.36)
1+ (21/s − 1)(T /TK )2
where G0 = 2e2 /h and s = 0.22 for S = 1/2. (Sometimes in this formula a
temperature-independent offset is included.) This dependence is schemat-
ically drawn in Fig. 15.14(c). Notice that the conductance increases loga-
rithmically with decreasing temperature and saturates at a value 2e2 /h at
the lowest temperatures in the case of symmetric lead-dot coupling. The
latter is commonly referred to as the Kondo effect in the unitary limit [671]
(see Exercise 8.10).
In a magnetic field the Zeeman splitting of the Kondo resonance leads
to the observation of two Kondo peaks symmetric in bias, separated by
twice the Zeeman energy. On the other hand, although we have been only
considering S = 1/2, it is important to note that other types of Kondo
systems are possible owing to orbital degeneracies [672] or triplet states
[673–675]; these can lead to a violation of the parity effect.

15.7 Single-molecule transistors: Experimental results

In this section we shall review some of most representative results obtained

in single-molecule transistors (SMTs). We also recommend the following
reviews on this subject [39, 603, 604, 606].
The question of whether charging effects play a major role in the con-
duction through molecular contacts arose immediately after the report of
the first transport measurements in single-molecule junctions. Thus for
instance, in the work of Reed and coworkers on benzenedithiol molecules
[16], see section 14.1.1, one may argue that the conductance gap observed
at low bias, see Fig. 14.1, is due to Coulomb blockade rather than to the
HOMO-LUMO gap of this molecule.13 Similar questions emerged in the
early work of Kergueris et al. [676]. In this case, the authors reported room
temperature measurements of the I-V characteristics of bisthiolterthiophene
molecules using the technique of microfabricated MCBJ. Zero-bias conduc-
13 The first maximum in the conductance in Fig. 14.1 at voltages above 1 V has a width
that is larger than the temperature. This indicates that, strictly speaking, this system
is not in the Coulomb blockade regime. Of course, this does not exclude that electronic
correlations of some sort are at work.
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Single-molecule transistors: Coulomb blockade and Kondo physics 457

Fig. 15.15 I-V curves recorded at room temperature in a gold-bisthiolterthiophene-gold

junction formed with the microfabricated MCBJ technique. Curves are shifted vertically
for clarity. Reprinted with permission from [676]. Copyright 1999 by the American
Physical Society.

tances were measured in the 10-100 nS range and different kinds of nonlinear
I-V curves with steplike features were reproducibly obtained. An example
of the results of these measurements can be seen in Fig. 15.15. Notice
that the I-V curves resemble very much the Coulomb staircase observed in
quantum dots. Indeed, the authors were able to fit the experimental results
within the framework of the ortodox Coulomb blockade theory described
in section 15.4, taking into account the discrete nature of the electronic
spectrum of the molecule. Let us mention that charging energies of the
order of 0.2 eV were used in the fits (see Ref. [676] for more details).
As we have discussed in previous sections, an unambiguous confirmation
that characteristics like the ones shown in Fig. 15.15 are a consequence of
the occurrence of charging effects requires the implementation of a gate elec-
trode, which is very challenging in the case of MCBJs. To our knowledge,
the first three-terminal single-molecule experiment was reported by Park et
al. [22] in 2000. These authors prepared single-C60 junctions by depositing
a dilute toluene solution of C60 onto a pair of connected gold electrodes
fabricated using e-beam lithography. A gap of 1 nm between these elec-
trodes was then created by electromigration [21]. The entire structure was
defined on a SiO2 insulating layer on top of a degenerately doped silicon
wafer which served as a gate electrode that modulates the electrostatic po-
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458 Molecular Electronics: An Introduction to Theory and Experiment

Fig. 15.16 I-V curves obtained at T = 1.5 K from a single-C60 transistor fabricated with
the electromigration technique. The curves corresponds to five different gate voltages.
The inset shows a schematic diagram of an idealized single-C60 transistor. Reprinted by
permission from Macmillan Publishers Ltd: Nature [22], copyright 2000.

tential of C60 . A schematic diagram of an idealized single-C60 transistor is

shown in the inset of Fig. 15.16.
Fig. 15.16 shows some typical I-V curves obtained in Ref. [22] at differ-
ent gate voltages. Notice that the device exhibited a strongly suppressed
conductance near zero bias voltage followed by step-like current jumps at
higher voltages. The voltage width of the zero-conductance region (con-
ductance gap) could be changed in a reversible manner by changing the
gate voltage. These transport features clearly confirm that the conduction
in this device is dominated by the Coulomb blockade effect and it can thus
be stated that this experiment constitutes the first true example of a SMT
reported in the literature.
A further confirmation of the underlying transport mechanism came
from the analysis of the stability diagrams. In Fig. 15.17 we reproduce
the results for the differential conductance as a function the bias and gate
voltages. As one can see, two diamond-like regions can be identified cor-
responding to two charge states of the C60 molecules. In these plots, the
peaks in the conductance, which correspond to the step-like features in
Fig. 15.16, show up as lines. As seen clearly in Fig. 15.17, the size of he
conductance gap and the peak positions evolve smoothly as the gate voltage
is varied. As the gate voltage is varied further away in both positive and
negative directions, the conductance gap continues to widen and exceeds
150 mV in some devices. This indicates that the charging energy of the C60
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Single-molecule transistors: Coulomb blockade and Kondo physics 459

Fig. 15.17 Different conductance plots as a function of the bias voltage V and the gate
voltage Vg obtained from four different devices. The dark triangular regions correspond
to the conductance gap, and the bright lines represent peaks in the differential conduc-
tance. The arrows mark the point where the conductance lines intercept the conductance
gap. Reprinted by permission from Macmillan Publishers Ltd: Nature [22], copyright
2000.

molecule in this geometry can exceed 150 meV. This value is much larger
than in semiconductor quantum dots.
Notice that in the stability diagrams of Fig. 15.17 there are running
lines that intersect the main diamonds or conductance gap regions. As
we explained in previous sections, this indicates the presence of internal
excitations of the C60 molecules. The energies of these excitations (of a few
meV) are too small to correspond to electronic excitations. Moreover, some
of these lines are observed for both charge states and multiple excitations
with the same spacing are observed (see Fig. 15.16). These observations
suggest that these lines may correspond to the excitation of vibration modes
of the C60 molecules. The lowest-energy mode is known to have around
33 meV and this could explain some of the lines seen in the experiment.
However, internal vibrational modes cannot account for the observed 5-
meV features in Fig. 15.17. The authors of Ref. [22] suggested that this
line could correspond to the excitation of the center-of-mass oscillation of
C60 within the confinement potential that binds it to the gold surface.14
14 The signatures of the excitation of vibration modes in the transport characteristics
will be discussed in detail in the next two chapters.
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460 Molecular Electronics: An Introduction to Theory and Experiment

After the first observation of the Coulomb blockade in molecular junc-

tions, it was clear that to observe other single-electron tunneling phenom-
ena was just a matter of time. Two years after the experiment on the
C60 transistor, the observation of the Kondo effect was reported simultane-
ously and independently by two groups [23, 24]. In the work of Park et al.
[23], two related molecules were examined containing a Co ion bonded to
polypyridyl ligands, attached to insulating tethers of different length. The
two molecules ([Co(tpy-(CH2 )5 -SH)2 ]2+ and [Co(tpy-SH)2 ]2+ ) differed by
a five-carbon alkyl chain within the linker molecules. These molecules were
selected because it is known from electrochemical studies that the charge
state of the Co ion can be changed from 2+ to 3+ at low energy. The role
of the linkers was to control the strength of the metal-molecule coupling. In
this work the SMTs were fabricated using the electromigration technique
with Au wires coated with the molecules. For the longer molecule, the
transport results at temperatures of ∼ 100 mK exhibited all the character-
istic features of the Coulomb blockade effect. In particular, they observed
two diamond-like regions which were associated to the two charge states of
the Co ion.
For the shorter molecule, a significantly larger conductance owing to the
shorter tether length was expected. The main results for this molecule are
summarized in Fig. 15.18. The differential conductance for one such device
is shown in Fig. 15.18(b). The most notable property is a peak at V = 0.
The peak has a logarithmic temperature dependence between 3 and 20 K,
see Fig. 15.18(c). The peak also splits in an applied magnetic field, as one
can see in Fig. 15.18(d), with a splitting equal to 2gµB H, where g ≈ 2 and
µB is the Bohr magneton.
As we explained in section 15.6.2, all these observations are consistent
with the occurrence of the (S = 1/2) Kondo effect. The observation of this
effect is consistent with the fact that the Co2+ ion has S = 1/2. By setting
the low-temperature full-width at half-maximum of the Kondo peak equal
to 2kB TK /e, where TK is the Kondo temperature, the authors estimated
that TK in different devices varied between 10 and 25 K. These large Kondo
temperatures indicate that the coupling between the localized state and the
electrodes is strong, consistent with the high conductances found for the
shorter linker molecule.
In the work of Liang et al. [24], the Kondo effect in SMTs was reported,
where an individual divanadium molecule served as a spin impurity. These
authors also used electromigrated break junctions to form the molecular
contacts. In Fig. 15.19 we show the results for the stability diagrams for
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Single-molecule transistors: Coulomb blockade and Kondo physics 461

Fig. 15.18 Observation of the Kondo effect in a single-molecule transistor fabricated

with the electromigration technique. (a) Breaking trace of a gold wire with adsorbed
[Co(tpy-SH)2 ]2+ at 1.5 K. After the wire is broken the current level suddenly increases
(dot) owing to the incorporation of a molecule in the gap. (b) Differential conductance
of a [Co(tpy-SH)2 ]2+ device at 1.5 K showing a Kondo peak. The inset shows ∂I/∂V
for bare gold point contacts for comparison. (c) The temperature dependence of the
Kondo peak for the device shown in (b). The inset shows the V = 0 conductance as a
function of temperature. The peak height decreases logarithmically with temperature
and vanishes around 20 K. (d) Magnetic-field dependence of the Kondo peak. The peak
splitting varies linearly with magnetic field. Reprinted by permission from Macmillan
Publishers Ltd: Nature [23], copyright 2002.

two single-molecule devices, designated as D1 and D2. Two distinct char-

acteristics are evident in the behavior of both devices. Each displays two
conductance-gap regions, I and II, bounded by two broad ∂I/∂V peaks that
slope linearly as a function of the gate voltage, Vg . These peaks cross at a
gate voltage at which the conductance gaps vanish. Moving away from this
point, the gaps in both regions continue to widen even beyond V = 100 mV.
Most significantly, the devices also exhibit a sharp zero-bias ∂I/∂V peak
in region I, whereas this peak is clearly absent in region II. This feature
strongly suggests the occurrence of the Kondo effect. In order to confirm
this impression, the authors carried out an analysis of the temperature and
magnetic field dependence of the differential conductance. Most of the re-
sults were relatively well explained in terms of the S = 1/2 Kondo effect,
but in particular the behavior of TK suggested that maybe also the orbital
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462 Molecular Electronics: An Introduction to Theory and Experiment

(a) (b)
V (mV)

V (mV)
Vg (V) Vg (V)

Fig. 15.19 Observation of the Kondo effect in a V2 single-molecule transistor fabricated

with the electromigration technique. The two panel show plots of differential conductance
∂I/∂V as a function of bias voltage (V ) and gate voltage (Vg ) obtained from two different
single-V2 transistors. Both measurements were performed at T = 300 mK. The values are
represented by the color scale, which changes in (a), from dark (0) to bright (1.55e2 /h)
and in (b), from dark (0) to bright (1.3e2 /h). The labels I and II mark two conductance-
gap regions, and the diagrams indicate the charge and spin states of the V2 molecule
in each region. Reprinted by permission from Macmillan Publishers Ltd: Nature [24],
copyright 2002.

degrees of freedom were playing an important role in the Kondo resonance

(due to the V ion spin structure), see Ref. [24] for details.
Most three-terminal single-molecule experiments have been carried out
with the electromigration technique. An interesting exception is the work
of Kubatkin and coworkers of Ref. [622], where the angle evaporation tech-
nique that we described in section 15.3 was employed. These authors
reported transport measurements through a single p-phenylenevinylene
oligomer, which has five benzene rings connected through four double
bonds (OPV5). The main experimental result of this work is reproduced in
Fig. 15.20, where one can see up to eight different diamonds in the stability
diagram. This suggests that the transport experiment had access to many
different charge or redox states, which is very unusual in molecular tran-
sistors. Electrochemistry confirms however that this molecule can indeed
have several stable redox states [677]. Even more surprising is the fact the
addition energies extracted from the stability diagrams differ largely from
those obtained from electrochemistry and computational methods. Spe-
cially dramatic is the deviation in the case of the neutral molecule. While
the spectroscopic HOMO-LUMO gap for this molecule is of the order of 2.5
eV, the extracted one from the central Coulomb diamond was one order of
magnitude smaller (∼ 0.2 eV). The authors argued that this discrepancy
is due to the fact that the intrinsic electronic levels of the molecules are
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Single-molecule transistors: Coulomb blockade and Kondo physics 463

Fig. 15.20 Experimental results of the transport characteristic of an OPV5 single-

molecule transistor. (a) Differential conductance as a function of bias voltage, Vs−d ,
and gate voltage, Vg . The full solid line at the top of the figure shows a representative
trace of the current versus Vg . (b) Examples of the I-V curves at different gate poten-
tials (T = 4.2 K). Curves are shifted vertically for clarity. Reprinted by permission from
Macmillan Publishers Ltd: Nature [622], copyright 2003.

significantly altered in the metallic junction. In particular, they suggested

that image charges generated in the source and drain electrodes by the
charges on the molecule are probably the origin of this effect. This is a
very interesting suggestion that may explain similar discrepancies in other
experiments [678, 679].
Surprisingly, this important issue has not received much attention. A
notable exception is the work of Kaasbjerg and Flensberg [646] in which a
realistic description of the screening environment in a SMT was combined
with quantum chemical calculations. These authors concluded that the
addition energies in a junction are indeed strongly reduced as compared
with naive expectations based on the ionization potentials and electron
affinities of the molecules in gas phase. They explained that this is a con-
sequence of both (a) a reduction of the electrostatic molecular charging
energy and (b) polarization-induced level shifts of the HOMO and LUMO
levels. These conclusions are at variance with most DFT-based calcula-
tions for two-terminal systems that suggest that the level spacing for small
molecules inside the junctions is still rather large and comparable to the
one of the isolated molecules. In this sense, it would be highly desirable
to have further theoretical and experimental work to clarify this important
issue.
SMTs have not only allowed to observe single-electron tunneling phe-
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464 Molecular Electronics: An Introduction to Theory and Experiment

nomena that were well-known in other nanodevices, but they have also
made possible to access new transport regimes and to discover novel physi-
cal phenomena. In the context of Kondo physics, we would like to mention
the work of Pasupathy et al. [663] where the Kondo effect in the presence
of ferromagnetism has been reported for the first time. In this work the
authors measured the transport through single-C60 transistors with ferro-
magnetic nickel electrodes. They showed that Kondo correlations persisted
despite the presence of ferromagnetism, but the Kondo peak in the differ-
ential conductance was split by an amount that decreased (even to zero)
as the spin polarizations in the two electrodes were turned from parallel to
antiparallel alignment. Although, the reported splitting was too large to
be explained by a local magnetic field, the voltage, temperature, and mag-
netic field dependence of the signal agreed with predictions for an exchange
splitting of the Kondo resonance [680, 681].
SMTs have also allowed to study the interplay between Kondo physics
and the electron-vibration interaction. The signatures of vibrational modes
have been shown to persist in the Kondo regime [617, 664, 682] and we
shall discuss this issue in certain detail in the next chapter. It is also worth
mentioning that although most of the experiments on the Kondo effect in
molecular junctions have been performed with the electromigration tech-
nique, the Kondo physics has also been studied with breakjunctions by
Ralph’s group [682]. Although in this case a gate electrode was not opera-
tive, the authors could tune the Kondo resonance in a single-C60 junction
by adjusting the metal-molecule distance that is a capability of the break-
junction technique that is lacking in electromigration-based experiments.
Changing the metal-molecule coupling the authors were able to tune the
Kondo temperature and showed that the temperature dependence of the lin-
ear conductance agreed with the scaling function expected for the S = 1/2
Kondo problem [682].
SMTs have also been used to explore other basic aspects of the Kondo
physics. Thus for instance, Roch et al. [684] have recently reported the
observation of a quantum phase transition between a singlet and a triplet
spin state at zero magnetic field in a single-C60 transistor. The analysis
of the transport through three-terminal molecular devices has also allowed
to study the fundamental scaling laws that govern the non-equilibrium the
standard S = 1/2 Kondo effect [683].
Another aspect to which SMTs have contributed enormously is the un-
derstanding of the role of vibrational modes in the transport through single
molecules. The signatures of the excitation of vibronic degrees of freedom
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Single-molecule transistors: Coulomb blockade and Kondo physics 465

Fig. 15.21 (a) Side view of a Mn12 molecule with tailor made ligands containing acetyl-
protected thiol end groups (R=C6 H4 ). The diameter of the molecule is about 3 nm.
(b) Schematic drawing of the Mn12 molecule (circle) trapped between electrodes. A
gate can be used to change the electrostatic potential on the molecule enabling energy
spectroscopy. (c) Scanning electron microscopy image of the electrodes. The gap is not
resolvable. Scale-bar corresponds to 200 nm. Reprinted with permission from [619].
Copyright 2006 by the American Physical Society.

are specially visible in the transport characteristics in the limit of weak cou-
pling between the molecule and the metallic electrodes [22, 685]. Moreover,
in this regime the electron-vibration interaction can lead to a great variety
of novel physical phenomena. This subject will be discussed in detail in the
next chapter.
We now turn to a class of experiments where the transport through
single-molecule magnets (SMMs) has been investigated (see Ref. [686] for a
progress article on this subject). This type of molecules exhibits magnetic
hysteresis due to their large spin and high anisotropy barrier, which ham-
pers magnetization reversal [687, 688]. The first transport experiment on
a SMM was performed by the group of van der Zant [619]. These authors
studied the prototypical SMM, Mn12 acetate, which has a total spin S = 10
and an anisotropy barrier of about 6 meV. The molecules that were inves-
tigated were [Mn12 O12 (O2 C-R-SAc)16 (H2 O)4 ] (Mn12 from now on), where
R={C6 H4 , C15 H30 }, see Fig. 15.21(a). These molecules were designed with
thiol groups in the outer ligand shell to ensure a strong affinity for gold
surfaces. On the other hand, the ligands are believed to serve as tunnel
barriers, so that the molecules are only weakly coupled electronically to the
gold and their magnetic properties are preserved. The molecules were in-
corporated in a SMT geometry with gold electrodes using electromigration,
see Fig. 15.21(b).
In Fig. 15.22(a) we reproduce some of the results of this work for the
differential conductance as a function of gate (Vg ) and bias voltage (Vb )
for one of the devices (T = 3 K, R=C6 H4 ). The lines separating the con-
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466 Molecular Electronics: An Introduction to Theory and Experiment

Fig. 15.22 (a) Differential conductance (gray-scale) as a function of gate voltage (Vg )
and bias voltage (Vb ) (T = 3 K, R=C6 H4 ). A region of complete current suppression
(left degeneracy point, arrow) and low-energy excitations with negative differential con-
ductance (right degeneracy point) are observed. The dashed line near the left degeneracy
point indicates the suppressed diamond edge. (Gray-scale from -0.8 nS [black] to 1.4 nS
[white]). (b) I − Vb at the gate voltage indicated in (a) with a line. NDC is clearly
visible as a decrease in |I| upon increasing |Vb |. Upon applying a magnetic field, current
is increased for negative bias. Reprinted with permission from [619]. Copyright 2006 by
the American Physical Society.

ducting regions from the diamond-shaped Coulomb blockade regions have

different slopes for the three different charge transport regions. Within the
orthodox Coulomb blockade theory this implies that the transport regions
belong to different quantum dots, since the capacitance to the environment
is assumed constant for each dot. However, for molecular quantum dots it
is not possible to rule out that these three regions come from three different
charge states of the same molecule.
The focus of the work of Ref. [619] was on transport features at low-
energy (. 5 meV): a region of complete current suppression (CCS) and a
strong negative differential conductance (NDC) excitation line in the sta-
bility diagrams. Both are visible in Fig. 15.22(a). At the left degeneracy
point in this figure the current is fully suppressed at positive bias voltage
above the left diamond edge (dashed line). Transport is restored beyond an
excitation that lies at 5 meV. Remarkably, the right diamond edge does con-
tinue all the way down to zero bias, defining a narrow strip (∼ 1 mV wide)
where transport is possible. In the right conductive regime in Fig. 15.22(a),
two excitations at an energy of 2 meV and 3 meV are the most pronounced
features. The 2 meV excitation is visible as a bright line with positive dif-
ferential conductance (PDC); the 3 meV excitation as a black line (NDC).
The strength of the NDC is clearly visible in the I-Vb plot in Fig. 15.22(b).
The observations of CSS and NDC lines at low energy do not follow in
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Single-molecule transistors: Coulomb blockade and Kondo physics 467

a straightforward way from conventional Coulomb blockade theory. The

authors of Ref. [619] explained qualitatively those features with the help
of sequential tunneling model that takes into account the high-spin ground
state and magnetic excitations of the molecule. They showed that sequen-
tial tunneling processes can result in spin blockade of the current, providing
a possible explanation for the observed NDC and CCS. This effect is differ-
ent from conventional spin-blockade [689], where there is no spin anisotropy.
The transport through similar Mn12 -based molecules, with short but
weak binding ligands, were studied independently by Jo et al. [620]. In par-
ticular, these authors presented an extensive analysis of the magnetic-field
dependence of the transport characteristics. They found two main signa-
tures of magnetic molecular states and magnetic anisotropy in the data: an
absence of energy degeneracy between spin states at zero magnetic field (B)
and a nonlinear evolution of energy level positions with B. The magnitude
of zero-field splitting between spin states was found to vary from device to
device, and they interpreted this as evidence for magnetic anisotropy varia-
tions upon changes in molecular geometry and environment. On the other
hand, they did not observe hysteresis in the electron-tunneling spectrum as
a function of swept magnetic field, as one might expect to find in analogy
to magnetization measurements on large ensembles of Mn12 molecules in
bulk crystals. They pointed out that the absence of hysterisis might be due
to the fact that sequential tunneling transitions can populate a sequence
of excited magnetic levels that surmount the anisotropy barrier and enable
rapid magnetic relaxation [690].
Another example of transport through individual magnetic molecules
has been reported by Grose et al. [691]. In this case, the authors fabricated
molecular transistors with individual molecules of the spin-3/2 endohedral
fullerene N@C60 and measured its spin excitations. N@C60 is an attrac-
tive model system because of its simple spin structure and because of the
possibility of doing control experiments with non-magnetic C60 molecules.
N@C60 molecules also have the advantage of being stable at the high tem-
peratures present during the electromigration process by which the molec-
ular junctions were formed in this work. In the experiments on SMMs that
we have reviewed above, the molecular magnetism was usually destroyed
during device fabrication. However, in the work of Ref. [691], it was ob-
served that the N@C60 devices exhibit clear magnetic character, meaning
that they exhibit a spin-state transition as a function of applied magnetic
field. The nature of this transition enabled the authors to identify the
charge and spin states of the molecule inside the junctions. The spectra
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468 Molecular Electronics: An Introduction to Theory and Experiment

of N@C60 also exhibited low-energy excited states and signatures of non-

equilibrium spin excitations predicted for this molecule [692]. The existence
of a spin transition in N@C60 accessible at laboratory magnetic fields was
associated with the scale of the exchange interaction between the nitrogen
spin and electron(s) on the C60 cage.
As mentioned above, in the transport experiments on SMMs reported
so far, no magnetic hysterisis has been observed, which is probably due to
structural deformations of the molecules. In this sense, it would be inter-
esting to test other (more robust) compounds. For instance, Mannini et al.
[693] have shown recently that tailor-made Fe4 complexes retain magnetic
hysterisis on gold surfaces. These results demonstrate that isolated SMMs
can be used for storing information and they open the way to address these
molecules individually in their blocked magnetization state.

15.8 Exercises

15.1 Rate equations in a single-level model: For those who are not familiar
with rate (or master) equations, it is convenient to start by analyzing the following
situation. Let us consider a quantum dot with a single (non-degenerate) level of
energy E1 , which is measured with respect to the equilibrium Fermi energy of
the leads, which we set to zero. (The energy E1 can depend on the gate voltage,
the exact electrostatic profile and the charging energy). This dot has only two
possible configurations with n1 = 0 (empty dot) and n1 = 1 (one electron in level
E1 ). We shall denote the corresponding probabilities as P0 and P1 , respectively.
As usual, we denote the left and right tunneling rates (in units of energy) as ΓL
and ΓR , respectively, and we assume them to be energy-independent.
(a) Write down the kinetic equation for the probability distribution and show
that in the stationary case the probabilities Pi are given by

ΓL f¯L + ΓR f¯R ΓL fL + ΓR fR
P0 = , P1 = .
ΓL + Γ R ΓL + Γ R

Here, fL,R = f (E1 ∓ eV /2), where V is the bias voltage and f (E) is the Fermi
function.
(b) Use the previous solution to show that the current through the dot can
be written as
e ΓL ΓR
I= [fL − fR ] .
~ ΓL + Γ R
Notice that this expression coincides with the expression for the current obtained
in the single resonant tunneling model in the limit of weak coupling.
(c) Using the previous expression, show that the linear conductance is given
by Eq. (15.26).
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Single-molecule transistors: Coulomb blockade and Kondo physics 469

15.2 Linear conductance in the Coulomb blockade regime: Derive the

formula of Eq. (15.26) for the linear conductance of a quantum dot in Coulomb
blockade regime. For this purpose, follow the next steps:
(a) In the linear response regime the distribution function can be written as

„ «
eV
P ({ni }) ≡ Peq ({ni }) 1 + Ψ({ni }) .
kB T

Linearize the detailed balance equation (15.23) and solve it to show that Ψ can
be written as

∞
!
(i)
X ΓR
Ψ({ni }) = constant + ni (i) (i)
−η ,
i=1 ΓL + Γ R

where the constant first term takes care of the normalization of P to first order
in V and it does not need to be determined explicitly. Hint: Use the following
relations:

1 − f (ǫ) = f (ǫ)eǫ/kB T , kB T f ′ (ǫ)(1 + e−ǫ/kB T ) = −f (ǫ),

Peq (n1 , . . . np−1 , 1, np+1 , . . .) = Peq (n1 , . . . np−1 , 0, np+1 , . . .)e−ǫ/kB T ,

where the prime symbol in the Fermi function stands for derivative with respect
to its argument.
(b) Linearize the formula for the current in Eq. (15.19) and use the expression
for Ψ({ni }) to obtain Eq. (15.26).
15.3 Coulomb oscillations, Coulomb staircase and stability diagrams:
The goal of this exercise is to compute transport characteristics in the Coulomb
blockade regime within the two-level model discussed in section 15.4.4.2.
(a) As a first step, compute the occupation probabilities of the four possible
configurations of the dot. For this purpose, show that the stationary kinetic
equation, Eq. (15.22), together with the normalization condition of Eq. (15.24)
can be written in the following matrix form: Ŵ p ~ = ~v . Here, p~ is the column
vector containing the probabilities of the four configurations of the dot, i.e. p ~T =
(P1 , P2 , P3 , P4 ), where 1 ≡ (0, 0), 2 ≡ (1, 0), 3 ≡ (0, 1) and 4 ≡ (1, 1). The vector
~v is simply given by ~v T = (1, 0, 0, 0) and the different elements of the matrix Ŵ
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470 Molecular Electronics: An Introduction to Theory and Experiment

adopt the form

W1i = 1 (i = 1, ..., 4), W23 = W32 = W41 = 0,

(1) (1)
W21 = ΓL f (E1i,l (N = 1)) + ΓR f (E1i,r (N = 1)),
W22 = −Γ f¯(E f,l (N = 1)) − Γ f¯(E f,r (N = 1))
(1) (1)
L 1 R 1
(2) (2)
−ΓL f (E2i,l (N = 1)) − ΓR f (E2i,r (N = 1)),
= ΓL f¯(E2f,l (N = 2)) − ΓR f¯(E2f,r (N = 2)),
(2) (2)
W24
(2) (2)
W31 = ΓL f (E2i,l (N = 0)) + ΓR f (E2i,r (N = 0)),
(1) (1)
W33 = −ΓL f (E1i,l (N = 1)) − ΓR f (E1i,r (N = 1))
−Γ f¯(E f,l (N = 1)) − Γ f (E f,r (N = 1))
(2) (2)
L 2 R 2
(1) ¯ (1) ¯
W34 = f,l
−ΓL f (E1 (N = 1)) − ΓR f (E1f,r (N = 1)),
(2) (2)
W42 = ΓL f (E2i,l (N = 1)) + ΓR f (E2i,r (N = 1)),
(1) (1)
W43 = ΓL f (E1i,l (N = 1)) + ΓR f (E1i,r (N = 1)),
−ΓL f¯(E1f,l (N = 2)) − ΓR f¯(E1f,r (N
(1) (1)
W44 = = 2))
−ΓL f¯(E2f,l (N = 2)) − ΓR f¯(E2f,r (N
(2) (2)
= 2)).

(p)
Here, the ΓL,R (p = 1, 2) are the tunneling rates, while the expressions for the
energies appearing in the arguments of the Fermi functions can be found in section
15.4.4.
(b) Solve numerically the 4 × 4 system Ŵ p ~ = ~v and use the expression of
Eq. (15.19) to reproduce the results of Figs. 15.8 and 15.9.
(c) In a molecular transistor the level splitting ∆E may be larger than the
charging energy, e2 /C. Study how the stability diagrams in this case differ from
those shown in Figs. 15.9. Choose for instance e2 /C = 30 meV and ∆ = 100
meV, while keeping the other parameters equal to those in the example of section
15.4.4.2.
(d) An important experimental issue is that for a particular charge state
lines are often only visible on one side of the Coulomb diamond. This is due to
an asymmetry in the coupling. Illustrate this fact with the example of section
15.4.4.2 by choosing very different tunneling rates for the left and right barriers.
15.4 Effects of inelastic scattering in the Coulomb blockade regime: In
the Coulomb blockade theory described in section 15.4.4 inelastic scattering was
assumed to take place exclusively in the reservoirs. One of the effects of inelastic
scattering in the dot is the thermalization of the electrons inside the dot. In the
limiting case of full thermalization, the probability distribution function P ({ni })
is given by the equilibrium expression of Eq. (15.20). Use this expression in the
example of section 15.4.4.2 (and of the previous exercise) to study the effect of
inelastic scattering in the different transport characteristics (Coulomb oscillations,
Coulomb staircase and stability diagrams).
15.5 Coulomb blockade theory for single-molecule transistors: The goal
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Single-molecule transistors: Coulomb blockade and Kondo physics 471

of this problem is to compute the different transport characteristics of a SMT in

the Coulomb blockade regime within the model of section 15.5.2. (a) Show first
that the transition rates of Eq. (15.31) are given by
r
W11 = −Γr [fr (E2 − E1 ) + fr (E3 − E1 )]
W12 = Γr f¯r (E2 − E1 ) = W13
r r

r
W21 = Γr fr (E2 − E1 )
r
= −Γr f¯r (E2 − E1 ) + fr (E4 − E2 )
ˆ ˜
W22
r
W24 = Γr f¯r (E4 − E2 )
r
W31 = Γr fr (E3 − E1 )
W33 = −Γr f¯r (E3 − E1 ) + fr (E4 − E3 )
r ˆ ˜
r
W34 = Γr f¯r (E4 − E3 )
r r
W42 = Γr fr (E4 − E2 ) = W43
W44 = −Γr f¯r (E4 − E2 ) + f¯r (E4 − E3 )
r ˆ ˜
r r r r
W14 = 0 = W23 = W32 = W41 ,

where r = L, R. The numbers 1 to 4 correspond to the four eigenstates of the

molecular Hamiltonian, as defined in section 15.5.2.
(b) Using the numerical values chosen in section 15.5.2, reproduce the results
of Fig. 15.10 and compute the corresponding stability diagram.
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472 Molecular Electronics: An Introduction to Theory and Experiment

January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Chapter 16

Vibrationally-induced inelastic
current I: Experiment

16.1 Introduction

In the previous chapter we discussed the transport phenomena that occur

in molecular junctions when the conduction is dominated by the Coulomb
interaction in the molecular bridge. We now want to focus on the corre-
sponding effects that originate from another inelastic interaction, namely
the electron-phonon interaction.1 When an electron proceeds through a
molecule, it can exchange energy by exciting its vibrational modes. De-
pending on the molecule, the energy of these modes ranges from a few meV
to several hundreds of meV [694]. This is comparable to the excess energy
of conduction electrons at the usual bias voltages applied in the junctions.
Thus, these internal degrees of freedom may influence the transport prop-
erties of molecular junctions. Indeed, the interplay between electronic and
nuclear dynamics does give rise to a great variety of transport phenomena,
as we shall show in this chapter.
When is the electron-phonon interaction expected to play an important
role in the electrical conduction through molecular junctions? As we ex-
plained in the introduction of the previous chapter, this will occur when
the time needed to interact with a vibrational
√ mode, ~/λ, becomes com-
parable to the traversal time, τ = ~/ ∆E 2 + Γ2 . Let us remind that
here, λ is the electron-phonon coupling constant, ∆E is the injection en-
ergy and Γ is the width of the molecular resonance (or strength of the
metal-molecule √ electronic coupling). In the limit of weak electron-phonon
coupling, λ ≪ ∆E 2 + Γ2 , the vibrational modes give rise to a small in-
elastic current that is superimposed in a background determined by the

1 The term “phonon” in this chapter is used for vibrational modes associated with any
nuclear motion.

473
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474 Molecular Electronics: An Introduction to Theory and Experiment

elastic contribution. This inelastic current has typically well-defined signa-

tures at energies that are basically the energies of the vibrational modes
of the neutral molecules inside the junctions. Thus, the analysis of the
inelastic current provides a local molecular spectroscopy and in turn, it
gives indirect information on the presence of the molecules, their structure,
orientation and coupling to the leads.
√ In the opposite limit of strong electron-phonon coupling, λ ≫
∆E 2 + Γ2 , vibronic effects can dominate the transport characteristics of
molecular junctions. Thus for instance, in a resonant situation (∆E ≈ 0)
and if the coupling Γ is not very large, as in the molecular transistors of the
previous chapter, the electron-phonon interaction can lead to pronounced
current steps, which contain valuable spectroscopic information about the
vibrational modes of the molecule in different charge states. On the other
hand, if the electron-phonon interaction is sufficiently strong, one can reach
a regime in which the vibrations make the electronic motion completely in-
coherent such that it can be described by successive classical rate processes,
usually referred to as hopping. The discussion of this hopping regime, where
the transport is mediated by thermally activated processes, will be deferred
until Chapter 18.
Apart from the energy scales mentioned in the previous paragraphs,
there are other important factors that determine the impact of vibrations
in the transport properties. Thus for instance, the temperature plays an
important role in determining the dominant transport mechanism. While
low temperatures favor the coherent transport, high temperatures reduce
drastically the inelastic scattering length by increasing the phonon popula-
tions and making the transport incoherent. On the other hand, the length
of the molecules is another important factor. Incoherent transport becomes
more important for longer molecules both because dephasing is more effec-
tive and (for off-resonant tunneling) because of the exponential fall off of
the coherent component.
We initiate here a series of two chapters in which we shall review our
present understanding of the role of molecular vibrations in the transport
properties of single-molecule junctions. In particular, we shall concentrate
on the analysis of their influence in the electrical current. The role of vi-
brational modes in other properties, including thermal transport, will be
discussed in the Chapter 19. We would like to remark that, following the
spirit of this monograph, we shall present a pedagogical introduction to this
subject, rather than a detailed review of the huge amount of work reported
in the last years. To be precise, after reading these two chapters the reader
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Vibrationally-induced inelastic current I: Experiment 475

should have a clear idea about: (i) what are the basic experimental signa-
tures of vibrational modes in the current through single-molecule junctions,
(ii) what are the physical mechanisms giving rise to those signatures and
(iii) what are the main open problems related to this subject.
In this first chapter we shall describe some of the main experiments that
have illustrated the role of vibrations in the electrical conduction through
molecular junctions. We have grouped these experiments in three differ-
ent categories. First of all, we shall discuss situations where the electron-
phonon interaction is weak, in the sense explained above, and the electron
tunneling is off-resonant. The analysis of the vibronic signatures in this
regime is known as inelastic electron tunneling spectroscopy (IETS) for the
historical reasons that will be explained in section 16.2. Then, we shall fo-
cus our attention in section 16.3 on the case of highly conductive junctions,
where the electron-phonon interaction is also weak. In this regime, and
again for historical reasons, the study of the vibrational modes is known as
point-contact spectroscopy (PCS). Section 16.4 is devoted to a discussion
of the relation between IETS and PCS. In section 16.5 we shall discuss the
third group of experiments that correspond to the regime sometimes known
as resonant inelastic electron tunneling. This regime corresponds to a situ-
ation where the transport is resonant and the electron-phonon interaction
can be very strong. This regime is realized, in particular, in the molecular
transistors described in the previous chapter. The discussion below will
end with a brief summary of the main vibrational signatures that can be
observed in the different transport regimes. If you are an impatient reader
(as we are), please feel free to jump directly to section 16.6 and then come
back to this point.
The recent progress in the understanding of vibrational effects in molec-
ular transport junctions has been thoroughly described by Galperin, Ratner
and Nitzan in the review of Ref. [695], which contains close to 500 references
related to the main subject of these two chapters. For those who prefer a
quick overview, we recommend them the shorter review of Ref. [696] of the
same authors.

16.2 Inelastic electron tunneling spectroscopy (IETS)

The first studies of the influence of the electron-phonon interaction on the

transport through molecules go back to the 1960’s. In a pioneering work,
Jaklevic and Lambe discovered in 1966 that vibrational spectra can be ob-
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476 Molecular Electronics: An Introduction to Theory and Experiment

Fig. 16.1 Recorded traces of d2 I/dV 2 versus voltage for three Al-Al oxide-Pb junctions
taken at 4.2 K. The zero of the vertical scale is shifted for each curve, and all three are
normalized to the same arbitrary units. The largest peaks represent increases of 1% in
the conductance. Also indicated are intervals associated with the energy of IR-active
molecular vibrational modes. Curve A is obtained from a “clean” junction. Curves B and
C are obtained from junctions exposed to propionic acid [CH3 (CH2 )COOH] and acetic
acid (CH3 COOH), respectively. The peaks positions are independent of the polarity.
Reprinted with permission from [697]. Copyright 1966 by the American Physical Society.

tained from molecules adsorbed at the buried metal-oxide interface of a

metal-oxide-metal tunneling junction [697]. In their experiment, the tun-
neling current I was measured as a function of the voltage V across the
junction. Small, but sharp increases in the differential conductance, dI/dV ,
were observed when the energy of the tunneling electrons reached the en-
ergy of a vibrational mode for molecules in the junction. These increases
represented changes in the differential conductance of about 1%. They
were interpreted as the result of electrons losing their energies to the vibra-
tional mode, giving rise to an inelastic tunneling channel, which is forbidden
when tunneling electrons have energies below the quantized vibrational en-
ergy. In the experiment, a peak at each vibrational energy was observed in
d2 I/dV 2 , see Fig. 16.1. This method, known as inelastic electron tunneling
spectroscopy (IETS), has been applied to a wide range of systems and has
led to a better understanding of molecules in the adsorbed state [698–704].
It is convenient for our discussions below to briefly review some of the
basic predictions of IETS theory concerning the following issues:
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Vibrationally-induced inelastic current I: Experiment 477

(1) Tunneling mechanism: The explanation for the appearance of the

peaks in the tunneling spectra is the following [697, 705]. As shown
schematically in Fig. 16.2, when the bias voltage applied to the junction
is increased and crosses the threshold for excitation of a vibrational
mode, electrons can tunnel either elastically or by emitting a vibrational
mode. The opening of this latter inelastic channel is accompanied by an
increase of the differential conductance (dI/dV ) at eV = ±~ω, where ω
is the frequency of the excited mode. As mentioned above, this change
is more clearly seen in the derivative of the conductance, d2 I/dV 2 ,
where the signature related to the excitation of a vibrational mode is
a peak (dip) for positive (negative) bias, see Fig. 16.2.

(a) (b) dI/dV

EF
−hω hω eV
hω eV
(c) 2 2
d I/dV
−hω
hω eV
Vacuum/
Tip Molecule Metal

Fig. 16.2 (a) Schematic representation of the inelastic tunneling above the threshold
for a vibrational excitation. An electron can tunnel losing part of its energy which is
employed to excite a vibration mode of energy ~ω. This process is only possible when
eV ≥ ~ω. (b) The opening of the inelastic channel gives rise to an increase in the
conductance at eV = ±~ω. (c) The onset of the inelastic process is seen in the second
derivative of the current, d2 I/dV 2 , as a peak (dip) for positive (negative) bias.

(2) Spectral linewidth: The full width at half maximum (FWHM) of the
d2 I/dV 2 vibrational peak is given by W = [(1.7Vm )2 + (5.4kB T /e)2 +
WI2 ]1/2 , where Vm is the modulation voltage in the lock-in technique,
kB is the Boltzmann constant, T is the temperature, and WI is the
intrinsic width (due to the finite phonon lifetime) [705, 706].
(3) Selection rules: Although there are no selection rules in IETS as there
are in infrared (IR) and Raman spectroscopy, certain selection prefer-
ences have been established. According to the IETS theory [707, 708],
molecular vibrations with net dipole moments perpendicular to the in-
terface of the tunneling junction have larger peak intensities than vibra-
tions with net dipole moments parallel to the interface (for dipoles close
to the electrodes). For a more complete description of the propensity
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478 Molecular Electronics: An Introduction to Theory and Experiment

Fig. 16.3 STM inelastic tunneling spectra of acetylene molecules. The plot shows back-
ground difference d2 I/dV 2 spectra for C2 H2 (1) and C2 D2 (2), taken with the same STM
tip. Notice the presence of peaks at 358 mV and 266 mV, respectively. The difference
spectrum (1 − 2) yields a more complete background subtraction. From [709]. Reprinted
with permission from AAAS.

rules, see Ref. [704].

Soon after the invention of the STM, it was clear that this tool could
serve to extend IETS all the way down to single molecules. However, this
turned out to be very challenging since it requires the use of low tempera-
tures (∼ 4 K) and very high mechanical stability. The breakthrough came
from Ho’s group that reported in 1998 the first study of the vibrational
spectra for a single molecule adsorbed on a solid surface [709]. To be pre-
cise, these authors measured the inelastic electron tunneling spectra for an
isolated acetylene (C2 H2 ) molecule adsorbed on the copper (100) surface
using a STM under UHV conditions at a temperature of 8 K. They observed
an increase in the tunneling conductance at 358 mV, which was attributed
to the excitation of the C-H stretch mode. The increase in conductance is
typically rather small (around 3-6% in these experiments depending on the
tip) and for this reason the features related to the vibrational modes are
better seen in the second derivative of the current, d2 I/dV 2 , where they
appear as peaks (for positive bias), very much like in the IETS in planar
tunnel junctions. We show an example of the original data in Fig 16.3.
To confirm the interpretation of the origin of the peak in d2 I/dV 2 , the
authors used isotopic substitution, i.e. they replaced the hydrogen atoms
by deuterium ones in the molecules. In the case of the deuterated acetylene
(C2 D2 ), they showed that the peak in d2 I/dV 2 is shifted to 266 mV, which
corresponds to the expected change in energy of the C-H stretch mode.
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Vibrationally-induced inelastic current I: Experiment 479

Fig. 16.4 Single-molecule vibrational spectra of oxygen molecules on an Ag(110) surface

measured with a STM. Curve a corresponds to 16 O2 , curve b to 18 O2 and curve c to
a clean Ag(110) surface. The difference spectra (curve a-c, curve b-c) are also shown.
Reprinted with permission from [711]. Copyright 2000 by the American Physical Society.

Indeed, these values are in close agreement with those obtained by electron
energy loss spectroscopy (EELS). This experiment inspired an enormous
amount of work in which the chemical sensitivity of the single-molecule
IETS has been exploited. This has led in the last years to a better under-
standing and control of surface chemistry at the atomic level. The activities
of the first years on STM-IETS have been reviewed by Ho in Ref. [710].
The first STM-IETS experiments raised several fundamental questions
related, for instance, to the selection (or propensity) rules that apply in
this case. With respect to the tunneling process that gives rise to the peaks
seen in the spectra, it was believed that there is no fundamental difference
with respect to the traditional IETS in oxide tunnel junctions. In other
words, the process responsible for the vibrational signatures was believed
to be the phonon emission process described in Fig. 16.2. However, it is
worth stressing that the electron-phonon interaction in these systems does
not always lead to an increase of the conductance at the phonon energies.
Thus for instance, Ho and coworkers have reported in Ref. [711] STM-
IETS studies that revealed two vibrational modes showing a decrease in the
conductance at 682.0 and 638.3 mV for single oxygen molecules chemisorbed
on the fourfold hollow sites of an Ag(110) surface at 13 K. These results can
be seen in Fig. 16.4, where one can observe the presence of two well-defined
dips at positive bias. It is worth remarking that in this case the change in
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480 Molecular Electronics: An Introduction to Theory and Experiment

Fig. 16.5 IETS spectrum of a C11 junction formed with gold cross wires. The dashed
line is a simple polynomial background and is presented as a guide to the eye. Mode
assignments are from comparison to previous experimental results. Reprinted with per-
mission from [712]. Copyright 2004 American Chemical Society.

the conductance at the vibrational energies continues to be rather small (a

few percent). Let us mention that also more complicated line shapes have
been observed in the context of STM-IETS studies [710].
The use of IETS to measure to the vibrational spectrum of metal-
molecule-metal junctions relevant to molecular electronics was first reported
in 2004 simultaneously by two different groups.2 Kushmerick and coworkers
presented in Ref. [712] in situ vibrational spectroscopy of metal-molecule-
metal junctions containing prototypical molecular wires: C11 (an alkane
chain with 11 carbon atoms), OPE, and OPV. The transport measure-
ments were performed with a cryogenic crossed-wire tunnel junction, where
one of the gold wires was coated with a monolayer of the molecule of inter-
est. The experiments were conducted at 4 K and standard ac modulation
techniques, along with two lock-in amplifiers, were utilized to measure di-
rectly both dI/dV and d2 I/dV 2 . An example of the results for C11 is
shown in Fig. 16.5. Here, the second derivative of the current is normalized
by the conductance. Notice that, as in the traditional IETS, the signature
of the molecular vibrations is a series of peaks, which were observed to have
2 These experiments were not the first ones to investigate the role of vibronic coupling

in molecular transport junctions, but they were the first ones that explored the regime
discussed in this section, where the transport through the junctions takes place in a
non-resonant manner and the current probes the vibrational modes of the ground state
of the molecule.
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Vibrationally-induced inelastic current I: Experiment 481

Fig. 16.6 IET spectrum of a C8 dithiol SAM measured with the nanopore technique.
The spectrum was obtained from lock-in second-harmonic measurements with an ac
modulation of 8.7 mV (rms value) at a frequency of 503 Hz (T = 4.2 K). Reprinted with
permission from [713]. Copyright 2004 American Chemical Society.

the same height in the positive and negative bias polarity.

Based on previous infrared, Raman, and high-resolution electron energy
loss spectroscopy studies of alkanethiolate monolayers, the authors were
able to assign the observed peaks in the C11 junction to specific molecular
vibrations. The C-H stretch at 362 mV is the most intense vibrational
mode observed, but they also observed a number of lower energy vibrations
in the region from 70 mV to 200 mV. An interesting observation in this
work was the fact that most, although not all, of the modes that were
identified corresponded to longitudinal molecular modes, which shows that
this type of modes couples more strongly to the tunneling electrodes.
Reed’s group reported simultaneously an IETS study of an alkanedithiol
self-assembled monolayer (SAM) using the nanopore technique [713]. The
second-harmonic signal d2 I/dV 2 was measured directly with a lock-in tech-
nique and an example of the results can be seen in Fig. 16.6. Notice that
in this case the IET spectrum exhibits peaks with shapes that clearly differ
from those of Fig. 16.5.3 As in Ref. [712], the authors used known results
from infrared, Raman, and high-resolution electron energy loss spectra of
SAM-covered gold surfaces to identify some of the vibrational modes. Some-
thing remarkable in this work is the fact that the authors were able to verify
3 Kushmerick and coworkers have argued that the discrepancies in the IET spectra
between these two experiments could be due to the presence of metal nanoparticles in
the nanopore devices of Wang et al. [713], see Ref. [714] for a detailed discussion of this
issue.
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482 Molecular Electronics: An Introduction to Theory and Experiment

Fig. 16.7 Single-molecule IETS measurements using STM break junctions. (a) Semilog
conductance histogram with a peak at G0 , and an additional peak at 6 × 10−3 G0 ,
which is attributed to the conductance of propanedithiol. (b) A conductance curve with
steps for a single molecule measurement. The four symbols represent four stretching
distances where the bias was swept and the I-V and first derivative were recorded. (c) The
corresponding four first derivative curves (offset for clarity), and (d) the corresponding
IET spectra obtained numerically. The curves are antisymmetric, and certain features
are very reproducible along the conductance plateau. Reprinted with permission from
[722]. Copyright 2008 American Chemical Society.

that the observed spectra were indeed valid IETS data by examining the
peak width as a function of temperature. This important test is usually
very difficult to carry out with other techniques.
IETS has become quite popular in the field of molecular electronics
over the last years and it has distinguished itself as a unique spectroscopic
probe of molecular junctions. From comparison between experiments and
computations, IETS can be useful for characterizing numerous aspects of
molecular junctions such as the confirmation of the presence of the molecule,
information on the nature of the interfaces, the orientation of the molecule
and even electronic pathways can be identified. For further experimental
examples of the use of the of IETS in the regime described in this section
see Refs. [574, 715–721].
The experiments that we have just described correspond to situations
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Vibrationally-induced inelastic current I: Experiment 483

where the transport is probed through an ensemble of molecules. In this

sense, it is highly desirable to perform similar experiments, but with single-
molecule junctions. However, such experiments in the off-resonant regime
that we are discussing in this section are rather scarce. The main problem is
to achieve the required stability, which can only be done by working at very
low temperatures. Recently, this difficulty has been overcome by Hihath et
al. [722] who reported IET spectra of a single 1,3-propanedithiol molecule
using an STM-break junction at cryogenic temperatures. In particular,
these authors were able to measure IET spectra at different stages of the
formation of the molecular contacts, see Fig. 16.7. This allows them to
correlate changes in the conductance with changes in the configuration of a
single-molecule junction. Moreover, the authors were able to do a statistical
analysis of the phonon spectra to identify the most relevant modes. Finally,
the vibrational modes found for propanedithiol matched well with IR and
Raman spectra and were described by a simple one-dimensional model.
This type of experiment provides very important information about the
formation of single-molecule junctions and in this sense, we are sure that
many more experiments of this kind will be reported in the near future (for
a more recent one see Ref. [723]).

16.3 Highly conductive junctions: Point-contact spec-

troscopy (PCS)

In this section we shall discuss the experimental signatures of the electron-

phonon interaction in the case of molecular junctions with a high conduc-
tance (close to G0 ). To be precise, the junctions discussed here are char-
acterized by the presence of a broad electronic resonance around the Fermi
energy with a width, Γ, considerably larger than the electron-phonon cou-
pling constant, λ. The analysis of the electron-phonon interaction in this
regime has its historical origin in the so-called point contact spectroscopy
(PCS) [724–726]. Thus, we shall start this section by briefly explaining the
basics of this technique.4
Many years before the rise of nanofabrication, ballistic metallic point
contacts were widely studied [724–726]. The fabrication principle was in-
troduced by Yanson in the 1970’s [727] and later developed by his group
and by Jansen et al. [728]. The technique has been worked out with var-
ious refinements for a range of applications, but essentially it consists of
4 Our discussion follows closely Ref. [15].
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484 Molecular Electronics: An Introduction to Theory and Experiment

d2 V/dI 2 (VA )
−2

Au

voltage (mV)

Fig. 16.8 An example of an electron-phonon spectrum measured for a gold point contact
by taking the second derivative of the voltage with respect to the current. The long-
dashed curve represents the phonon density of states obtained from inelastic neutron
scattering. Reprinted with permission from [728]. Copyright 1980 IOP Publishing Ltd.

bringing a needle of a metal gently into contact with a metal surface. With
this technique stable contacts are typically formed having resistances in
the range from ∼ 0.1 to ∼ 10 Ω, which corresponds to contact diameters
between d ≃ 10 and 100 nm. The elastic and inelastic mean free path
can be much longer than this length d, when working with clean metals
at low temperatures, and the ballistic nature of the transport in such con-
tacts has been demonstrated in many experiments. The main application of
the technique has been to study the electron-phonon interaction in metals.
Here, one makes use of the fact that the (small but finite) probability for
back-scattering through the contact is enhanced as soon as the electrons ac-
quire sufficient energy from the electric potential difference over the contact
that they are able to excite the main phonon modes of the material. The
differential resistance, dV /dI, of the contact is seen to increase at the char-
acteristic phonon energies of the material. Notice that this is at variance
with the typical signature in IETS. A spectrum of the energy-dependent
electron-phonon scattering can be directly obtained by measuring the sec-
ond derivative of the voltage with respect to the current, d2 V /dI 2 , as a
function of the applied bias voltage. An example is given in Fig. 16.8.
Peaks in the spectra are typically observed between 10 and 30 mV, and
are generally in excellent agreement with spectral information about the
phonons of the corresponding metal obtained from other experiments (e.g.
neutron scattering), and with calculated spectra.
Traditionally, electron-phonon spectroscopy in large metallic contacts
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Vibrationally-induced inelastic current I: Experiment 485

µ2
µ1
µ1
kF

µ2 kF

Fig. 16.9 Electron distribution function in the vicinity of the orifice. Here, kF is the
equilibrium Fermi wave vector, µ1 and µ2 are the chemical potentials for each side,
which, far from the orifice and in the presence of an applied potential V , are equal to
EF − eV /2 and EF + eV /2, respectively.

is described by considering the non-equilibrium electron distribution near

the contact that results from the applied bias voltage, as illustrated in
Fig. 16.9 [727–729]. Electrons that arrive at the left electrode, coming from
the right, are represented in a Fermi surface picture by a cone with an angle
corresponding to the solid angle at which the contact is viewed from that
position in the metal. These electrons have eV more energy that the other
Fermi surface electrons, and they can be scattered inelastically to all other
angles outside the cone. Only those that scatter back into the contact will
have a measurable effect on the current.
As the energy difference eV increases, this backscattering increases due
to the larger phonon density of states, which will be observed as a decreas-
ing conductance. Ignoring higher order processes, the decrease of the con-
ductance comes to an end for energies higher than the top of the phonon
spectrum, which is typically 20–30 meV. By taking the derivative of the
conductance with respect to the voltage one obtains a signal that directly
measures the strength of the electron-phonon coupling. An example for
gold is illustrated in Fig. 16.8. Several authors have derived an expression
for the spectrum [728, 730], which adopts the following form
d2 I 4 e3 m2 vF 3 2
2
= a α Fp (eV ), (16.1)
dV 3π ~4
where a is the contact radius, vF is the Fermi velocity, m the electron mass
and the function α2 F is given by
m2 v F
Z Z
2
α Fp (E) = d n d2 n′ |gnn′ |2 η(θ(n, n′ ))δ(E − ~ωnn′ ). (16.2)
2
4πh3
Here, the integrals run over the unit vectors of incoming and outgoing
electron wave vectors (n = k/k), gnn′ is the matrix element for the electron-
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

486 Molecular Electronics: An Introduction to Theory and Experiment

1
conductance (2e /h)
2

dG / dV (arb. units)
0 .9 9 T
L

0 .9 8 L
T

-4 0 -2 0 0 20 40
voltage (m V)
0 .9 7
-0.4 -0 .2 0 0 .2 0.4
voltage (V)

Fig. 16.10 Differential conductance as a function of the applied bias voltage for a one-
atom Au contact at 4.2 K. The contact was tuned to have a conductance very close
to 1 G0 , which suppresses the amplitude of the conductance fluctuations. This allows
the observation of a phonon signal, which is seen as a maximum at zero bias. Inset: By
taking the derivative of the conductance the transverse (T) and longitudinal (L) acoustic
branches can be recognized symmetrically positioned around zero. Note the expanded
scale of the voltage axis in the inset. Reprinted with permission from [731]. Copyright
2000 by the American Physical Society.

phonon interaction, and η is a function of the scattering angle that takes the
geometry into account, such that only backscattering through the contact is
effective, η(θ) = (1 − θ/ tan θ)/2. From this expression, and by considering
Fig. 16.9, one can see that the contribution of scattering events far away
from the contacts is suppressed by the effect of the geometric angle at
which the contact is seen from that point. The probability for an electron
to return to the contact decreases as (a/d)2 , with a the contact radius
and d the distance from the contact. This implies that the spectrum is
dominantly sensitive to scattering events within a volume of radius a around
the contact, thus the effective volume for inelastic scattering in the case of
a clean opening (the contact) between two electrodes is proportional to
a3 . Clearly, this effective volume must depend on the geometry of the
contact. For a long cylindrical constriction, the electrons scattered within
the constriction will have larger return probability, the effective volume, in
this case, increases linearly with the length [724].
The point-contact spectroscopy has been extended in recent years to
atomic-sized contacts. As the contact becomes smaller, the signal comes
from scattering on just a few atoms surrounding the contact. The spectrum
no longer measures the bulk phonons, but rather local vibrational modes of
the contact atoms. In attempting to measure the phonon signal for small
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Vibrationally-induced inelastic current I: Experiment 487

1.01

2
a 1.00 d

G (2e /h)
short wire long wire 0.99

2
conductance (2e /h)

L
2

0.98
M
0.97
1
3
long wire (L)

rupture rupture
S
2

0
0 5 0 5 10 15 20 25
displacement (Å) 1

G0 dG/dV (V )
-1
1.01
0.92

0.91
b 1.00 c 0
G0 dG/dV (V ) G (2e /h)

0.99
2

0.90

-1
0.98 -1
0.89 short wire (S) short wire (M)
0.97
-1

0.8 0.8
0.4 0.4 -2

0.0 0.0
-0.4 -0.4 -3
-1

-0.8 -0.8
-30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30
bias voltage (mV) bias voltage (mV) bias voltage (mV)

Fig. 16.11 Point contact spectroscopy of gold atomic chain. (a) Short and long atomic
wire, ∼ 0.4 and ∼ 2.2 nm, respectively, as given by the length of the conductance plateau.
Panels (b–d) show the differential conductance and its derivative at points S, M , and
L, respectively. The various curves in (b–d) were acquired at intervals of 0.03, 0.03 and
0.05 nm, respectively. Note that the vertical scales for the last thee panels are chosen
to be identical, which brings out the relative strength of the electron-phonon interaction
for the longer chains. The wire in (d) has a length of about 7 atoms. Reprinted with
permission from [732]. Copyright 2002 by the American Physical Society.

contact sizes one encounters the problem that the phonon signal intensity
decreases, according to Eq. (16.2), while the amplitude of the conductance
fluctuations5 remains roughly constant, or slightly increases. The result
is that the phonon signal is sometimes hidden in the conductance fluctua-
tions for the smallest contacts. A solution to this problem is obtained for
the special and interesting case of a contact made up of a single channel
with nearly perfect transmission probability, where these fluctuations are
suppressed. Under these conditions the features due to phonon scattering
become clearly visible. This is illustrated in Fig. 16.10 where we show an
example of the point contact spectrum of a gold one-atom contact [731].
Surprisingly, one observes a spectrum (see inset of the figure) that still
closely resembles the bulk phonon spectrum, although the relative intensi-
ties of the features in the spectrum are different.
5 These fluctuations were described in detail in section refsec-cond-fluct.
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488 Molecular Electronics: An Introduction to Theory and Experiment

The point contact spectroscopy was pushed to study the phonon modes
in Au atomic chains by Agraı̈t and coworkers [732, 733]. As we described
in section 11.8, atomic chains of certain metals can be formed with the
STM and break junction techniques. These chains constitute in some sense
the simplest molecules that one can think of. Thus, the PCS of gold chains
of Refs. [732, 733] is of special interest for us. In these experiments, the
differential conductance was measured using a lock-in detection with a 1
mV modulation voltage, from which dG/dV was calculated numerically.
The energy resolution was limited by the temperature of 4.2 K to 2 meV.
The results for the differential conductance and its derivative for a long
atomic chain (∼ 7 atoms) are shown in Fig. 16.11(d). Notice that at ±15
mV bias the conductance exhibits a rather sharp drop by about 1%. In the
second derivative d2 I/dV 2 this produces a pronounced single peak, point-
symmetric about zero bias. The chains of Au atoms have the fortuitous
property of having a single nearly perfectly transmitted conductance mode,
which suppresses conductance fluctuations that would otherwise mask the
phonon signal. Some asymmetry that can still be seen in the conductance
curves is attributed to the residual elastic scattering and interference con-
tributions.
The fact that only one conductance drop is clearly seen was interpreted
by the authors as follows. By energy and momentum conservation the
signal can only arise from electrons that are back-scattered, changing their
momentum by 2kF . With ~ω2kF the energy for the corresponding phonon,
the derivative of the conductance is expected to show a single peak at
eV = ±~ω2kF . The transverse phonon mode cannot be excited in this one-
dimensional configuration and only the longitudinal mode is visible. We
shall see in the next chapter that this argument is, strictly speaking, only
valid for infinite chains, while it is approximate for the finite chains realized
in the experiments.
Another interesting feature of the point-contact spectra of gold atomic
chains is that the position of the peak in dG/dV shifts as a function of the
strain in the wire. As one can see in Fig. 16.11(d), the frequency of the
mode associated to the peak decreases as a function of the tension because of
the decreasing bond strength between the atoms. However, the amplitude
(peak height) increases, until an atomic rearrangement takes place, signaled
by a small jump in the conductance (not shown here). At such points the
amplitude and energy of the peak in dG/dV jump back to smaller and
larger values, respectively. This is consistent with the phonon behavior
of Au atomic chains found in ab initio calculations [360]. The growing
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Vibrationally-induced inelastic current I: Experiment 489

50
D
40 2

30
dI/dV (2e /h)

20
0.98
2

10
dI/dV (2e /h)

0.96
2

0.95
60
0.94
50 HD
0.96
40
0.93
(arb.unit)

Counts
0.1
30
0. 0
20
d I/dV (arb.unit)

0.2 -0.1
2

10
d I/dV

-0.2
-80 -60 -40 -20 0 20 40 60 80
2

Energy (meV)

0. 0 30 H
2

20
2

-0.2 10

-80 -60 -40 -20 0 20 40 60 80

2

0 20 40 60 80 100
Energy (meV) Energy (meV)

Fig. 16.12 Left panel: Differential conductance curve for D2 contacted by Pt leads.
The dI/dV curve (top) was recorded over 1 min, using a standard lock-in technique with
a voltage bias modulation of 1 meV at a frequency of 700 Hz. The lower curve shows the
numerically obtained derivative. The spectrum for H2 in the inset shows two phonon
energies, at 48 and 62 meV. Right panel: Distribution of vibrational energies observed
for H2 , HD, and D2 between Pt electrodes, with a bin size of 2 meV. The peaks in
the distribution for H2 are marked by arrows and their widths byp error margins. These
p
positions and widths were scaled by the expected isotope shifts, 2/3 for HD and 1/2
for D2 , from which the arrows and margins in the upper two panels have been obtained.
Reprinted with permission from [567]. Copyright 2005 by the American Physical Society.

amplitude is due to the softening of the phonon modes with tension.

The first application of point-contact spectroscopy to the characteriza-
tion of a molecular junction was carried out by Ruitenbeek’s group in their
study of the transport through hydrogen molecules that we discussed in
section 14.1.3. The original study of Ref. [127] was extended in Ref. [567]
with a thorough analysis of the stretching behavior of point-contact spectra
as well as DFT calculations of the vibrational modes of the Pt-H2 junctions.
The left panel of Fig. 16.12 shows examples for Pt-H2 and Pt-D2 junc-
tions at a conductance near 1 G0 . The conductance is seen to drop by
about 1 or 2%, symmetrically at positive and negative bias, very muck like
in the Au atomic chains just described. The energies of the conductance
drops are in the range of 50-60 meV, well above the Debye energy of ∼
20 meV for Pt. A high energy for a vibrational mode implies p that a light
element is involved, since the frequency is given by ω = κ/M with κ
an effective spring constant and M the mass of the vibrating object. The
proof that the spectral features are indeed associated with hydrogen vibra-
tional modes came from further experiments where H2 was substituted by
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

490 Molecular Electronics: An Introduction to Theory and Experiment

the heavier isotopes D2 and HD. The positions of the peaks in the spec-
tra of d2 I/dV 2 vary within some range between measurements on different
junctions, which can be attributed to variations in the atomic geometry of
the leads to which the molecules bind. Fig. 16.12 (right panel) shows his-
tograms for the vibrational modes observed in a large number of spectra for
each of the three isotopes. Two pronounced peaks are observed in each of
the distributions, that scale approximately as the square root of the mass of
the molecules, as expected. The two modes can often be observed together,
as in the inset of the left panel of Fig. 16.12. For D2 an additional mode
appears near 90 meV. This mode cannot easily be observed for the other
two isotopes, since the lighter HD and H2 mass shifts the mode above 100
meV where the junctions become very unstable. For a given junction with
spectra as in Fig. 16.12 (left panel), it is often possible to stretch the contact
and follow the evolution of the vibrational modes. The frequencies for the
two lower modes were seen to increase with stretching, while the high mode
for D2 is seen to shift downwards. This unambiguously identifies the lower
two modes as transverse modes and the higher one as a longitudinal mode
for the molecule. This interpretation agrees well with DFT calculations for
a configuration of a Pt-H-H-Pt bridge in between Pt pyramidally shaped
leads [567, 571]. The fact that the vibrational modes observed for HD that
are intermediate between those for H2 and D2 confirms that the junction
is formed by a molecule, not an atom.
A drop in the conductance as a fingerprint of the presence of a molecule
in highly conductive junctions has also been reported, for instance, in
Ref. [385]. In this work, PCS was used to identify the presence of oxygen
intercalated in Au atomic chains. More recently, similar vibration-induced
steps down in the conductance have been also observed in various small
molecules directly bonded to Pt electrodes [474].

16.4 Crossover between PCS and IETS

As we have seen in the previous two subsections, electron-phonon inter-

action leads to an increase in the conductance for junctions in the tunnel
regime (e.g. IETS done in STM); however, it decreases the conductance
for junctions in the contact regime (e.g. PCS across a Pt-H2 junction). In
spite of this difference, √
all these physical systems have in common that the
traversal time, τ = ~/ ∆E 2 + Γ2 , is much smaller than the time that it
takes to interact with a vibrational mode, ~/λ. In IETS this is due to the
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Vibrationally-induced inelastic current I: Experiment 491

fact that the tunneling is typically off-resonant, and therefore the injection
energy ∆E is rather large. In the PCS case, however, this occurs because
the molecule is strongly coupled to the leads and thus Γ is very large. In
view of this similarity, one may wonder whether there is any fundamen-
tal difference between IETS and PCS in molecular junctions. As we shall
discuss in the next chapter, recent theoretical work has shown that IETS
and PCS are indeed two sides of the same coin and they can be described
in a unified manner. In other words, these two techniques are based on
the same underlying physics and they simply refer to two different limiting
cases depending on the junction transparency.
In recent years, different experiments on highly conductive single-
molecule junctions, but with conductances not to close to 1 G0 , have clearly
suggested the idea that there is a smooth crossover between IETS and PCS.
Thus for instance, experiments on Pt-H2 junctions [734], Ag atomic wires
decorated with oxygen [605] and Pt-benzene junctions [473] with conduc-
tances between 0.1 and 0.4 G0 have shown that the signature of vibrational
modes is a step up in the conductance at the vibrational energies, i.e. exactly
like in the standard IETS case. The experiment that has finally clarified
this issue was reported recently by Tal et al. [735] and we now proceed to
describe it in certain detail.
In Ref. [735] the authors presented PCS and shot noise measure-
ments across a single-molecule junction formed by Pt electrodes and H2 O
molecules. The Pt/H2 O molecular junctions were formed using a MCBJ
setup at about 5 K. The formation of a clean Pt contact was verified by
conductance histograms, which exhibited a single peak around 1.4 G0 , pro-
viding so a fingerprint of a clean Pt contact [127]. Water molecules were
then introduced to the junction through a heated capillary, while the Pt
junction was broken and formed repeatedly. Following the introduction of
water, the typical Pt peak in the conductance histogram was suppressed
and contributions from a wide conductance range were detected with mi-
nor peaks around 0.2, 0.6, and 1.0 G0 . The continuum in the conductance
counts implies a variety of stable junction configurations that the authors
exploited for spectroscopy measurements on junctions with different con-
ductance.
In Fig. 16.13 we reproduce the results for the differential conductance as
a function of the voltage across the Pt/H2 O junction at two different linear
conductance values: 1.02 ± 0.01 G0 (a) and 0.23 ± 0.01 G0 (b). Junctions
with different zero-bias conductance were formed by altering the distance
between the Pt contacts or by re-adjusting a new contact. The steps in
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492 Molecular Electronics: An Introduction to Theory and Experiment

(a) (b)

1. 02 0.28 Step up
dI/dV [G0]

dI/dV [G0]
0.26
1.00

0.24

0.98
Step down
0.22
-80 -60 -40 -2 0 0 20 40 60 80 -80 -60 -40 -20 0 20 40 60 80
Bias Voltage [mV] Bias Voltage [mV]

Fig. 16.13 Differential conductance (dI/dV ) as a function of the bias voltage for two
different Pt-H2 O-Pt junctions with linear conductance of 1.02 ± 0.01 G0 (a) and 0.23
± 0.01 G0 (b). Reprinted with permission from [735]. Copyright 2008 by the American
Physical Society.

the conductance that appear at 46 mV in Fig. 16.13(a), and 42 mV in

Fig. 16.13(b) indicate the onset of a vibrational excitation at these volt-
ages. Notice that while in (a) the differential conductance is decreased
(“step down”), the curve (b) taken at lower linear conductance shows an
increase in the differential conductance (“step up”). These two examples
demonstrate that both conductance suppression and enhancement can be
observed at a relatively high conductance (much higher than the typical
tunneling conductance).
As we shall show in the next chapter, the theory predicts that in the
regime of weak electron-phonon coupling, the transition from a step down
to a step up in the conductance occurs at a transmission equal to 0.5 for
a single channel model [736, 737]. In order to confirm these predictions
Tal and coworkers collected many dI/dV spectra at different zero-voltage
conductance values. They found that curves with steps up appear below
0.57 ± 0.03 G0 and curves with steps down were detected only above 0.72 ±
0.03 G0 . Thus, they demonstrated that the crossover between conductance
enhancement and conductance reduction by the electron-vibration inter-
action occurs between these two values. Since more than one conduction
channel can contribute to the conductance in these junctions, the authors
carried out shot noise measurements to determine the number of channels
and their transmission probabilities. They concluded that there were typ-
ically two conduction channels. More importantly, they showed that the
dominant channel had a transmission 0.51 ± 0.01 at the crossover con-
ductance, which is nicely consistent with the predictions of single-channel
models (see Ref. [735] for more details).
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Vibrationally-induced inelastic current I: Experiment 493

16.5 Resonant inelastic electron tunneling spectroscopy

(RIETS)

In this section we shall discuss the signatures of the electron-phonon in-

teraction in the case of resonant situations, when the traversal time is not
small in comparison with the time ~/λ, where λ is the electron-phonon
coupling constant. This occurs when the injection energy, ∆E, is rather
small and the molecular orbital width, Γ, is not too large. In this regime,
strong vibronic effects are expected and the transport characteristics pro-
vide in this case what is sometimes referred to as resonant inelastic electron
tunneling spectroscopy (RIETS) [695]. This physical situation is realized,
in particular, in the single-molecule transistors (SMT) discussed in the pre-
vious chapter. In these junctions the electronic states can be brought close
to the chemical potentials of the reservoirs by means of a gate voltage.
The additional flexibility provided by the third electrode together with the
strong electron-phonon coupling give rise to a rich phenomenology that we
now want to describe.
The experiment performed by Park et al. [22] that we described at the
beginning of section 15.7 was also the first SMT experiment that revealed
vibronic effects. Let us remind that in this work, the transport through
a single C60 molecule was studied in a three-terminal device. In this case
the fingerprint of the vibrational modes can be seen directly in the I-V
characteristics, see Fig. 15.16. In that figure one can see that (for positive
bias) the I-V curves exhibit a first step that corresponds to the crossing of an
electronic resonance. There is a second step that is separated from the first
one by a distance of around 5-10 mV. As discussed in section 15.7, there are
good reasons to attribute that signature to the excitation of a vibrational
mode that corresponds to the center-of-mass oscillation of C60 . Notice that,
contrary to the cases discussed so far, the signature of a vibrational mode
is now a step in the I-V curves (or a peak in the differential conductance).
Moreover, this feature does not appear at a voltage equal to ~ω/e, where
ω is the vibration frequency, but rather at a bias voltage that is equal to
the bias that is necessary to cross the electronic resonance (at a given gate
voltage) plus a voltage of the order of ~ω/e.6
What is the origin of this peculiar signature? This will be explained
in detail in the next chapter, but let us briefly say that this feature origi-
nates from the same inelastic tunneling process (phonon emission) discussed
6 The exact distance between the Coulomb peak and the first sideband depends on the
voltage profile, i.e. on how the resonant level is shifted by the bias voltage.
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

494 Molecular Electronics: An Introduction to Theory and Experiment

above, see Fig. 16.2. The difference is now that this process is much more
probable when the energy of the electron surpasses the energy of the elec-
tronic level by an amount that is equal to ~ω (corrected by a factor that
depends on how the voltage drops across the junction). The reason is that
an electron with that energy may lose an energy equal to ~ω (by emitting
a vibrational mode) and then it crosses the molecule exactly at resonance
with the molecular level. The enhanced probability of this inelastic process
gives rise to a peak in the differential conductance at a bias voltage of the
order of ~ω/e away from the Coulomb blockade peak. From this argument,
it is also easy to understand that in order to observe a pronounced current
step, the width of the electronic resonance, Γ, must be smaller than ~ω. In
any case, the bias voltage must be larger than ~ω for this inelastic process
to take place.
As it was also explained in section 15.7, the signature of a vibrational
mode can be also seen in the stability diagrams, see Fig. 15.17. In these
plots, the peaks in the conductance, which correspond to the step-like fea-
tures in Fig. 15.16, show up as lines. In particular, the vibration mode
with energy 5 meV appears there as running lines that intersect the main
diamonds or conductance gap regions. The energy of this excitation is too
small to correspond to an electronic excitation. Moreover, some of these
lines are observed for both charge states, which would be very unlikely for
an electronic excitation. Even more convincing is the fact that multiple
excitations with the same spacing are observed, see Fig. 15.17(d). This
corresponds most likely to the excitation of several vibrational quanta of
the same mode, i.e. multi-phonon processes. Let us also say to conclude
this discussion that signatures of intrinsic vibrational modes of the C60
molecules were also observed in the stability diagram of some devices, see
in particular Fig. 3 in Ref. [22].
The experiment just described was followed by other experiments with
weakly coupled molecules where signatures of the vibrational modes in
the transport characteristics were also observed. For instance, Zhitenev et
al. [738] reported transport measurements through a small self-assembled
monolayer of thiolated organic molecules in which the conductance exhib-
ited a series of equally spaced peaks, the position of which could be con-
trolled by a gate voltage. These peaks were attributed to the lowest molec-
ular vibrations of the molecules. The most surprising thing in this exper-
iment was the observation of a large number of conductance peaks with
slowly decreasing amplitudes. This would mean that phonon processes of
very high-order were taking place in these junctions. On the other hand,
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Vibrationally-induced inelastic current I: Experiment 495

Fig. 16.14 Stability diagrams (dI/dV vs. V and Vg ) for four C140 SMTs fabricated
with the electromigration technique. White arrows indicate excited levels at 11 and
22 meV. dI/dV is represented by a color scale from black (zero) to white (maximum),
with maximum values 200 nS (device I), 600 nS (II), 15 nS (III), and 100 nS (IV).
Measurements were done at 1.5 K for I-III and 100 mK for IV. Reprinted with permission
from [685]. Copyright 2005 American Chemical Society.

Park et al. [23] observed low-lying excitations in the stability diagrams of

SMTs based on coordination complexes with Co ions. These excitations ap-
peared in the Coulomb blockade regime in the two charge states observed
in the diagrams, which clearly suggested that they might correspond to
vibrational modes.
In the previous experiments it was difficult to determine the precise
nature of the vibrational modes. In the transistors made from C60 [22]
the mode observed was not intrinsic to the molecule itself. In this sense,
experiments like the one of Pasupathy et al. [685] were important. In this
case, the authors reported the study of single-molecule transistors made
using a C140 molecule, in which it was possible to clearly identify low-
energy internal vibrational modes. Such modes were clearly visible in the
stability diagrams, see Fig. 16.14, and an excitation at 11 ± 1 meV was
seen in most devices. By means of a detailed molecular modeling, it was
possible to identify this mode as an internal stretching mode of the molecule.
The modeling also explained the strong coupling of this mode to tunneling
electrons, relative to other molecular modes.
An impressive example of resonant inelastic electron tunneling has
been reported by Osorio et al. [678]. These authors performed trans-
port measurements in electromigrated single-molecule junctions based on
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

496 Molecular Electronics: An Introduction to Theory and Experiment

Fig. 16.15 Stability diagrams of a three-terminal junction with OPV-5, measured at 1.6
K. Plotted in (a) is dI/dV as measured with a lock-in technique (modulation amplitude
0.4 mV) and in (b) the numerically calculated second derivative, which serves to highlight
the fine structure of the excitations. The current levels are the same near both degeneracy
points, which is a strong indication that they belong to the same molecule. Three
different charge states are probed. The N + 1 state is not indicated; for low bias voltages
it starts at gate voltages larger than 2.2 V. The data yield an addition energy of 210
meV and a gate coupling of 0.05. Reproduced with permission from [678]. Copyright
Wiley-VCH Verlag GmbH & Co. KGaA.

an oligophenylenevinylene derivative (OPV-5). An example of the stabil-

ity diagrams obtained in these experiments is shown in Fig. 16.15. This
diagram clearly shows the presence of sets of excitation lines for all three
charge states accessible in the experiment. A close inspection reveals that
the point of intersection between the lines and the diamond edge are sym-
metric with respect to the bias polarity, and that their position is almost
independent of the charge state. This observation makes it unlikely that
the excitations are a result of electronic states, because these are expected
to depend strongly on the charging of the molecule. Moreover, the 17 exci-
tations present in the experimental data are unlikely to reflect precisely 17
available electronic states that differ by only 5-10 meV in energy. Therefore,
the excitations were attributed to the vibrational modes of the single OPV-5
molecule trapped between the electrodes. The authors compared the vibra-
tional modes probed in the transport experiment with those probed with
light by using Raman and IR spectroscopy and found a good agreement for
the ones with the highest energies (see Ref. [678] for further details).
Molecular junctions offer the possibility to examine transport regimes
that are difficult to access in other systems. In particular, single-molecule
transistors are ideal systems where to study the interplay between vibronic
effects and Kondo physics. As we explained in section 15.7, the Kondo
effect have been observed in SMTs by several groups. Already in the first
observations of this effect there were clear hints of the coexistence of the
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Vibrationally-induced inelastic current I: Experiment 497

Fig. 16.16 Maps of d2 I/dVSD 2 as a function of V

SD and VG at 5 K for two Co-ion-based
SMTs fabricated with the electromigration technique. Brightness scales are −8 × 10−5
A/V2 (black) to 3 × 10−5 A/V2 (white), and −2 × 10−5 A/V2 (black) to 2 × 10−5
A/V2 , respectively. The zero-bias features correspond to Kondo peaks in ∂I/∂VSD .
Prominent inelastic features are indicated by black arrows. In both devices, when the
inelastic features approach the boundaries of the Coulomb blockade region, these levels
shift and alter the line shape (white arrows). Black dashed line in left map traces
an inelastic feature across the Coulomb blockade region boundary and into the Kondo
regime. Reprinted with permission from [617]. Copyright 2004 by the American Physical
Society.

Kondo resonance and vibrational sidebands [23, 662]. The first work in
which this coexistence was studied in detail was reported by Natelson’s
group [617]. Using electromigration-based SMT junctions they analyzed
the transport through a molecule comprising a single Co ion coordinated
by conjugated ligands. In many devices they observed the Kondo effect
and a Kondo temperature of ∼ 40 K was deduced from the temperature
dependence of the zero-bias conductance. Moreover, in some cases, the
conductance in the classically blockaded region and/or outside the Kondo
resonance was large enough to allow clean measurements of ∂ 2 I/∂VSD 2
. In
Fig. 16.16 we reproduce results from Ref. [617] where maps of this quan-
tity are shown as a function of VSD (source-drain voltage) and VG (gate
voltage) in two different devices at 5 K. The left panel shows mainly a
diamond-like region corresponding to a charge state exhibiting standard
Coulomb blockade, while the right one focuses on the next diamond where
the Kondo resonance is visible at zero bias. Two prominent features within
the blockaded (Kondo) regime are indicated with black arrows. Features in
∂ 2 I/∂VSD
2
of opposite sign are symmetrically located around zero source-
drain bias, consistent with inelastic tunneling expectations.
The ∂ 2 I/∂VSD2
features in the blockaded region occur at essentially con-
stant values of VSD until VG is varied such that the feature approaches the
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

498 Molecular Electronics: An Introduction to Theory and Experiment

edge of the blockaded region. This independence of VG resembles the signa-

ture of inelastic cotunneling (see section 15.6.1) which has been observed,
for instance, in semiconductor single-electron devices [650]. The inelastic
modes occur at energies low compared to the expected level spacing (> 100
meV), implying that the modes being excited are unlikely to be electronic.
Indeed, the authors compared the energies of these features with Raman
and IR data and found a nice correlation supporting the idea that they
correspond to vibration modes.
Another work in which the interplay between the Kondo physics and
vibration-assisted tunneling was investigated is that of Parks et al. [682]. In
this experiment the Kondo resonance in a C60 junction was tuned mechan-
ically using the MCBJ technique. They also observed pronounced peaks
in the differential conductance at symmetric values of the bias voltage (see
Fig. 4 in Ref. [682]). The main observed feature appeared at ± 33 mV,
which was attributed to the lowest intracage vibrational mode of a C60
molecule. Additionally, as the electrodes were pulled apart, the energies of
the modes were observed to shift due to the change in the strength of the
metal-molecule coupling.
Let us mention that a Kondo resonance accompanied by vibrational
sidebands has also been observed in STM experiments on the transport
through a single molecular layer of a purely organic charge-transfer salt
grown on a metal surface [739].
To conclude this section, we now want to briefly mention two experi-
ments of special relevance. In the first one, Dekker and coworkers stud-
ied the current through a suspended single-wall carbon nanotube injected
from a STM tip [740, 741]. They showed that the current exhibited usual
features of the Coulomb blockade, i.e. a series of peaks in the differen-
tial conductance. Moreover, they found that these peak were accompanied
not only by the usual RIETS satellite peaks on the right hand side of the
Coulomb peaks (for positive bias), but also by peaks on the left hand side.
The satellite peaks on the right are a signature of phonon emission (the
mode excited in this experiment was believed to be the radial breathing
modes of the tube). The peaks on the left were interpreted as a fingerprint
of phonon absorption. Since the bath temperature of the experiment was
much smaller than the energy of the modes (and therefore they could not be
excited thermally), it was concluded that these anomalous peaks were the
signature of nonequilibrium phonons that are created by the electrical cur-
rent. This experiment illustrates that “hot” (or nonequilibrium) phonons
can play an important role in the transport through a molecular structure.
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Vibrationally-induced inelastic current I: Experiment 499

(a) IETS: (b) PCS: (c) RIETS:

Weak e−ph coupling Weak e−ph coupling Strong e−ph coupling
Off−resonant tunneling High conductance (~ G0 ) Weak electronic coupling
dI/dV dI/dV I

hω
− hω hω eV − hω hω eV eV

2 2
2
d I/dV
2 d I/dV dI/dV hω
− hω hω
hω eV − hω eV eV

Fig. 16.17 Summary of the main vibrational signatures in the transport characteristics
of molecular junctions in various transport regimes.

The other experiment that we want to briefly comment was reported

by Ho’s group [742]. The experiments described so far in this section were
mainly performed with the electromigration technique and with the aid of
a third terminal. However, this is not the only way to reach the transport
regime that we are discussing. For instance, in Ref. [742] a STM was used
to define a double-barrier junction by positioning the STM tip over an indi-
vidual copper phthalocyanine molecule adsorbed on a thin (approximately
0.5 nm) insulating Al2 O3 film grown on the NiAl(110) surface. The two
tunnel barriers in the junction were the vacuum gap between the STM tip
and the molecule, and the oxide film between the molecule and NiAl. The
current through this double junction was found to exhibit clear signatures
of molecular vibronic states that were observed to change dramatically by
varying the tip-molecule separation, which in turn controls the ratio of
electron tunneling rates through the two tunnel barriers.

16.6 Summary of vibrational signatures

Let us now briefly summarize the main vibrational signatures that we have
shown to appear in the different transport regimes (see Fig. 16.17):

• In off-resonant situations (low transmissive junctions), in which the

electron-phonon interaction is weak, the typical signature of vibrational
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500 Molecular Electronics: An Introduction to Theory and Experiment

modes is a small increase of the differential conductance at the mode

energies. These features are usually better seen in the second derivative
of the current (d2 I/dV 2 ), where they appear as peaks (for positive
bias), see Fig. 16.17(a). This regime is realized in STM junctions in the
tunneling limit and in strongly coupled contacts where the conductance
is smaller than approximately 0.5 G0 . Let us stress that in the case
of STM tunnel junctions, the observation of dips in d2 I/dV 2 is also
possible.
• In the case of strongly coupled metal-molecule-metal junctions with
conductances close to G0 and weak electron-phonon interaction, the
vibrations are manifested in the conductance as small drops at the mode
energies and therefore, as dips in d2 I/dV 2 , see Fig. 16.17(b).7 This
regime is realized, for instance, in atomic gold chains and hydrogen-
based junctions.
• In the case of resonant transport, if the metal-molecule coupling is
not very strong, like it is usually the case in SMTs, the excitation of
vibrational modes leads to steps in the current versus the bias voltage,
or a series of peaks in the dI/dV , see Fig. 16.17(c). If a gate electrode is
available, the vibronic excitations can be seen in the Coulomb blockade
regime as running lines in the stability diagrams. In the Kondo regime
the modes induce sidebands at the phonon energies. These sidebands
are seen in the stability diagrams as horizontal lines (i.e. independent
of the gate voltage), parallel to the zero-bias Kondo resonance.

The reader should bear in mind that this summary is slightly oversim-
plified and more complex signatures are also possible. Thus for instance,
Thijssen et al. [743] have reported the observation of anomalous spikes in
the differential conductance of a variety of junctions, which were attributed
to vibrationally induced two-level systems. On the other hand, vibronic
effects can also be responsible for other strong non-linearities in the I-V
characteristics (see section 8 of Ref. [695] for a detailed discussion of this
issue).

7 Here, we are assuming that the conductance is dominated by highly transmissive

conduction channels. However, one can have situations in which several channels combine
to give a conductance close to G0 . In this case, the signature of the vibrational modes
can be a step down in the conductance, depending on the precise value of transmission
coefficients.
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Chapter 17

Vibrationally-induced inelastic
current II: Theory

This chapter is devoted to the theoretical description of the vibrational

effects detailed in the previous one. In particular, our main goal is to
explain the origin of the different signatures summarized in section 16.6
(see also Fig. 16.17).
At the moment, there is no unified theory covering all the different
regimes explored experimentally. However, a lot of progress has been made
in several important limiting cases that will the main subject of our discus-
sion here. The first one corresponds to the limit of weak electron-phonon
coupling, in the sense explained in the introduction of the previous chapter.
In this case, a perturbative approach has been quite successful in explain-
ing the basic experimental observations. In the opposite limit of strong
electron-phonon interaction, when the electronic metal-molecule coupling
is weak, it is possible to describe the physics in terms of the rates equations
that take into account the vibronic effects in a non-perturbative manner.
In between these two extreme limits there is a loosely-defined crossover
regime of intermediate electron-phonon coupling. The next three sections
are devoted to the analysis of these three different regimes, and we shall
finish this chapter with some comments and a brief discussion of the basic
open problems for both theory and experiment.

17.1 Weak electron-phonon coupling regime

In this section we shall address the limit in which the traversal time is much
smaller than the time needed for an electron to feel the molecular vibrations.
In this case, the usual approach is to treat the electron-electron interaction
at a mean field level and to make a perturbative expansion in the electron-
phonon interaction. Our discussion of this regime will be divided into two

501
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502 Molecular Electronics: An Introduction to Theory and Experiment

subsections. In the first one, we shall discuss in detail the results obtained
from the resonant tunneling model including in addition the coupling to a
single phonon mode. This model will help us to understand the origin of the
different vibrational signatures in this regime. Then, the next subsection
will be devoted to a description of the ab initio methods that have been
developed so far to elucidate the propensity rules in this regime and to
establish a quantitative comparison with experimental results.

17.1.1 Single-phonon model

In this subsection we shall discuss the predictions of a toy model for the
regime of weak electron-phonon interaction. As we shall see, this simple
model explains the origin of the different experimental signatures described
in sections 16.2-16.4 and provides a deep insight into the tunneling pro-
cesses responsible for these signatures. We find this subsection particularly
important and in order to make it accessible to everybody, we have avoided
very technical discussions.1
The simplest model to study the electron-phonon interaction in a molec-
ular junction is a natural extension of the resonant tunneling model, which
includes the interaction with a single vibrational mode. Let us recall that
in the resonant tunneling model an electronic level with energy ǫ0 is cou-
pled to two metallic reservoirs. The strength of this coupling is described
by the scattering rates ΓL and ΓR , where L and R denote the left and
right leads, respectively. For the sake of simplicity, we shall assume here
that these rates are energy-independent. In order to describe the role of
the electron-phonon interaction, we now assume that this resonant level is
also coupled to a single vibrational mode of energy ~ω (see Fig. 8.2). The
Hamiltonian describing this system has the following form

H = He + ~ω b† b + 1/2 + λd† d b† + b .
¡ ¢ ¡ ¢
(17.1)

Here, He describes the electronic part of this problem as it is given by

Eq. (7.93). The second term corresponds to the vibrational (or phonon)
mode, which is described here as a simple harmonic oscillator. The oper-
ators b† and b are the creation and annihilation operators related to the
phonon mode, and they satisfy the bosonic commutation relations. Finally,
the last term describes the electron-phonon interaction in the molecule,
where λ is the electron-vibration coupling constant and d† and d are the
1 The technical details of the calculations reported in this subsection can be found in
section 8.2.1.
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Vibrationally-induced inelastic current II: Theory 503

fermionic operators related to the electronic level.2 Let us remark that we

ignore here the electron-electron interaction in the molecule. This model is
sometimes referred to as the (single-level) Holstein model.
This model has been analyzed in the last years by numerous authors
to study different aspects of the problem that we are addressing here
[736, 744–751]. In spite of its apparent simplicity, there is no known ex-
act solution for this model and approximations have to be made. To keep
our discussion as simple as possible, in what follows, unless we state oth-
erwise, we shall make use of the following two approximations: (i) the
electron-phonon interaction is treated perturbatively and we include only
the lowest-order corrections (second order in λ) and (ii) we assume that
the phonons are in thermal equilibrium at the bath temperature. The first
approximation is referred to as the lowest-order expansion (LOE). The sec-
ond one means that the phonon mode is occupied according to the Bose
function and it requires the existence of a mechanism that equilibrates the
local vibrations (e.g. coupling to bulk phonons). Later, we shall discuss the
consequences of relaxing these two approximations.
Our goal is to compute the I-V characteristics when a constant bias
voltage is applied. The details of the calculation can be found in section
8.2.1 and we concentrate here on the analysis of the results. In the absence
of electron-phonon interaction, the transport characteristics of this model
have been discussed in sections 7.4.3 and 13.2. Within the LOE approxi-
mation, i.e. collecting all the contributions up to order λ2 , the current can
be written as (see section 8.2.1)
0
I = Iel + δIel + Iinel . (17.2)
0
Here, Iel is the elastic current in the absence of electron-vibration interac-
tion (see section 13.2). The other two terms constitute the correction to
the current due to the electron-vibration interaction and we now proceed
to explain their physical meaning.
The term Iinel is the inelastic contribution coming from the emission
and absorption of a single vibrational mode. At temperatures much lower
than ~ω/kB , the emission process dominates. This latter process is exactly
the one considered in the standard IETS (see section 16.2) and we show
it again schematically in Fig. 17.1(a). At zero temperature the emission
process has a threshold voltage equal to ~ω/e below which it cannot occur.
Above this voltage this term gives always a positive contribution to the
2 The spin does not play any role in this problem and we have dropped it in the previous
expression.
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504 Molecular Electronics: An Introduction to Theory and Experiment

(a) Phonon emission (b) Elastic correction

hω eV hω eV
L L
ε0 R ε0 R

Metal Molecule Metal Metal Molecule Metal

Fig. 17.1 Second-order inelastic processes contributing to the low-temperature current
in a molecular junction due to electron-phonon interaction: (a) phonon emission and (b)
elastic correction.

current, which means that it contributes to a step up in the conductance

at a voltage equal to ~ω/e.
The term δIel corresponds to the contribution of a process which in-
volves the emission and re-absorption of a virtual vibrational mode, see
Fig. 17.1(b). In this process, which was first discussed by Davis back in
1970 [752], there is a net conservation of the energy of the electrons and
for this reason we shall refer to its contribution as elastic correction. This
process has in general no threshold voltage and it gives a contribution to
the current that can be positive or negative depending on the voltage,
transmission and other factors, as we shall show below. This process has
traditionally been ignored in the context of IETS and also in many publi-
cations related to vibronic effects in molecular junctions, which has led to
some confusion.
The additional elastic contribution δIel can be interpreted as arising
from the interference between the zero order elastic amplitude and the
second order amplitude of the process in which a phonon is created and
destroyed [752]. The idea goes as follow. The total quantum-mechanical
amplitude of an electron tunneling event in the presence of the electron-
phonon interaction can be written as a series: A = A(0) + A(1) + A(2) + · · · ,
where the superindex indicates the order of the contribution in the electron-
phonon coupling constant, λ. The corresponding probability is obtained by
taking the modulus square. Thus, collecting all the terms up to second
order one gets3 |A|2 ≈ |A(0) |2 + |A(1) |2 + 2Re{A(0) A(2) }. The term |A(1) |2
corresponds to the processes involving the emission or absorption of a sin-
gle phonon, while the last one arises from the interference mentioned above
3 Theterm proportional to λ in this series vanishes because it does not conserve the
number of phonons in the system.
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Vibrationally-induced inelastic current II: Theory 505

kBT = 0.0
_
1 kBT = 0.01hω
G/G0 and Gel/G0

_
kBT = 0.025hω
1 _
0.995 kBT = 0.05hω
_

G/G0
kBT = 0.2hω
Gel
0.99 G 0.99

(a) 0.985
(c)
0.98
0 0.5 1_ 1.5 2 0.5 0.75 1_ 1.25 1.5
eV/hω eV/hω
δGel/G0 and δGinel/G0

0
2
(1/eG0) d I/dV
0.01
δGinel -0.05
0
2

-0.1
-0.01
δGel
-0.02 (b) -0.15 (d)
0 0.5 1_ 1.5 2 0.5 0.75 1_ 1.25 1.5
eV/hω eV/hω
Fig. 17.2 Results of the single-phonon (or Holstein) model for a highly transmissive
contact: ǫ0 − EF = 0, λ = 2~ω and ΓL = ΓR = 10~ω. (a) Zero-temperature total
conductance (G = dI/dV ) and elastic conductance (Gel = dIel 0 /dV ) as a function of the

bias voltage. (b) Elastic (δGel = dδIel /dV ) and inelastic (δGinel = dIinel /dV ) conduc-
tance corrections vs. voltage for the parameters of (a). (c) Temperature dependence of
the total conductance. (d) The corresponding d2 I/dV 2 vs. voltage for the temperatures
considered in (c).

and it is the origin of the elastic correction. Depending on whether this

interference is constructive (enhancing the forward scattering probability)
or destructive (enhancing the backscattering probability), this process can
give a positive or a negative contribution to the conductance, respectively.
So in short, the actual signature of the vibration modes observed in an ex-
periment in the weak electron-phonon regime is a result of the competition
between the emission term and the elastic correction.
Now we turn to the analysis of the results of this model. These results
have been calculated numerically using the formulas detailed in section
8.2.1. Let us start by discussing the case of a highly conductive junction in
the spirit of PCS, see section 16.3. In Fig. 17.2(a) we present the results
for the differential conductance for an on-resonant situation where ǫ0 = 0
(measured with respect to the Fermi energy) and ΓL = ΓR = 10~ω. With
these values the conductance is equal to G0 at zero bias and it shows a
very weak voltage dependence. As one can see in Fig. 17.2(a), the zero-
temperature conductance (sum of the elastic and inelastic contributions)
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506 Molecular Electronics: An Introduction to Theory and Experiment

exhibits an abrupt step down (of about 1%) at eV = ~ω. This result re-
produces the typical signature observed in the gold atomic chains or in the
Pt-H2 junctions discussed in section 16.3. As one can see in Fig. 17.2(b),
the step down in the conductance is due to the dominant negative con-
tribution coming from the elastic correction (δGel = dδIel /dV ). In other
words, the elastic correction gives rise in this limit to a finite backscatter-
ing that reduces the conductance of the junction [745]. After all, this is
natural because the (elastic) transmission is already close to one and thus,
an incoming electron can only be backscattered.
In Fig. 17.2(c) and (d) we show for this high transmission case the tem-
perature dependence of the differential conductance and the corresponding
d2 I/dV 2 , respectively. First, notice that the signature of the inelastic cur-
rent in d2 I/dV 2 is a dip and second, notice also that for temperatures of
the order of 0.2~ω/kB the signature is no longer visible.
We now consider a low-transmissive situation by simply shifting the level
away from the Fermi energy (ǫ0 − EF = 80~ω), but keeping the values of
the scattering rates of the previous example unchanged. Thus, the (elastic)
zero-bias conductance is equal to 0.059 G0 . In Fig. 17.3(a) we show the
contributions δGel and δGinel versus the voltage, as well as the sum of the
two (δG). In this case, we have assumed that the temperature is kB T =
0.05~ω. Notice that there several basic differences with respect to the
previous example. First, the change in the conductance at eV = ~ω is
dominated this time by the phonon emission process giving rise to a step
up. Second, the contribution δGel is positive for every voltage, but it
decreases slightly at the phonon energy. Third, the emission term has no
abrupt onset because of the finite temperature.
As one can see in Fig. 17.3(b), the signature of the vibrational mode is
barely visible in the differential conductance and one has to resort to its
derivative to see it clearly, see Fig. 17.3(c). Of course, the order of mag-
nitude of the inelastic current depends primarily on the electron-phonon
coupling constant, λ, which we have chosen small in comparison with the
scattering rates to ensure the validity of the perturbative approach. On the
other hand, notice that d2 I/dV 2 exhibits a linear background, typically
seen in the experiments, which is due to the contribution of the elastic
current, which contains a tiny cubic term (∝ V 3 ).
The model also describes the crossover between the two situations just
described, as we illustrate in Fig. 17.4. In this example, we have kept
constant the values of the scattering ΓL = ΓR = 10~ω (symmetric junction)
and changed the level position. As one can see in this figure, the vibrational
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Vibrationally-induced inelastic current II: Theory 507

0.062

G/G0 and Gel/G0

0,0001 (a) (b)
δG/G0
0.06
G
Gel
7.5e-05 δGel/G0
0.058
0 0.5 1_ 1.5 2
5e-05
eV/hω
0.0016
(1/eG0) dG/dV (c)
δGinel/G0 0.0015
2.5e-05

0.0014
0
0 0.5 1_ 1.5 2 0 0.5 1_ 1.5 2
eV/hω eV/hω
Fig. 17.3 Results of the single-phonon model for a low transmissive contact: ǫ0 −
EF = 80~ω, λ = 2~ω, ΓL = ΓR = 10~ω and kB T = 0.05~ω. (a) Elastic (δGel ),
inelastic (δGinel ) and total (δG = δGel +δGinel ) conductance corrections versus voltage.
(b) Corresponding total conductance and elastic conductance versus voltage. (c) The
corresponding d2 I/dV 2 .

signature in d2 I/dV 2 evolves from dip in an on-resonant situation to a peak

in an off-resonant case. It is also worth stressing that the crossing point of
this transition occurs exactly at an (elastic) transmission equal to 0.5.
It is possible to get an analytical insight into the previous results by
assuming that the elastic transmission is energy-independent. This is a
good approximation in two cases: (i) when the coupling to the leads is so
strong that the broadening of the resonant level (ΓL + ΓR ) is much larger
than ~ω, eV and |EF − ǫ0 | or (ii) when the resonant level is far away from
the Fermi energy, i.e. |ǫ0 − EF | ≫ ΓL,R , eV, ~ω. As we have shown in sec-
tion 8.2.1, under this assumption, one can prove that the zero-temperature
conductance for a symmetric contact (ΓL = ΓR = Γ) exhibits a jump at
eV = ±~ω given by (λ2 /Γ2 )τ 2 (1 − 2τ )/4, where τ is the transmission of
the contact. This result suggests that in a symmetric situation the con-
ductance shows a step down for τ > 1/2 and a step down for τ < 1/2,
while the signature vanishes for τ = 1/2. This result, which has been
coined as the 1/2 rule, was first derived by Paulsson et al. [736] using the
model that we are discussing and by de la Vega et al. [737] using an al-
ternative model. Both models differ in the exact transmission dependence
of the conductance jump, but both of them predict a crossover at exactly
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508 Molecular Electronics: An Introduction to Theory and Experiment

0
(a) ε0 = 0.0 0.03 _
-0.02 (b) ε0 = 10hω
0.02
Gel(V=0) = G0 Gel(V=0) = 0.8G0
-0.04 0.01
0 0.5 1_ 1.5 2 0 0.5 1_ 1.5 2
eV/hω eV/hω
2
(1/eG0) d I/dV

0.028 _
(c) ε0 = 20hω
2

0.027
0.026
Gel(V=0) = 0.5G0
0.025
0 0.5 1_ 1.5 2
eV/hω
_
(d) ε0 = 40hω
_
(e) ε0 = 80hω
0.009
0.0015
0.0085 Gel(V=0) = 0.2G0
0.0014 Gel(V=0) = 0.059G0
0.008
0 0.5 1_ 1.5 2 0 0.5 1_ 1.5 2
eV/hω eV/hω
Fig. 17.4 Results of the single-phonon model for the crossover between PCS and IETS:
λ = 2~ω, ΓL = ΓR = 10~ω and kB T = 0.05~ω. Second derivative of the current as a
function of the voltage for different values of the level position (measured with respect
to EF ) as indicated in the different panels. We also indicate in the panels the value of
the zero-bias elastic conductance.

τ = 1/2. Moreover, one can show that while the phonon emission term
gives a contribution equal to +(λ2 /Γ2 )τ 2 /4 to the conductance jump, the
elastic correction gives a negative contribution equal to −(λ2 /Γ2 )τ 3 /2. No-
tice that this result suggests that for very low transparencies, and if one
is only interested in the signature at the phonon energy, the contribution
of the elastic correction can be ignored, which is usually done in the IETS
context.
With respect to the temperature dependence of the phonon signature,
Paulsson et al. [736] have shown that for a symmetric contact, and ignoring
the energy dependence of the elastic transmission, the full width at half
maximum (FWHM) of the peak in d2 I/dV 2 is approximately 5.4kB T , i.e.
like in the standard IETS case [705].
Let us address now the typical situation realized in the STM contacts,
where there is a large asymmetry in the couplings between the molecule
and the surface and the molecule and the STM tip. In Fig. 17.5 we show
IET spectra for a junction in which ΓL = 10~ω and ΓR = 0.01ΓL . In
this figure the level position has been varied from a resonant case in panel
(a) to an off-resonant situation in panels (c) and (d). As one can see in
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Vibrationally-induced inelastic current II: Theory 509

0
(a) 0.0022 (b)
2

-0.002
(1/eG0) d I/dV

ε0 = 0.0 ε0 = 10hω
_
-0.004
0.002
2

-0.006
(c) (d)
0.0008 0.00013
_
_
ε0 = 20hω 0.00012 ε0 = 40hω
0.0007
0.00011
0 0.5 1_ 1.5 2 0 0.5 1_ 1.5 2
eV/hω eV/hω

Fig. 17.5 Results of the single-phonon model for a very asymmetric contact simulating
the situation typically realized in STM experiments: λ = 2~ω, ΓL = 10~ω, ΓR = 0.01ΓL
and kB T = 0.05~ω. Second derivative of the current as a function of the voltage for
different values of the level position (measured wit respect to EF ) as indicated in the
different panels.

the latter panels, the vibrational mode is manifested in the d2 I/dV 2 as

a peak at the mode energy, as it is usually observed in most STM-IETS
experiments. Notice, however, that when the level is brought close to the
Fermi energy, the signature progressively changes into a dip, as it is shown
in panel (a). As we explained in section 16.2, a dip is sometimes observed in
STM experiments and these results nicely clarify the necessary conditions
for the observation of dips. The origin of this crossover is the same as for
symmetric junctions, i.e. the elastic correction gives a dominant negative
contribution in the resonant case, while the phonon emission dominates in
off-resonant situations leading to a peak in the spectra.
To our knowledge, Persson and Baratoff were the first to point out the
possibility of a decrease in the conductance of a molecule in a STM experi-
ment due to resonant tunneling [753]. The issue of peaks and dips observed,
in particular, in the STM experiments has been revisited by Galperin and
coworkers going beyond the LOE [748, 749]. More recently, Egger and
Gogolin have reported analytic results for the zero-temperature inelastic
current in a molecule within the LOE approximation [750]. They have es-
tablished the criteria for the sign change of the step in the conductance.
In particular, they have shown that this transition, in general, not only
depends on the transmission of the junction, but it is governed by essen-
tially all system parameters (scattering rates and level position), as we have
shown here.
The single-phonon model is able to describe in a unified manner the ba-
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510 Molecular Electronics: An Introduction to Theory and Experiment

Fig. 17.6 Phase diagram for the single-phonon level model discussed in this section
(inset) illustrating the sign of the conductance change at the onset of phonon emission.
At a given asymmetry factor α the elastic transmission τ has an upper bound τmax (solid
line), and the inelastic conductance change undergoes a sign change at τcrossover =
τmax /2 (dashed line). Reprinted with permission from [751]. Copyright 2008 by the
American Physical Society.

sic vibrational signatures observed in the experiments in which the electron-

phonon interaction is weak. We can summarize the results discussed so far
following Paulsson et al. [751] with the phase diagram of Fig. 17.6. This
diagram describes the parameter range in which an increase or decrease of
the conductance is expected due to the phonon mode. This diagram has
been constructed assuming that the transmission can be considered energy-
independent (see discussion above). The diagram is plotted for the ratio
of the coupling to the two leads α = ΓR /ΓL and the transmission τ at EF .
In this model the maximal transmission is τmax = 4α/(1 + α)2 correspond-
ing to the on-resonance case. Notice that the crossover from a decrease to
an increase in the conductance is given by the 1/2 rule [735–737], i.e. at
τcrossover = τmax /2.
So far, we have assumed that the phonon mode in this model is in ther-
mal equilibrium at the bath temperature. In principle, the current flow can
drive this mode out of the equilibrium creating a finite population even at
zero temperature (as long as eV > ~ω). From a technical point of view, the
correct description of this nonequilibrium effect requires the evaluation of
the “phonon self-energies” that contain the information about the phonon
occupation and the phonon lifetime [748, 749]. This is a complicated task
that it is not easy to carry out in a consistent manner. For this reason
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Vibrationally-induced inelastic current II: Theory 511

we will follow here Paulsson et al. [736] and describe this effect at a phe-
nomenological level. The simplest way to include non-equilibrium heating
is to write down a rate equation for the phonon occupation, n, including
an external damping rate γd of the phonons [736]
P
ṅ = + γd [nB (~ω) − n] , (17.3)
~ω
where P is the power dissipated into the phonon mode and nB is the
Bose function. The external damping can be due to either the interac-
tion with the phonons of the electrodes or the electron-phonon interaction
in the molecule. From this equation, the steady state occupation n is easily
found. To complete the calculation we need now an expression for both the
power and the current in terms of the nonequilibrium phonon occupation.
Assuming that the transmission is energy-independent and considering a
symmetric junction (ΓL = ΓR = Γ), Paulsson et al. [736] showed that these
quantities can be expressed within the LOE approximation as follows
γeh π~
P LOE = γeh ~ω [nB (~ω) − n] + P, (17.4)
4 ~ω
2e2 1 − 2τ π~ Sym
I LOE = τ V + eγeh I , (17.5)
h 4 e~ω
where γeh = (ω/π) λ2 τ 2 /Γ2 is the electron-hole damping rate.4 Here, P and
I Sym are universal functions of the voltage, phonon frequency, temperature
and phonon occupation given by
h ³ ´ i ³ ´ ³ ´
eV ~ω eV
~ω cosh kB T − 1 coth 2kB T ~ω − eV sinh kB T
P= ³ ´ ³ ´ (17.6)
π~ ~ω
cosh kB T − cosh kB T eV

à !
Sym 2e ~ω − eV ~ω + eV
I = 2eV n + ~ω−eV − ~ω+eV . (17.7)
h e kB T − 1 e kB T − 1
Eq. (17.5) reproduces the zero-temperature transmission dependence
of the conductance jump discussed above. However, there is a small dis-
crepancy between these two results, namely the zero-temperature inelastic
conductance in Eq. (17.5) vanishes for eV < ~ω, while this is not the case
in the results presented above. The origin of this little difference is unclear
to us.
Eqs. (17.3)-(17.5) were used by Paulsson et al. in Ref. [736] to fit the
experimental results of Pt-H2 junctions [127, 567]. As one can see in
4 There is difference of a factor 4 in the expression of γeh with respect to Ref. [736]
because of the different definition of the scattering rates.
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512 Molecular Electronics: An Introduction to Theory and Experiment

dI/dV (G0) 0.98 a) Pt D2

Exp.
γd = 0
0.97 γd = γeh
0.96 γd = 10 γeh

60 40 20 0 20 40 60
1 b) Au
dI/dV (G0)

Exp.
γd = 0
γd = 3 γeh
0.98
γd = 10 γeh

-20 -10 0 10 20
Bias (mV)

Fig. 17.7 Single level model [Eqs. (17.4) and (17.5)] fitted to the experimentally mea-
sured conductance through a deuterium molecule [567]. The parameters used for the
fit are ~ω = 50 meV, τ = 0.9825, γeh = 1.1 × 1012 s−1 , and T = 17 K. (b) A simple
model (see Ref. [736] for details) fitted to the measured conductance through an atomic
gold wire (experimental data from Ref. [732]). The fit yields the following parameters:
~ω = 13.8 meV, T = 10 K, γeh = 12 × 1010 s−1 , and γd = 3γeh . Reprinted with
permission from [736]. Copyright 2005 by the American Physical Society.

Fig. 17.7(a), an excellent fit of the experimental data can be achieved by

using γeh and γd as adjustable parameters. The best fit was obtained using
a negligible external damping of the phonon mode (γd ≪ γeh ), which can
be understood physically from the mass difference between the hydrogen
molecule and the platinum atoms of the break junction. The nonequilibrium
occupation gives rise to the conductance slope that is seen in the experi-
ments for eV > ~ω. This feature in absent in the model with thermalized
phonons.
Using a similar single-phonon model designed for the atomic gold chains,
Paulsson et al. were also able to fit the experimental results of Ref. [732],
as one can see in Fig. 17.7(b). In this case the external damping γd =
3γeh is not negligible in contrast to the hydrogen case. This indicates that
presumably there is a strong interaction between the modes of the gold
chains and the phonons of the leads.

17.1.2 Ab initio description of inelastic currents

Although the simple model discussed in the previous section has proven to
be very useful, there are still many basic questions that are out of its scope.
Probably the most important one is related to the issue of the selection
or propensity rules. An understanding of the factors that determine why
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Vibrationally-induced inelastic current II: Theory 513

certain vibrational modes show up in transport experiments, while others

remain hidden, requires a microscopic modeling of the problem. In this
section we shall briefly review the work done in this direction in recent
years.
Probably the first ab initio calculations to investigate the inelastic cur-
rent through molecules were carried out by Lorente and Persson [754, 755].
They used a combination of DFT and Green’s function techniques to in-
terpret the STM experiments of Ho and coworkers [709, 756, 757]. In par-
ticular, Ref. [755] represents a first attempt to formulate propensity rules
for inelastic tunneling spectra. Since then, numerous authors have applied
microscopic methods to the description of vibrationally-induced inelastic
currents in molecular transport junctions with different levels of sophisti-
cation [202, 566, 751, 758–779].5 In what follows, we shall first formulate
the general problem of electron-phonon interaction in molecular junctions.
This will serve us to appreciate the ingredients that are required to calculate
vibrationally-induced inelastic currents. Then, we shall briefly comment on
the different approximations that have been put forward to perform these
calculations in realistic systems. Additionally, we shall describe the propen-
sity rules that have been derived so far and we shall show some examples
of the comparison between experiment and theory.6

17.1.2.1 Formulation of the problem

The general objective is to describe the effect of vibrations on the transport
through molecular junctions, when a voltage is applied. The coupled sys-
tem of electrons and vibrations in a molecular contact can be generically
modeled by the following Hamiltonian: H = He + Hvib + He−vib , where
X †
He = di Hij dj
ij
X
~ωα b†α bα + 1/2
¡ ¢
Hvib =
α

d†i λα
XX
†
He−vib = ij dj (bα + bα ). (17.8)
ij α
Here ωα are the vibrational frequencies, Hij = hi|H|ji are the matrix ele-
ments of the single-particle electronic Hamiltonian H in the atomic-orbital
5 This list is by no means complete, but it should be easy to trace back from it the

whole relevant literature on this subject.

6 What follows is more technical than usual and it is meant for the theoretical reader-

ship. The reader not interested in this theoretical discussion can jump directly to the
description of the propensity rules in section 17.1.2.4.
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514 Molecular Electronics: An Introduction to Theory and Experiment

basis {|ii}, and λα

ij are the electron-vibration coupling constants. The index
i denotes collectively the atomic sites and orbitals, and α runs from 1 to
3Nvib , where Nvib is the number of atoms in the system, which are allowed
to vibrate. The creation and annihilation operators for vibrational modes
b†α and bα satisfy the bosonic commutation relation [bα , b†β ] = δαβ . The
electronic basis is in general non-orthogonal, with overlap matrix elements
Sij = hi|ji. The calculation of the electronic structure of the junction, i.e.
the determination of the Hamiltonian He , is usually done within the DFT
framework (see Chapter 10) or with sophisticated tight-binding parameter-
izations (see Chapter 9).
In practice, the calculation of the vibrational modes is restricted to a
central region that includes the molecule and a small portion of the elec-
trodes. In principle, one should also describe how these central (or primary)
vibrations are coupled to the phonons in the electrodes. This is very dif-
ficult to do in a rigorous manner and such a coupling is usually taking
into account by means of a phenomenological parameter that enters as a
broadening in the density of states of the primary vibrations [695].
The solution of the inelastic transport problem involves a few rather
separate sub-problems: (i) the optimization of the geometry and evaluation
of the vibrational modes, (ii) computation of the electron-vibration coupling
constants and (iii) the calculation of the transport. Let us now discuss these
sub-problems in certain detail.

17.1.2.2 Vibrational modes and electron-vibration coupling con-

stants
The calculation of the vibrational modes requires knowledge of the total
ground-state energy of the system as a function E(R ~ k ) of the ionic coor-
~
dinates Rk with k = 1, . . . , Nvib . This energy is usually determined in the
framework of DFT. This energy needs to be minimized in order to find
the equilibrium configuration {R ~ (0) }. Now consider small displacements
k
~k = R ~k − R
~ (0)
Q k around the equilibrium positions. The Hamiltonian (in
first quantization) describing the oscillations of the ions around R ~ (0) is
k
given in the harmonic approximation by
1X 1 X
Hion = Mk Q̇2kµ + Hkµ,lν Qkµ Qlν , (17.9)
2 2
kµ kµ,lν

where Mk are the ionic masses, µ, ν = x, y, z denote the Carte-

sian components of vectors and H is the Hessian matrix: Hkµ,lν =
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Vibrationally-induced inelastic current II: Theory 515

∂ 2 E/∂Rkµ ∂Rlν . This matrix can be diagonalized by the transformation

P3Nvib
Qkµ = α=1 Akµ,α qα , where qα are the normal coordinates. Thus, we
obtain Hion = 21 α (q̇α2 + ωα2 qα2 ), where ωα (α = 1, . . . , 3Nvib ) are the
P

vibrational frequencies. The transformation matrix A is normalized ac-

cording to AT M A = 1, M being the diagonal mass matrix Mij = Mi δij .
1/2
Using the canonical quantization prescription qα = (~/2ωα ) (b†α + bα )
1/2 †
and q̇α = i (~ωα /2) (bα − bα ), one finally obtains Hvib in Eq. (17.8).
The electron-vibration interaction may be derived as follows [174, 202].
Assume that the electronic single-particle Hamiltonian H is a function
of the ionic coordinates, denoted collectively as R. ~ Then, we may ex-
~ (0) ~ ≈ H(R ~ )+
(0)
P ~ ~ . Defining H′e =
pand H(R + Q) k Qk · ∇k H|Q=0
~
P † ~ (0) + Q)|jid
~
d hi|H(R
ij i j , inserting the expansion, and using the canoni-
cal quantization for qα again, one finds H′e = He + He−vib . Here He and
He−vib are given by Eq. (17.8), with H being given by H(R ~ (0) ) and the
electron-vibration coupling constants by
µ ¶1/2 X
~ kµ
λα
ij = Mij Akµ,α , (17.10)
2ωα
kµ

kµ
where Mij = hi|∇kµ H|Q=0
~ |ji. From Eq. (17.10) one can see that the
calculation of the coupling constants requires to compute derivatives of the
Hamiltonian matrix elements with respect to the atomic position. Indeed,
since the employed basis sets are usually nonorthogonal, things are slightly
more complicated and the coupling constants are often calculated using the
ideas of Head-Gordon and Tully [780], see e.g. Ref. [775].

17.1.2.3 Inelastic current

The electric current is usually computed making use of the nonequilibrium
Green’s function (NEGF) techniques that we have described in Chapter 7.
In section 8.2 we derived the general current expression for an interacting
junction. As we explained there, there are indeed several possibilities for
this formula. Following Caroli et al. [209], we write the current as the sum
of two contributions, I = Iel + Iinel , where [see Eqs. (8.34)-(8.35)]7
8e ∞
Z
Iel = dE Tr [Gr ΓR Ga ΓL ] (fL − fR ), (17.11)
h −∞
7 Here the current has been evaluated at the left interface. Let us recall that to compute
the current one first divides the system into three parts: the leads (L and R) and a central
region (C), which contains the molecule and part of the electrodes. The electron-phonon
interaction is assumed to be restricted to this central part.
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516 Molecular Electronics: An Introduction to Theory and Experiment

∞
4ie
Z
dE Tr Ga ΓL Gr (fL − 1)Σ+− −+
© £ ¤ª
Iinel = e−vib − fL Σe−vib ,
h −∞

where fL,R (E) = f (E − µL,R ), f (E) = [1 + exp(βE)]−1 is the Fermi func-

tion and β = 1/kB T is the inverse temperature. Here, the full retarded
and advanced Green functions Gr,a are given by Gr = [ESCC − HCC
−ΣrL − ΣrR − Σre−vib ]−1 and Ga = [Gr ]† . On the other hand, Σr,a L,R are
the electronic self-energies that describe the electronic coupling between
the central region and the leads. The imaginary part of the advanced self-
energies are the corresponding scattering rate matrices, ΓL,R . The self-
energies Σre−vib and Σ±∓ e−vib are due to the electron-vibration interaction in
the central region. Since they vanish in the absence of λα , we call the Iinel
part an “inelastic” current, while Iel is the “elastic” part.
Up to now the expression of the current is exact, but we must now spec-
ify an approximation for the electron-vibration self-energies. Most of the
realistic calculations done so far have been carried out within the lowest-
order expansion (LOE), which we already used in the single-phonon model
above [736, 202]. More accurate approximations, like the so-called self-
consistent Born approximation (SCBA), have also been used [763], but
this latter approximation is computationally very costly. In the LOE ap-
proximation the Green’s functions are expanded to second order in λα , i.e.
Gr = G̃r + G̃r Σre−vib G̃r + · · · . In this way the elastic current is split into
0
two parts as Iel = Iel + δIel , where δIel is an “elastic correction”. We find
8e
Z
0
Iel = dE Tr[G̃r ΓR G̃a ΓL ](fL − fR ) (17.12)
h
16e
Z
δIel = dE ReTr[ΓL G̃r Σre−vib G̃r ΓR Ga ](fL − fR )
h
4ie
Z
Iinel = dE Tr{G̃a ΓL G̃r [(fL,R − 1)Σ+− −+
e−vib − fL,R Σe−vib ]}.
h
Notice that this division was also made in the analysis of the single-phonon
model. The expressions of the second-order self-energies can be found, for
instance, in Appendix C of Ref. [202], and they are natural extension of
those in Eq. (8.43). It is worth mentioning that within this approximation
one can rigorously prove the conservation of the current.
Even in the LOE, the current formulas [see Eqs. (17.12)] involve double
energy integrals which can be very cumbersome to evaluate. A further
simplification is achieved by assuming that the elastic transmission has no
pronounced energy dependence in the energy window where the vibrational
modes show up in the current. This approximation is not valid in the case
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Vibrationally-induced inelastic current II: Theory 517

where sharp resonances are present, but it turns out to be quite reasonable
in many situations of interest. With this assumption, the retarded and
advanced Green’s functions, as well as the scattering rates, can be evaluated
at the Fermi energy and some of the integrals can be done analytically,
which simplifies enormously the calculations. The detailed formulas for the
current within this approximation can be found in Refs. [736, 202].
Another important issue is the expression of the phonon occupation
that enters in the current formula via the electron-vibration self-energies.
The simplest approximation, which is fully consistent with the LOE, is to
assume that the phonons are in thermal equilibrium at the bath tempera-
ture. Heating effects, due to the nonequilibrium established at finite bias,
can be described in a various ways. For instance, as we explained for the
single-phonon model, the authors of Refs. [763, 736, 775] determine the
phonon occupation in a self-consistent manner by imposing that the power
transferred by electrons from the leads into to the device is balanced by the
power transferred from the device electrons to the phonons. Another phe-
nomenological way of introducing the nonequilibrium effects is discussed in
Ref. [777].

17.1.2.4 Propensity rules

One of the most surprising aspects revealed by the experiments is the fact
that only a small number out of the many possible vibrational modes gives
a signal in the transport characteristics. Motivated by this fact, many re-
searchers have employed the formalism detailed above, or variations of it,
to establish the rules that govern the contribution of a mode to the inelas-
tic signal. Let us emphasize that there are no strict selection rules like in
optical spectroscopies, but rather propensity rules. It is also worth remark-
ing that the contribution of a mode to the inelastic spectra does not only
depend on the symmetry of the mode itself, but also on the nature of the
orbitals that contribute to the current. In the search for these rules, some
general trends have been identified. For instance, the most significant con-
tributions typically come from modes with large longitudinal component,
i.e. motion along the tunneling direction [760, 766, 768]. On the other
hand, the calculations indicate a high sensitivity of the computed spectra
to the structure of the molecular bridge [760, 767, 768].
One of the most systematic studies of the propensity rules has been
carried out by Troisi and Ratner [765, 769, 770]. These authors have de-
veloped a simplified computational method that, although it does not allow
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518 Molecular Electronics: An Introduction to Theory and Experiment

them to compute the line shapes of the IET spectra, it provides a conve-
nient way to determine the intensities of the peaks in off-resonant situations.
More importantly, this appealing formulation has allowed the authors to
get a deeper insight into the propensity rules. With this approach, Troisi
and Ratner have again emphasized the importance of modes with large
component in the tunneling direction. Thus for instance, they have shown
that for a linear chain with one orbital per atom, only totally symmetric
modes contribute to IETS signal. For molecules with side chains any nor-
mal mode dominated by side chain motion will contribute only weakly to
IETS. The authors have also employed group theory to identify the main
normal modes for planar conjugated molecules with C2h symmetry.
Gagliardi et al. [776] have also presented a detailed study of the propen-
sity rules in the case of low-transmissive junctions, extending the work of
Troisi and Ratner. The approach of this work is based on the idea that
both the elastic and inelastic current can be expressed as the sum of a
small number of essentially noninteracting paths or conduction channels
through the device.
More recently, Paulsson et al. [751] have reported a method to determine
the propensity rules in junctions with arbitrary transparency (within the
weak electron-phonon coupling regime). Similar to Ref. [751], the key idea
in this work is to analyze the inelastic transport in terms of just a few
selected electronic scattering states, namely those belonging to the most
transmitting channels at the Fermi energy. These scattering states typically
have the largest amplitude inside the junction and thus account for the
majority of the electron-phonon scattering.

17.1.2.5 Quantitative comparison with experiments

The theory has been quite successful in reproducing the experimental in-
elastic spectra in the limit of weak electron-phonon interaction. By now,
there are many examples of satisfactory agreement between experiment and
theory. It is impossible review all these examples and here we shall just
mention a few illustrative cases.
One of the first comparisons was reported by Pecchia et al. [762],
who found reasonable agreement between their calculations on Au-
octanethiolate-Au junctions and the IETS results of Ref. [713]. Frederiksen
et al. [763] reported quantitative agreement between their calculation of the
IETS signal for atomic gold wires and experimental results [732]. These
authors found that the modes responsible for the inelastic signal are lon-
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Vibrationally-induced inelastic current II: Theory 519

Fig. 17.8 IET spectra of an anthracene thiol junction. The upper curve corresponds to
the experimental spectra and the lower curve to the computed one. Labels refer to the
normal modes of the molecule computed in the absence of metal and numbered from the
lowest energy vibration. Reprinted with permission from [718].

gitudinal ones with alternating bond length. Moreover, their calculations

showed the decrease in conductance with increase in the inelastic signal
and softening of the modes resulting from straining the wire. These cal-
culations were extended by Viljas et al. [202] who studied systematically
how the position and height of the conductance steps vary as a gold wire is
stretched and more atoms are added to it, and found good agreement with
the experiments.
Troisi and Ratner have applied their approach to several experimental
examples and they have found consistently a good agreement in all these
cases [765, 574, 718, 720]. In Fig. 17.8 we show a comparison between
theory and experiment for an anthracene thiol junction that was reported
in Ref. [718].
Another impressive example of agreement between theory and experi-
ment has been reported by Paulsson and coworkers [751]. These authors
studied very different model systems that range from atomic gold chains,
as an example of highly conductive junction, to off-resonant situations typ-
ically realized in STM-IETS experiments, see Fig. 17.9. The satisfactory
agreement with the experiment in these very different cases nicely illustrates
the level of understanding achieved in the weak electron-phonon coupling
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520 Molecular Electronics: An Introduction to Theory and Experiment

Fig. 17.9 Calculated (black lines) and experimental (blue lines) IETS. (a) OPE molecule
with Au(111) leads, (b) Au chain connected to Au(100) leads, (c) O2 molecule on
Ag(110), and (d) CO molecule on Cu(111). In case (c) the Fermi energy has been
shifted manually to match the experiment (dashed red line). The experimental data
originates from Refs. [712, 732, 711, 781]. For the STM configurations (c) and (d), the
calculated IETS is compared with a rescaled d2 I/dV 2 . Reprinted with permission from
[751]. Copyright 2008 by the American Physical Society.

regime. This agreement of the calculated and measured IET spectra makes
this spectroscopy, in combination with theory, a very useful diagnostic tool.

17.2 Intermediate electron-phonon coupling regime

The perturbative methodology discussed above describes correctly off-

resonant situations encountered in standard IETS experiments as well as
the resonant tunneling regime in cases where weak vibronic coupling re-
sults from strong electronic coupling to the leads (large electronic width
Γ) that ensures short electron lifetime on the bridge. The electronic trans-
port through a junction with a strong electron-phonon interaction is very
different from the weak coupling limit. Physically, in the course of the
transmission process the electron occupies the bridge long enough to affect
polarization of the bridge and its environment. In the ultimate limit of
this situation, decoherence and thermal relaxation are sufficient to render
the processes of bridge occupation and de-occupation, and often also trans-
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Vibrationally-induced inelastic current II: Theory 521

mission between different sites on the bridge, independent of each other.

This makes it possible to treat the transmission process as a sequence of
consecutive statistically decoupled kinetic events. In this section we want
to discuss intermediate situations where effects of transient polaron forma-
tion on the bridge have to be accommodated, however dephasing is not fast
enough to make a simple kinetic description possible.
In this intermediate regime, the LOE fails and new theoretical ap-
proaches are necessary. A strategy is to improve systematically the per-
turbation theory by including higher orders. Thus for instance, differ-
ent authors have used the self-consistent Born approximation (SCBA)
[746–749, 782]. In this case, the lowest-order Feynman diagrams taken
into account in the LOE are “dressed” by using the full Green’s functions.
Additionally, the description of the phonons can be improved by including
the corresponding phonon self-energies that describe the renormalization of
the phonon energies and their finite lifetimes. This method provides a way
to sum up certain diagrams up to infinite order, but it misses important
contributions of other high-order diagrams (vertex corrections). In this
sense, its validity is restricted to rather weak electron-phonon coupling.
In recent years, many other theoretical schemes have been introduced
to describe this intermediate regime [783, 785–791]. It is important to
emphasize that the application of all these methods has been restricted
to model Hamiltonians, in particular, to the single-phonon mode dis-
cussed above. Some of these works are based on the NEGF methodol-
ogy [783, 786–788, 790, 791] and others are based on an extension of the
equation-of-motion (EOM) method described in section 5.4.3 to include the
phonon dynamics [785, 789]. A central idea in most of these approaches is
the application of the polaron (or Lang-Firsov) transformation [792, 174] to
the single-phonon model. This transformation, which will be described be-
low, replaces the additive electron-phonon coupling [last term in Eq. (17.1)]
by a renormalization of the electronic coupling elements by phonon dis-
placement operators. The renormalized electronic coupling contains then
the effects of electron-phonon interaction to all orders and the transformed
Hamiltonian provides a more adequate starting point for situations where
the electron-phonon coupling is rather strong.
All the different approaches reported so far are approximate and the ex-
act description of vibrational effects for arbitrary strength of the electron-
phonon interaction, even ignoring electron-electron interaction, remains
as an open problem, at least in nonequilibrium situations. However, the
works mentioned above have been extremely useful to elucidate the essential
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522 Molecular Electronics: An Introduction to Theory and Experiment

-7
( 10 )

-8
( 10 )
5

dI/d (A/V)
2
dI/d (A/V)

1
3

3.0 3.1 3.2 3.3 3.4 3.5

(V)

3.0 3.5 4.0 4.5 5.0

(V)

Fig. 17.10 Differential conductance versus source-drain voltage calculated with the
EOM method applied to the single-phonon model of Eq. (17.1). The parameters have
the following values: ΓL = ΓR = 0.02 eV, T = 10 K, ǫ0 = 2 eV, ~ω = 0.2 eV, and
λ = 0.01 eV. The solid line corresponds to the self-consistent result and the dashed line
to the zero-order result (see Ref. [789] for details). The inset shows a blow-up of the
phonon absorption peak that appears on the left of the main resonance. Reprinted with
permission from [789]. Copyright 2006 by the American Physical Society.

physics in the intermediate regime. In particular, these approaches nicely

describe the appearance of phonon sidebands, which is the main vibrational
signature in resonant situations (when the metal-molecule coupling is not
too strong). As an illustration, we reproduce in Fig. 17.10 results reported
by Galperin et al. [789]. These authors applied the EOM method to the
Holstein model of Eq. (17.1). In particular, Fig. 17.10 shows the differential
conductance as a function of the bias voltage for a set of parameter val-
ues that corresponds to the case of a relatively narrow electronic resonance
(see figure caption). The first thing to notice in this figure is the appear-
ance of a main conductance peak at (at ∼ 3.6 V). This is the usual elastic
peak that appears when the resonant level crosses the chemical potential
of one of the reservoirs. In the absence of electron-vibration coupling this
peak would appear at 4 eV in this example because the voltage was ap-
plied symmetrically. As one can see in Fig. 17.10, the position of the level
has been renormalized by the interaction with the vibrational mode. The
most important consequence of phonon-assisted resonant tunneling is the
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Vibrationally-induced inelastic current II: Theory 523

(a) Resonant phonon emission (b) Resonant phonon absorption

hω eV ε0 hω
eV
L L
ε0
R R

Metal Molecule Metal Metal Molecule Metal

Fig. 17.11 Resonant inelastic tunneling processes in molecular junctions: (a) resonant
phonon emission and (b) resonant phonon absorption. This latter process does not have
a threshold voltage and it requires a finite occupation of the vibration modes due to a
finite temperature or to nonequilibrium phonon generation.

appearance of satellite peaks on the right hand side of the elastic peak.
Notice that these peaks are separated by a distance ∼ 2~ω/e. The factor 2
is again due to the choice of the voltage profile.
As we explained earlier, the fact that vibrations are in this case mani-
fested as peaks in the conductance, rather than peaks in d2 I/dV 2 , is simply
due to the fact that we are now dealing with a resonant situation. As shown
schematically in Fig. 17.11(a), the probability of the phonon emission tun-
neling process is greatly enhanced when the energy of an incoming electron
is such that by emitting a phonon it loses exactly the energy necessary
to cross the molecular level on resonance. This implies the appearance of
a peak in the differential conductance when the bias exceeds the voltage
necessary to see the resonant level in a quantity equal to ~ω/e times a cor-
rection factor that accounts for the shift of the level due to the voltage (this
factor equals 2 in Fig. 17.10). This argument applies for a single-phonon
process. If the electron-phonon coupling is large enough, emission of sev-
eral vibrational quanta becomes possible and it results in the appearance of
additional peaks in the conductance separated in this example by a voltage
equal to ∼ 2~ω/e, see Fig. 17.10. It is important to emphasize that in order
to resolve such satellite vibronic peaks, both the electronic coupling (Γ) and
the thermal energy (kB T ) must be smaller than ~ω, as in the example of
Fig. 17.10. Of course, at very low temperatures the voltage must be larger
than ~ω/e for the emission process to happen at all.
Another remarkable feature in Fig. 17.10 is the appearance of an ad-
ditional peak at Φ ∼ 3.25 V (see inset). As we show schematically in
Fig. 17.11(b), the resonant absorption of phonon could lead in this case
to the appearance of a peak on the left side of the elastic one. However,
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524 Molecular Electronics: An Introduction to Theory and Experiment

at the temperature of the calculation, the probability to thermally excite

phonons is negligible. Therefore, such feature must be a result of the heat-
ing of the phonon subsystem by electron flux. In other words, it probably
originates from the absorption of nonequilibrium phonons generated by the
current flow. As we explained at the end of section 16.5, such nonequilib-
rium absorption peaks were reported by LeRoy and coworkers in tunneling
experiments with suspended carbon nanotubes [740, 741].

17.3 Strong electron-phonon coupling regime

The impact of the vibrations on the transport characteristics increases with

both the strength of the electron-phonon interaction and the time that elec-
trons reside in the molecule. The former factor is not easy to tune, while
the second one can be controlled via the length of the molecule or the
metal-molecule coupling. In particular, when the electronic coupling be-
tween the molecule and the electrodes is weak, as in the case of molecular
transistors, the electrons in the molecule have sufficient time to interact
strongly with vibrations leading to the polaron formation (a mixed state
in which an electron is “dressed” by a phonon cloud [174]). What compli-
cates the theoretical description of this strong coupling regime is the fact
that the vibronic effects coexist with strong electronic correlations due to
the Coulomb interaction. Thus, electron-phonon interaction and electron-
electron interaction must be described in an equal footing.8 In this section
we shall review the present status of the theoretical understanding of this
strong coupling regime, which is realized in molecular transistors. We have
divided the discussion into two main parts. First, we shall consider vi-
bronic effects in the Coulomb blockade regime and then, we shall focus in
the interplay between Kondo physics and electron-phonon interaction.

17.3.1 Coulomb blockade regime

Let us now consider the case in which a molecule is weakly coupled to
metallic electrodes so that the transport is dominated by the Coulomb
blockade effect. In this regime the coupling to vibrational degrees of free-
dom leads to the emergence of sidebands in the I-V characteristics. This
phenomenon, which we have already discussed in the previous section, was
8 Notice that in the previous sections the electron-electron interaction was either ignored
or just described at a mean-field level.
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Vibrationally-induced inelastic current II: Theory 525

already described at the end of the 1980’s [794, 795]. In the context of
molecular transistors, Boese and Schoeller [796] were the first to analyze
vibronic effects in the Coulomb blockade regime. Motivated by the ex-
periments on C60 SMTs by Park et al. [22], these authors generalized the
many-body master equations described in section 15.5.1 to include vibronic
effects. Since then, many authors have used rate equations to study differ-
ent aspects of this problem, see e.g. Refs. [747, 797–807]. In what follows,
we shall first describe how this transport problem is formulated in terms of
rate equations and then, we shall briefly discuss some of the main physical
effects that have been predicted to occur in this regime.
The different formulations of rate (or master) equations differ only in
minor details and we have chosen to follow Ref. [747]. The starting point
is the single-phonon model (Holstein model) that we have extensively dis-
cussed in previous sections, but now with the inclusion of the electron-
electron interaction in the molecule. In this model, often referred to as
Anderson-Holstein model, the transport through the molecule is assumed
to be dominated by a single level of degeneracy dg with energy ε in the pres-
ence of one vibrational mode with frequency ω0 . This system is described
by the Hamiltonian H = Hmol + Hleads + Ht , where9

U
Hmol = εnd + nd (nd − 1) + λ~ω0 (b† + b)nd + ~ω0 (b† b + 1/2),
2
X X
Hleads = ǫp c†apσ capσ ,
a=L,R p,σ
X X¡
ta c†apσ diσ + h.c. .
¢
Ht = (17.13)
a=L,R; i=1,dg p, σ

Here, Hmol describes the molecular degrees of freedom, Hleads the leads and
Ht the tunneling between the leads and the molecule. The Coulomb block-
ade is taken into account via the charging energy U . We focus on the regime
of strong Coulomb blockade, U → ∞, appropriate when eV, kB T ≪ U . The
operator diσ (d†iσ ) annihilates (creates) an electron with spin projection σ
on degenerate level i of the molecule and nd = i=1,dg ; σ d†iσ diσ denotes
P

the corresponding occupation-number operator. Similarly, capσ (c†apσ ) an-

nihilates (creates) an electron in lead a (a = L, R) with momentum p and
spin projection σ. Notice that now the strength of the electron-phonon
interaction is measured in units of ~ω0 and it is characterized by the di-
mensionless constant λ.
9 Here, we ignore the dependence of the hopping integrals (ta ) on the indexes i and p.
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526 Molecular Electronics: An Introduction to Theory and Experiment

It is convenient to choose a representation which is diagonal in the

molecule degrees of freedom. In the present model this is achieved via the
polaron or Lang-Firsov canonical transformation [792, 174]. Defining S =
λ( i,σ d†i,σ di,σ ) b† − b and transforming all operators O via eS Oe−S
P ¡ ¢

leads to a transformed Hamiltonian H′ = H′mol + Hleads + H′t with

Ũ
H′mol = ε′ nd + ~ω0 (b̃† b̃ + 1/2) + nd (nd − 1) (17.14)
2
X X¡
′ †
¢
Ht = ta Xcapσ diσ + h.c. , (17.15)
a=L,R;i p,σ

where the transformed phonon operator b̃ = b − λ i,σ d†iσ diσ , so that the
P

phonon ground state depends on the dot occupancy. Moreover ε′ = ε−λ2 ω0

is the “polaron shift” in the energy for adding one electron to the molecule
and the interaction parameter U is also renormalized: Ũ = U − 2λ2 ~ω0 .
This renormalization will not be important below, since we shall focus here
on the limit U → ∞. The crucial phonon renormalization of the electron-
lead coupling is given by
h ³ ´i
X = exp −λ b̃† − b̃ . (17.16)

We are now in a position to write rate (master) equations for the

electron-phonon joint probabilities, which take the form10
Xn (n−1)
Ṗqn = fa ((q − q ′ )~ω0 + U (n − 1)) Γaq,q′ Pq′
a,q ′
(n+1)
+ [1 − fa ((q ′ − q)~ω0 + U n)] Γaq,q′ Pq′
− [1 − fa ((q − q ′ )~ω0 + U (n − 1))] Γaq′ ,q Pqn
−fa ((q ′ − q)~ω0 + U n) Γaq′ ,q Pqn .
ª
(17.17)
Here, Pqn is the probability to find the molecule with n electrons n (n =
0, · · · , 2dg ) and q phonons, while fa (x) is a short form for the Fermi function
f (x + ε′ − µa ), µa being the chemical potential of lead a.
Thus, the rate for going from a state with n electrons and q phonons on
n→n−1
the molecule to a state with n − 1 electrons and q ′ phonons is Wq→q ′ =
′ a a
P
a=L,R a f ((q − q )~ω 0 + U (n − 1)) Γq,q ′ , where Γ q ′ ,q represents the transi-
tion rate involving hopping of an electron from the dot to lead a by chang-
ing the phonon occupancy from q (measured relative to the ground state
of H′mol with occupancy n) to q ′ (measured relative to the ground state
10 These equations are a straightforward generalization of those discussed in sections
15.4.4 and 15.5.1 in the absence of the electron-phonon interaction.
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Vibrationally-induced inelastic current II: Theory 527

of H′mol with occupancy n − 1). This rate is equal to the transition rate
involving hopping of an electron from the lead a to the dot by changing the
phonon occupancy from q (measured relative to the ground state of H′mol
with occupancy n − 1) to q ′ (measured relative to the ground state of H′mol
with occupancy n). More explicitly
2
Γaq′ ,q = Γa |hq ′ |X|qi| . (17.18)
The matrix elements hq ′ |X|qi are known as the Franck-Condon matrix ele-
ments because they also govern the transitions between different vibrational
states in molecular physics. They can be computed by standard methods
[174] and their absolute value |hq|X|q ′ i|2 ≡ Xqq2
′ , which are symmetric un-
′
der interchange of q and q , are given by (see Exercise 17.1)
¯X (−λ2 )k (q!q ′ !)1/2 λ|q−q′ | e−λ2 /2 ¯2
¯ q ¯
2
Xq<q′ = ¯ ¯ . (17.19)
¯ ¯
¯ (k)!(q − k)!(k + |q ′ − q|)! ¯
k=0

It is interesting to write down explicitly a few elements:

2 λn 2
X0n = e−λ /2 √ ; X11 = 1 − λ2 e−λ /2
¡ ¢
(17.20)
n!
√ λ2 λ4
µ ¶ µ ¶
2 2
X21 = 2λ 1 − e−λ /2 ; X22 = 1 − 2λ2 + e−λ /2 .
2 2
Notice that for certain values of λ some of the matrix elements vanish.
This unusual behavior is an interference phenomenon. A state which has q
phonons excited above the ground state of the system with n = 0 electrons
is a superposition (with varying sign) of many multi-phonon states, when
viewed in the basis which diagonalizes the n = 1 electron problem, and
therefore the transition described by Xqq′ is really a superposition of many
different transitions, which for some values of λ may destructively interfere.
The current through the lead a in terms of the joint probability distri-
bution functions is given by
X
Ia = (2dg − n)Pqn fa ((q ′ − q)~ω0 + U n) Γaq,q′ (17.21)
n,q,q ′

−(n + 1)Pqn+1 [1 − fa ((q − q ′ )~ω0 + U n)] Γaq′ ,q ,

where the sum on n is from 0 to (2dg − 1), 2dg being the maximum occu-
pation of the dot.
Eq. (17.17) describes the nonequilibrium dynamics of the molecular vi-
brations. We shall now discuss the opposite limit, of phonons equilibrated
to an independent heat bath, assumed to be at the same temperature as
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528 Molecular Electronics: An Introduction to Theory and Experiment

the leads. To implement this, one forces the probability distributions on

the right hand side of Eq. (17.17) to have the phonon-equilibrium form
Pqn = P n e−q~ω0 /kB T (1 − e−~ω0 /kB T ). In the U → ∞ limit this Ansatz
implies that the probability P 0 that the molecule is empty is given by
a −q~ω0 /kB T ¯
P
0 a,q,q ′ Γq,q′ e fa,q,q′
P =P , (17.22)
a
a,q,q ′ 2Γq,q′ e
−q ′ ~ω 0 /k B T fa,q,q + Γq,q′ e−q~ω0 /kB T f¯a,q,q′
′
a

where f¯a,q,q′ = 1 − fa ((q − q ′ )~ω0 ), fa,q,q′ = 1 − f¯a,q,q′ and P 1 = 1 − P 0 .

For both equilibrated and unequilibrated cases the rate equations may
be written in the matrix form
Ṗ = M̂ P. (17.23)
Therefore under steady state conditions (Ṗn = 0), the problem reduces to
finding the eigenvector corresponding to the zero eigenvalue of the matrix
M̂ , which is easy to do numerically.
Let us now turn to analysis of the results of this approach. In Fig. 17.12
we reproduce some results of Ref. [747] where the current is depicted as a
function of the source-drain voltage (Vsd = (µL −µR )/e). In these examples
the level position was assumed to be ε′ = 0, i.e. the electronic level is at
resonance at zero bias. The two panels in Fig. 17.12 correspond to two
different values of the gate voltage defined as Vg = (µL + µR )/e. The upper
panel corresponds to Vg = 0 (µL = −µR ), while the lower one corresponds
to Vg = Vsd /2 (µR = 0). In both cases the results are shown for equilibrated
and unequilibrated phonons.
As one can see in Fig. 17.12, steps (broadened by the temperature)
in the current associated with “phonon sidebands” are observed when the
source-drain voltage passes through an integer multiple of the phonon fre-
quency. As we explained in the previous section, these steps originate from
resonant phonon emission processes. Notice that these I-V characteristics
reproduce the main features observed in the experiments in this regime,
see e.g. Fig. 15.16. In the linear response limit Vsd → 0 (not shown here),
as Vg is varied one finds one main step in the I-V curves. This is natural
since, as explained above, the appearance of phonon sidebands requires a
bias voltage larger than ~ω0 .
Fig. 17.12 also reveals that in some cases the current is larger for equili-
brated phonons than for the unequilibrated case. This is surprising because
one expects that in the unequilibrated case the phonons arrange themselves
so as to maximize the current. The authors of Ref. [747] attributed this
behavior to the special dependence of the Franck-Condon matrix elements
on the coupling constant λ.
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Vibrationally-induced inelastic current II: Theory 529

0.1

0.05
Equilibrated Phonons
I Unequilibrated phonons
0

−0.05

−0.1
−5 −4 −3 −2 −1 0 1 2 3 4 5
0.03
0.02
Equilibrated Phonons
0.01 Unequilibrated Phonons
I 0
−0.01
−0.02
−0.03
−5 −4 −3 −2 −1 0 1 2 3 4 5
Vsd

Fig. 17.12 Current (I) vs source-drain voltage Vsd for coupling constant λ = 1.0, ~ω0 =
1 and kB T = 0.05. Upper panel is for Vg = 0.0, while lower panel is for Vg = Vsd /2, µR =
0. I is in units of ekB T /~. Reprinted with permission from [747]. Copyright 2004 by
the American Physical Society.

The steps in current may be conveniently parameterized by the height

(or the area, as the width is simply proportional to T ) of the corresponding
peaks Gmax in the differential conductance G = dI/dV . Ratios of peak
heights (or areas) provide a convenient experimental measure of whether the
phonons are in equilibrium. At low T , the equilibrium phonon distribution
corresponds to occupancy only of the n = 0 phonon state, so the n-th
sideband involves a transition from the 0 phonon to the n phonon state.
Therefore the ratios of the peak heights or areas are controlled by ratios of
|Xn0 |2 . In particular, Eqs. (17.20) and (17.21) imply that if µL = −µR and
kB T ≪ ~ω0 ,
¯
Gnmax ¯¯ |Xn0 |2 λ2n
= = . (17.24)
G0max ¯ 2|X00 |2 2(n!)
¯
eq

This equation also gives a simple rule of thumb to estimate how many
phonon sidebands are expected for a given coupling constant λ. In particu-
lar, multiple steps arise only if λ is of the order of 1 or larger. Let us mention
that Sapmaz et al. [793] reported I-V characteristics of suspended single-
wall carbon nanotube quantum dots exhibiting a series of steps equally
spaced in voltage. These features were attributed to the excitation of the
stretching mode of the nanotubes. By comparing the I-V curves with the
model above for equilibrated phonons, a reasonable agreement was found
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530 Molecular Electronics: An Introduction to Theory and Experiment

with coupling constants of order unity.

Now that we have described the basic formalism, let us briefly discuss
some of the main physical effects that have been predicted in this regime:
Negative differential conductance (NDC).– It has been shown by several
authors that the interaction with vibronic degrees of freedom can lead to
negative differential conductance (NDC). This was first discussed by Boese
and Schoeller [796] and then by others [797, 801, 803, 804]. See in partic-
ular Ref. [804] for a detailed discussion of the conditions for the appearance
of NDC in this regime. This phenomenon has indeed been reported in
tunneling experiments with suspended carbon nanotubes [793].
Franck-Condon blockade.– When the electron-phonon interaction is very
strong, the Franck-Condon physics leads to a significant current suppres-
sion at low bias voltages, which has been termed as Franck-Condon (FC)
blockade [801, 805]. This phenomenon is illustrated in Fig. 17.13 where
we reproduce the results reported by Koch and von Oppen in Ref. [801].
This figure illustrates the strong dependence of the transport characteris-
tics on the electron-phonon coupling strength λ. In particular, Fig. 17.13(a)
shows the I-V curves for λ = 1 (intermediate coupling) and λ = 4 (strong
coupling), as obtained from the rate-equation approach. These results cor-
respond to ε′ = 0, i.e. the molecular single-particle level and the lead Fermi
energies are aligned at zero bias voltage. Notice that for λ = 1, the current
increases sharply due to resonant tunneling when switching on a small bias
voltage, and it exhibits the characteristic steps. In contrast, for λ = 4 the
current is significantly suppressed at low bias voltages.
The current suppression originates from the behavior of the FC matrix
elements determining the rates of phononic transitions q1 → q2 . For weak
coupling, λ ≪ 1, transitions mainly occur along the diagonal q1 → q1 . For
intermediate coupling, λ ≈ 1, the distribution of transition rates becomes
wider, and transitions slightly off-diagonal are favored. For strong electron-
phonon coupling, λ ≫ 1, the distribution widens considerably and a gap
of exponentially suppressed transitions between low-lying phonon states
opens, see Fig. 17.13(b). Finally, let us mention that the observation of
FC blockade has recently been reported in the context of suspended carbon
nanotube quantum dots [808].
Pair-tunneling.– The coupling to molecular vibrations induces a polaron
shift and can lead to a negative effective charging energy. In this case a
ground state with even number of electrons is favored. Moreover, the charge
transport through such molecules can be dominated by tunneling of electron
pairs and the I-V characteristics can exhibit striking differences from the
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Vibrationally-induced inelastic current II: Theory 531

Fig. 17.13 (a) I-V characteristics for intermediate (λ = 1) and strong (λ = 4) electron-
phonon coupling for ε′ = 0 and kB T = 0.05~ω0 for equilibrated and unequilibrated
phonons. The strong electron-phonon coupling leads to a significant current suppres-
sion at low bias voltages. This Franck-Condon blockade arises from the behavior of the
Franck-Condon rates for phonon transitions q1 → q2 plotted in (b). The rates Γq1 q2 ,
shown for λ = 1 (left) and λ = 4 (right), are given in units of the ordinary electronic
coupling Γ (here a symmetric junction is considered). For strong electron-phonon cou-
pling, transitions between low-lying phonon states are exponentially suppressed. The
corresponding current suppression cannot be lifted by a gate voltage, which may serve
as a fingerprint of FC blockade. This is depicted in the plot of dI/dV in the V –Vg plane
for unequilibrated phonons with λ = 4 (c). The case of intermediate coupling with λ = 1
(d) is shown for comparison. Reprinted with permission from [801]. Copyright 2005 by
the American Physical Society.

conventional Coulomb blockade. For a discussion of this phenomenology,

see Ref. [809].
Absorption sidebands.– As we discussed in the previous section, the cur-
rent flow can drive the vibrational modes far out of thermal equilibrium,
which will, in turn, act back on the current. This can be reflected in the
transport, in particular, with the appearance of vibrational sidebands in
the differential conductance on the left side of the Coulomb peaks (for pos-
itive bias). This is due to resonant absorption of nonequilibrium phonons
generated by the current, see Fig. 17.11(b). This phenomenon, observed in
suspended nanotubes [740], has also been studied in the Coulomb blockade
regime, see Refs. [810, 811].
Vibrational nonequilibrium effects with multiple electronic states.– The
phenomena discussed above referred to a situation where the transport was
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532 Molecular Electronics: An Introduction to Theory and Experiment

assumed to be dominated by a single electronic molecular level. Härtle

et al. [812] have shown recently that if multiple electronic states of the
molecular bridge are involved in the transport, a number of additional vi-
bronic processes take place and they may have a profound influence on the
current-voltage characteristics.
To conclude this section, let us say that most of the theoretical investi-
gations on vibrational effects in the Coulomb blockade regime have so far
concentrated on model systems with only one vibrational mode. Only re-
cently, several groups have started to combine ab initio methods with rate
equations to investigate more realistic systems. Thus for instance, Chang
et al. [806] reported the calculation of various phonon overlaps and their
corresponding phonon emission probabilities for the problem of an elec-
tron tunneling onto and off of the fullerene-dimer molecular quantum dots
C72 and C140 . In their approach, they do not assume that the vibrational
modes are identical for different charge states, as it is usually done. An-
other example along this direction is the work of Seldenthuis et al. [807]. In
this case, the authors have developed a method to calculate the vibrational
spectrum of a sizable molecule in the sequential tunneling regime, based
on DFT calculations to obtain the vibrational modes in a three-terminal
setup. This method takes the charge state and contact geometry of the
molecule into account and predicts the relative intensities of vibrational
excitations. In addition, transitions from excited to excited vibrational
state are accounted for by evaluating the Franck-Condon factors involving
several vibrational quanta. Thus, this method can predict qualitatively dif-
ferent behavior compared to calculations that only include transitions from
ground state to excited vibrational state.

17.3.2 Interplay of Kondo physics and vibronic effects

When the metal-molecule coupling is not too weak, high-order tunneling
processes become possible and their interplay with the Coulomb repulsion in
the molecules can lead to many-body phenomena like the Kondo effect (see
section 15.6.2). As we discussed in section 16.5, different experiments have
shown that the Kondo effect can coexist with vibronic effects. In this section
we shall present a brief discussion of the theoretical work done to clarify
the interplay between Kondo physics and electron-phonon interaction in
molecular junctions.
Since the Kondo effect is a coherent many-body phenomenon, one may
wonder under which circumstances this effect can survive in the presence of
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Vibrationally-induced inelastic current II: Theory 533

vibrationally-induced inelastic scattering. This question has been answered

to a large extend by Cornaglia and coworkers in a series of papers [813–817].
These authors have applied the numerical renormalization group (NRG) to
the Anderson-Holstein model in order to study the ground state and linear
conductance for a broad range of parameters. They have found that at low
temperatures and weak electron-phonon coupling (2λ2 ~ω0 ≪ U ) the prop-
erties of the conductance can be explained in terms of the standard Kondo
model with renormalized parameters. In particular, the electron-phonon in-
teraction leads surprisingly to an increase of the Kondo temperature in this
regime. In the limit of strong electron-phonon interaction (2λ2 ~ω0 ≫ U )
the problem can be mapped onto an anisotropic Kondo model where the
Kondo temperature decreases as λ increases [813].
Cornaglia et al. [817] also applied NRG to the Anderson-Holstein model
to explain the anomalous gate voltage dependence of the Kondo temper-
ature (TK ) found by Yu et al. [664] in SMTs based on transition metal
complexes. They found that, as the frequency of the vibrational mode
decreases, an anomalous gate dependence of TK and of the transport prop-
erties emerges. This effect arises because soft vibrational modes in the
molecular transistor drive the system into a new regime where the charac-
teristic energy scales for spin and charge fluctuations are not related as in
the conventional theory of the Kondo effect.
As shown in section 16.5, the clearest signature of the coexistence of
Kondo physics and vibronic effects is the appearance of sidebands at the
vibrational energies in the differential conductance versus the bias voltage.
This is in clear contrast to what it is found the Coulomb blockade regime.
What does the theory say about this nonequilibrium effect? This is an
extremely challenging problem since, even in absence of electron-phonon
interaction, there is no exact description of the finite-bias Kondo effect. In
this context, we would like to mention the work of Paaske and Flensberg
[818] where this problem has been addressed. These authors studied the
Anderson-Holstein model for a very asymmetric contact (ΓL ≫ ΓR ) and
found that the nonlinear conductance exhibits Kondo sidebands located
at bias voltages equal to multiples of the vibrational frequency. Moreover,
due to selection rules, the side-peaks were found to have strong gate-voltage
dependences. An example of the results is shown in Fig. 17.14. The left
panel shows a gray-scale plot of ∂ 2 I/∂V 2 as a function of bias-voltage V
and mean occupation number (gate voltage) N = Cg Vg /e. The right panel
shows three cuts revealing the side-band resonances on the flanks of the
central zero-bias resonance.
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534 Molecular Electronics: An Introduction to Theory and Experiment

1 0.04

G/ (ΓR / ΓL ) 2 e 2 / h
eV / ω 0

0 0.035

−1 0.03 c
a b c
−2
b
0.025 a

0.8 1 1.2 −2 −1 0 1 2
(Vg Cg /e) / (1−λ2ω 0 / Ec ) eV /ω0

Fig. 17.14 Left panel: ∂ 2 I/∂V 2 vs. bias and gate voltage, for λ2 = 3, N (0)|tL |2 =
0.1~ω0 , U = 16~ω0 , and kB T = 0.01~ω0 . The junction is considered to be very asym-
metric (ΓL ≫ ΓR ). Black/white indicates large negative/positive values. Right panel:
Conductance vs. bias voltage for three values of Vg corresponding to the vertical black
lines (a,b,c) in the upper panel. The lower curve (a) corresponds to the symmetric point
N = 1 − λ2 ~ω0 /EC . Reprinted with permission from [818]. Copyright 1999 by the
American Physical Society.

We conclude here our brief discussion of this transport regime by rec-

ommending Refs. [819–826] for further details on this problem.

17.4 Concluding remarks and open problems

Although we have discussed many different vibrationally-induced transport

phenomena in molecular junctions, it must be clear that our list is by no
means complete. For instance, we have not touched at all the dramatic
nonlinear effects that might appear due to a strong electron-phonon cou-
pling. It has been predicted by Galperin et al. [509] that the charging of
a molecular bridge (stabilized by the electron-phonon interaction) can lead
to a modification of the molecular geometry and in turn to effects like neg-
ative differential conductance, multistability and hysteresis. Such issues
have been addressed by several authors [827–829].
As we have seen in this chapter, the role of vibrations in the transport
through molecular junctions is one of the most studied topics in molecular
electronics. There several good reasons for that. On the one hand, as we
have shown throughout this chapter, the understanding of the vibrational
signatures in the transport characteristics is crucial for the detection of
internal modes of the junctions. In turn, the observation of these modes
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Vibrationally-induced inelastic current II: Theory 535

provides a valuable in situ characterization of the contacts. The modes

contain information not only about the presence of the molecule, which is
a non-trivial issue with most experimental techniques, but also about the
orientation of the molecule, its structure, the presence of defects and many
other aspects. On the other hand, molecular junctions provide an ideal
system to investigate new transport regimes where vibronic effects play an
essential role. In this sense, molecular transport junctions are becoming an
endless source of new physical phenomena.
The progress made in the last years in the understanding of vibronic
effects in molecular electronics is certainly remarkable. However, there are
still many challenges and basic open problems. Let us just mention a few of
them. From the experimental side, it would highly desirable to have more
IETS-like experiments with single-molecule junctions. Such experiments
could be very important to obtain structural information which could help
us to understand how the junctions are actually formed. In the context
of three-terminal devices, it would be interesting to characterize in more
detail the vibronic features in the Kondo regime. There are by now clear
predictions, for instance, about how the Kondo temperature is affected
by the electron-phonon interaction or about the gate dependence of the
Kondo sidebands at finite bias. These predictions await for experimental
confirmation.
The major open problem for the theory is the development of methods
that are able to interpolate between the different transport regimes that we
have discussed in this chapter. On the other hand, most theoretical models
avoid considering the back action produced by the excitation of vibration
modes. Such excitation, especially at high bias, may lead to structural
changes that can affect dramatically the transport properties. With respect
to the strong-coupling regime, little has done to describe microscopically
how the energy and coupling of the modes depend on the charge state of
the molecule. Of course, the description of the nonequilibrium “vibronic”
Kondo effect needs further investigation. Finally, irrespective of the trans-
port regime, more work is required to understand the role and signatures
of anharmonicity.

17.5 Exercises

17.1 Franck-Condon matrix elements: Show that the Franck-Condon matrix

elements in the Anderson-Holstein model are given by Eq. (17.19).
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536 Molecular Electronics: An Introduction to Theory and Experiment

17.2 Phonon sidebands in the Coulomb blockade regime: Solve the rate
equations both for equilibrated and unequilibrated phonons to reproduce the
results of Fig. 17.12. Using the parameters of the upper panel of Fig. 17.12,
compute the corresponding stability diagram.
17.3 Franck-Condon blockade: Use the rate-equation formalism described
in section 17.3.1 to study the Franck-Condon blockade in the regime of strong
electron-phonon coupling. In particular, solve the master equations for both
for equilibrated and unequilibrated phonons to compute the different transport
characteristics and reproduce the results of Fig. 17.13.
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Chapter 18

The hopping regime and transport

through DNA molecules

In Chapters 15-17 we have discussed how the electronic transport is modi-

fied when the quantum coherence is partially destroyed by either Coulomb
correlations or the excitation of molecular vibrations. One of the central
subjects of this chapter will be the analysis of the charge transport in sit-
uations in which this coherence is completely lost. As we explained in
previous chapters, this incoherent regime is realized when the tunneling
traversal time is considerably larger that the time scales associated to the
inelastic interactions. Obviously, this becomes more likely as the length
of a molecular bridge increases. In the extreme case in which the inelas-
tic scattering time is much smaller than the tunneling time, the current
is transported by electrons that hop sequentially from one segment of the
molecule to another. For this reason this transport regime is also referred
to as the hopping regime.
In long molecules, especially in biological ones, there are additional
issues that should be considered when exploring the electronic transport
through them. Thus for instance, the environment (solvent, atmosphere,
etc.) in which the experiments are carried out plays a decisive role. In order
to illustrate these issues, we shall also discuss in this chapter the transport
through DNA molecules, which is one of the most emblematic and difficult
topics in the field of molecular electronics.
The two main goals described above will be addressed in the following
sections. First, we shall discuss in section 18.1 the characteristic signatures
of the hopping transport regime. Then, in section 18.2 we shall describe
some representative examples of experiments in which the hopping regime
has been realized. Finally, section 18.3 is devoted to a brief review of the
recent activities on the electronic transport through DNA-based molecular
junctions.

537
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538 Molecular Electronics: An Introduction to Theory and Experiment

1 2 N−1 N
∆E

L R

Fig. 18.1 Schematic representation of the model discussed in text to describe the inco-
herent tunneling through a molecular junction. Here, N sites with the same energy are
connected via nearest-neighbor transfer rates kj,j±1 . The continua on the left and right
correspond to the metallic states in the electrodes and ∆E is the activation energy.

18.1 Signatures of the hopping regime

The question that we want to address in this section is: How can we identify
the occurrence of the hopping regime in an experiment? As we saw in
Chapter 13, the coherent transport in off-resonant situations is manifested
in the linear conductance as an exponential dependence on the length of
the molecule and as an independence on the temperature. The hopping
regime is however characterized by the following two main signatures:

• The conductance decays linearly with the length of the molecular wire.
• The conductance depends exponentially on the temperature as
exp(−∆E/kB T ), where ∆E is an activation energy that depends on
the system under study.

Following the spirit of this monograph, we now proceed to discuss a sim-

ple model that illustrates how these two signatures come about. The model
for a metal-molecule-metal junction, which is borrowed from the field of
electron transfer [830, 831], is schematically represented in Fig. 18.1. Here,
the molecular bridge has N sites (or states) and the incoherent tunneling
between them is described by the transfer rates ki,j (from state j to state
i).1 For the sake of simplicity, we assume that all the states in the wire have
the same energy, which differs by ∆E from the equilibrium Fermi energy
of the leads. The quantity ∆E, which is nothing but the injection energy,
plays here the role of an activation energy.

1 We assume that only nearest-neighbor sites are directly connected, i.e. the only non-
zero rates are kj,j±1 .
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The hopping regime and transport through DNA molecules 539

In this model the current between sites j and j + 1 is determined by the

occupations Pj and Pj+1 in those two sites as follows2
Ij = e (kj+1,j Pj − kj,j+1 Pj+1 ) . (18.1)
Assuming that the tunneling between the different sites is incoherent, the
occupations Pj fulfill then the following classical kinetic equations
Ṗ1 = −(k0,1 + k2,1 )P1 + k1,0 P0 + k1,2 P2
..
. (18.2)
Ṗj = −(kj−1,j + kj+1,j )Pj + kj,j−1 Pj−1 + kj,j+1 Pj+1
..
.
ṖN = −(kN −1,N + kN +1,N )PN + kN,N −1 PN −1 + kN,N +1 PN +1 ,
where Ṗj stands for dPj /dt, P0 = fL and PN +1 = fR , fL,R being the
Fermi functions describing the electron occupations on the left and right
electrodes. We are interested in a stationary situation where Ṗj = 0. In
this case, the previous kinetic equations reduce to the following algebraic
equations
(k0,1 + k2,1 )P1 = k1,0 P0 + k1,2 P2 (18.3)
..
.
(kj−1,j + kj+1,j )Pj = kj,j−1 Pj−1 + kj,j+1 Pj+1
..
.
(kN −1,N + kN +1,N )PN = kN,N −1 PN −1 + kN,N +1 PN +1 .
As a further simplification, we assume that all the internal rates in the
bridge are equal: kj,j±1 = k. Moreover, the detailed balance condition
leads to the following relations for the rates involving the leads3
k1,0 = kL e−(∆E−eV )/kB T ; k0,1 = kL (18.4)
−∆E/kB T
kN,N +1 = kR e ; kN +1,N = kR . (18.5)
Here, we have taken into account the influence in the activation energy, ∆E,
of the bias voltage, V , which we assume to be applied in the left electrode.
2 The current is, of course, conserved and therefore, it is irrelevant where it is evaluated.
3 In equilibrium the current must vanish and this leads to the relations: kj+1,j Pjeq =
eq
kj,j+1 Pj+1 , known as detailed balance conditions. Here, Pjeq is the occupation probabil-
eq
ity of the site j in equilibrium. Therefore, kj+1,j /kj,j+1 = Pj+1 /Pjeq = exp[−(Ej+1 −
Ej )/kB T ], if Ej+1 > Ej and 0 otherwise.
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540 Molecular Electronics: An Introduction to Theory and Experiment

It is straightforward to solve Eqs. (18.3) and to show that the charge

current is given by (see Exercise 18.1)
e−∆E/kB T
I=e [eeV /kB T fL − fR ]. (18.6)
[1/kL + 1/kR + (N − 1)/k]
Therefore, the corresponding linear conductance can be expressed as
e2 e−∆E/kB T
G= . (18.7)
kB T [1/kL + 1/kR + (N − 1)/k]
Here, for the sake of simplicity, we have neglected the temperature depen-
dence coming from the Fermi functions of the leads.
From Eq. (18.7) one can deduce the two signatures described at the be-
ginning of this section. First, notice that the conductance decays linearly
with the number of sites (or incoherent segments) and therefore with the
length of the molecular bridge. This is nothing else but the classical Ohm’s
law, which is a consequence of the loss of quantum coherence. Notice that
if we ignore the activation process, the conductance simply adopts the stan-
dard expression of the conductance of a combination of resistors in series.4
On the other hand, the conductance depends exponentially on the tem-
perature, as in any thermally activated process. In our particular model,
this process takes place at the metal-molecule interfaces, but in general it
can occur at any point along the junction and there can even be several
activation centers with their corresponding activation energies.
It is worth stressing that this model just provides a simple argument to
understand the origin of the main signatures of the hopping regime, but one
cannot expect quantitative predictions from it. An important issue that this
model fails to describe is the transition from the coherent to the incoherent
regime as function of the temperature and the length of the molecular
bridge. Such transition, which is a key signature in the experiments (see
next section), has been described by several authors using, for instance,
the reduced density matrix formalism [830, 832–834]. In these models, the
dephasing and relaxation is provided by a generic thermal bath. For a
discussion on the unified description of coherent tunneling and the hopping
mechanism, see Ref. [835].
The problem with these simple bath models is that they do not shed light
on the microscopic origin of the loss of coherence. The main physical mech-
anism that makes the transport incoherent is believed to be the electron-
phonon interaction inside the molecular bridge. In particular, when the
4 Here, the resistors are the two metal-molecule interfaces, with resistances

kB T /(e2 kL,R ), and the N − 1 connections between the bridge sites, with a resistance
kB T /(e2 k) each.
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The hopping regime and transport through DNA molecules 541

coupling between different segments of a molecule is weak, the charge car-

riers can be localized over a single or a few segments. In this case, the
molecule tends to change its conformation in order to lower its energy when
charged. This process is known as polaron formation. As explained in the
previous chapter, a polaron is a combination of a charge carrier and lo-
calized deformation. At room temperature, charge transport can be then
dominated by incoherent hopping of polarons along the molecule. Such
incoherent hopping gives rise to the exponential temperature dependence
described above, with an activation energy which is the polaron binding
energy.
The polaronic mechanism has been extensively studied in the context
of conduction in solids [836]. This mechanism plays a fundamental role, for
instance, in the conduction properties of organic materials used in organic
electronics. In principle, the polaronic effects in molecular junctions can
be described with an extension of the theoretical methods discussed in the
previous chapter. However, such an extension is not straightforward. In
practice, the polaron formation have mainly been analyzed with the help of
single-level models [509, 827, 837], and polaron hopping in long molecules
has typically been described using simple rate equations [838, 839]. For a
discussion on the difficulties in describing polaron formation and hopping
in molecular junctions and on recent advances in the treatment of this
problem, see Ref. [840] and references therein.

18.2 Hopping transport in molecular junctions: Experi-

mental examples

Experiments showing clear indications of the occurrence of hopping trans-

port are rather scarce, especially at the level of single molecules. The main
reason for this is the difficulty of measuring the temperature dependence
of the transport characteristics. In this section we shall review a couple of
representative examples in which the observation of hopping transport has
been claimed.
The observation of thermally activated transport in single-molecule
junctions was first claimed by Selzer et al. [841]. These authors
studied the transport through individual 1-nitro-2,5-di(phenylethynyl-4′ -
mercapto)benzene molecules, see inset of Fig. 18.2(a), with gold electrodes
using the electromigration technique. In this experiment, I-V measurements
were taken at a temperature range of 13-296 K over a ±1 V bias range.
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542 Molecular Electronics: An Introduction to Theory and Experiment

(a) (b)

Fig. 18.2 (a) A set of I-V curves measured at different temperatures in Au-molecule-Au
junctions fabricated with the electromigration technique. The molecule is shown in the
upper inset, where a junction is schematically represented. (b) Arrhenius plots of Ln
current (amperes) versus inverse T (K−1 ) at different bias voltages showing a transition
in conductance from T -independent tunneling behavior at low T to a thermally activated
process at high T . The bias increment between curves is 0.1 V, and the bias of the lowest
curve is 0.1 V. The transition temperatures between coherent and incoherent behavior
are marked by the intersection between lines; see, for example, the arrow for 0.3 V.
Reprinted with permission from [841]. Copyright 2004 American Chemical Society.

A representative set of I-V curves for different temperatures is shown in

Fig. 18.2(a), where one can see that the current is quite sensitive to the
bath temperature. An Arrhenius plot for a typical junction is shown in
Fig. 18.2(b). Here, one can clearly see the transition from temperature-
independent behavior at low T , where the conduction is dominated by
coherent tunneling, to temperature-independent hopping behavior at high
T , where presumably the transport is incoherent (hopping regime).
As one can see in Fig. 18.2(b), the transition from coherent to inco-
herent behavior is shifted to lower temperatures with increasing bias. As
pointed out by the authors, there may be two complementary reasons for
this behavior. First, the activation energy ∆E for hopping decreases as
a function of bias. As the current in the hopping mechanism is propor-
tional to exp(−∆E/kB T ), it is initiated at a lower bath temperature as
∆E decreases. Second, due to heat dissipation, the effective temperature
of the molecule increases with bias, which can also induce a transition to
incoherent tunneling at a lower bath temperature.
Let us also mention that the activation energy at zero bias was found to
be 0.13 eV, which is probably too small to correspond to the injection energy
(distance between the Fermi energy and the closest molecular level) for
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The hopping regime and transport through DNA molecules 543

Fig. 18.3 Measurements of the temperature dependence of the current in three-terminal

molecular junctions with sulfur end-functionalized tercyclohexylidenes. The different
curves correspond to four values of the source-drain voltage as denoted in the figures.
The panel (a) corresponds to a gate voltage of -1.0 V and the panel (b) to 1.0 V. The
solid lines represent the best fits to the resonant tunneling model (see text). Reprinted
with permission from [471]. Copyright 2006 American Chemical Society.

this molecule. This suggests that the rate-limiting process in the hopping
mechanism is not thermal population of electrons/holes from the electrodes
into the first hopping site, but rather an intramolecular hopping process
(along the molecule).
Although the evidence presented in Ref. [841] is rather convincing, one
cannot completely exclude an interpretation of the data of Fig. 18.2 in terms
of coherent tunneling. As we explained in section 13.2, coherent tunneling
can also lead to a pronounced temperature dependence of the I-V charac-
teristics (see discussion below). This has been illustrated by Poot et al.
[471], who reported data similar to those of Fig. 18.2(b) in a three-terminal
device fabricated with the electromigration technique. In particular, these
authors investigated the gate and temperature dependence of the current
in molecular junctions containing sulfur end-functionalized tercyclohexyli-
denes. In Fig. 18.3 we reproduce some of the results of Ref. [471] in which
one can see the current as a function of temperature for four different bias
voltages at two gate voltages on a semilog scale. Notice that at low bias
the curves of Fig. 18.3 show thermally activated transport at high tem-
perature and temperature-independent transport at low temperature, i.e.,
very much like in Fig. 18.2(b). The crossover temperature is about 150
K in Fig. 18.3(a) and it decreases slightly as the bias is increased. The
slope of the exponential increase above this crossover temperature yields
and activation energy of 120 meV at low bias and this value decreases with
increasing bias.
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544 Molecular Electronics: An Introduction to Theory and Experiment

The authors of Ref. [471] use the simple resonant tunneling model de-
scribed in detail in section 13.2 to analyze their data. Let us recall that in
this model the temperature dependence comes from the Fermi distribution
function in the leads and that the current becomes temperature-dependent
when kB T is not too small in comparison with the injection energy (or level
position measured with respect to the Fermi energy), which is the case in
this experiment and in the previous one described above. As one can see
in Fig. 18.3, the authors were able to fit the experimental data with this
model using as adjustable parameters the level position, ǫ0 , and the scatter-
ing rates ΓL and ΓR . It is important to remark that the two gate voltages
in Fig. 18.3 were far away from the degeneracy points, i.e. the transport is
not completely at resonance. In the fits, the values found for ǫ0 were very
similar to the activation energy mentioned above of 120 meV, which shows
the consistency of the fits. The total broadening of the level, Γ = ΓL + ΓR ,
was found to be in the range from 0.1 to 5 meV and it increased with
increasing bias voltage.5 In addition, the ratio Γ/ǫ0 was found to range
between 10−3 and 10−2 .6
The previous discussion shows that an unambiguous identification of
the hopping regime requires additional information beyond the tempera-
ture dependence of the current. As we explained in the previous section,
another key signature of the hopping regime is the linear decay of the con-
ductance/current with the length of the molecule. To our knowledge, this
signature, which is well-known in the context of electron transfer (see e.g.
Ref. [842]), has not yet been reported in single-molecule junctions. How-
ever, Choi et al. [843] have reported recently the transition from coherent
to hopping regime as a function of the molecular length in junctions based
on monolayers of conjugated oligophenyleneimine (OPI) molecules ranging
in length from 1.5 to 7.3 nm. The OPI wires were grown on a gold substrate
and contacted by a metal-coated AFM as a second electrode. In Fig. 18.4(a)
we reproduce the results of this experiment concerning the resistance (R)
versus molecular length (L) for a series of OPI molecules with different
numbers of phenyl units (n). As one can see, there is a clear transition of
the length dependence near 4 nm (OPI 5). In short wires, the linear fit in
Fig. 18.4(a) indicates that the data are well described with the standard
formula of coherent non-resonant tunneling: R = R0 exp(βL). The β value

5 Here, Γ corresponds to the full width of the resonance at half maximum, while in
section 13.2 it represents the half width at half maximum.
6 The temperature dependence of the current within the resonant tunneling model is

further discussed in Exercise 18.2.

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The hopping regime and transport through DNA molecules 545

Fig. 18.4 Measurements of molecular wire resistance with a conducting probe AFM. A
gold coated tip was brought into contact with an OPI monolayer on a gold substrate.
(a) Semilog plot of R versus L for the gold/wire/gold junctions. Each data point is the
average differential resistance obtained from 10 I-V traces in the range -0.3 to +0.3 V.
Straight lines are linear fits to the data according to R = R0 exp(βL). The inset shows
a linear plot of R versus L, demonstrating linear scaling of resistance with length for the
long OPI wires. (b) Arrhenius plot for OPI 4, OPI 6, and OPI 10. Each data point is
the average differential resistance obtained at six different locations on samples in the
range -0.2 to +0.2 V. Straight lines are linear fits to the data. From [843]. Reprinted
with permission from AAAS.

was found to be 0.3 Å−1 , which is within the range of β values of typical
conjugated molecules.
For long OPI wires, there is a much flatter resistance versus molecular
length relation (β ∼ 0.09 Å−1 ). The extremely small β suggests that the
principal transport mechanism is hopping. As one can see in the inset of
Fig. 18.4(a), a plot of R versus L for long wires is linear, which is consis-
tent with hopping. The change in transport mechanism was also verified
by the temperature dependence. Fig. 18.4(b) shows that the resistance
for OPI 4 is independent of temperature from 246 to 333 K, as expected
for non-resonant coherent tunneling. However, both OPI 6 and OPI 10
display the strongly thermally activated transport that is characteristic of
hopping. The activation energies determined from the slopes of the data
are identical at 0.28 eV for both OPI 6 and OPI 10. Concerning the ques-
tions on the nature of the hopping sites and the origin of this activation
energy, the authors suggested that three-repeat conjugated subunits are
the charge-hopping sites in the long wires and that the hopping activation
energy corresponds to the barrier for rotation of the aromatic rings, which
transiently couples the conjugated subunits.
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546 Molecular Electronics: An Introduction to Theory and Experiment

18.3 DNA-based molecular junctions

When the transport through a long molecule is investigated additional in-

gredients not discussed so far in this book, such as the presence of a solvent
or the interaction with a substrate, may play a decisive role. This is spe-
cially clear in the case of biological molecules, where these factors can affect
dramatically their conduction properties. In order to illustrate these ideas,
we shall discuss in this section the transport through DNA molecules.
The great interest in the DNA molecule as a possible component of
molecular electronic devices is due to its unique recognition and self-
assembling properties. These properties offer in principle the possibility
to build complex circuits with a bottom-up approach using this biological
molecule as a building block. Obviously, the understanding of the electron
transport in DNA molecules is a necessary prerequisite for the develop-
ment of a DNA-based molecular electronics [844–847]. For this reason, a
great effort has been devoted in the last 20 years to elucidate the transport
properties of this molecule. However, due to the complexity of DNA, there
are still many open questions in this subject. In this section, and taking
into account the scope of this book, we shall briefly review the experiments
in which the electronic transport through single DNA molecules has been
investigated. For the vast literature on multi-molecule measurements we
refer to the review articles by Porath, Cuniberti and Di Felice [848] and by
Endres, Cox, and Singh [849].
Let us remind that natural DNA consists of two long polymers of simple
units called nucleotides, with backbones made of sugars and phosphate
groups joined by ester bonds. Each of these strands is called single-stranded
DNA (ssDNA). These two strands form a double helix with the backbones
pointing outwards, the so-called double-stranded DNA (dsDNA). Attached
to each sugar is one of four types of molecules called bases: adenine (A),
cytosine (C), guanine (G) and thymine (T). Each type of base on one
strand forms a bond with just one type of base on the other strand. This is
called complementary base pairing or Watson-Crick pairing. In particular,
A binds only to T, while C binds only to G. This arrangement of two
nucleotides binding together across the double helix is called a base pair.
Double-stranded DNA exists in several conformations, among which the
B-conformation is the natural one, which is, however, only stable in aqueous
environment. In the B-conformation CG and AT pairs are stacked above
each other at a distance of 3.4 Å between each pair, see Fig. 18.5. Each
strand is stabilized by the backbone keeping the bases at this distance. In
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The hopping regime and transport through DNA molecules 547

1 turn = 10 base pairs = 3.4 nm

2 nm
major groove minor groove

Fig. 18.5 Double helical structure of DNA in B-conformation. Taken from Wikipedia
Commons.

the B-conformation the long axis of neighboring base pairs are twisted with
respect to each other by an average angle of 36o such that 10 base pairs
make a full turn. This conformation is stabilized by water molecules and
other counter ions. In dry conditions (less than 5 H2 O molecules per base
pair) the stable conformation is the so-called A-conformation that differs
from the B-conformation by an inclination of the base pairs with respect
to the molecule’s axis and a somewhat weaker twist [850].
With respect to the conduction mechanism in DNA, it is generally ac-
cepted that the electron transfer in DNA takes place via the overlap between
the π-orbitals of neighboring base pairs. This is similar to what happens
in certain stacked aromatic crystals, like the Beechgard salts, which are in-
deed metallic. This electron transfer mechanism in DNA suggests that the
base sequence can be very important since the π-system of the individual
bases may be different. Moreover, the conformation is important because
it determines the overlap between the base pairs. Finally, in order to have
a measurable electrical current in a DNA junction, it is crucial to make
sure that the π-system hybridizes strongly with the metallic states of the
electrodes.
Now, we turn to the discussion of the transport experiments in DNA-
based junctions. Let us start by summarizing the main findings. Most of
the transport measurements on single DNA molecules reported so far can be
divided into three classes. First, there are experiments showing that DNA
is an insulator for lengths larger than 40 nm at room temperature, with
essentially no discernible conductance up to 10 V. This suggests that the
electronic states of DNA are completely localized [851, 852]. Second, some
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548 Molecular Electronics: An Introduction to Theory and Experiment

experiments show that it is possible to transport charge through short DNA

molecules of up to 20 nm with currents of the order of 1 nA at 1 V. This
suggests that short DNA pieces behave as large-bandgap semiconductors
[853, 854]. Third, some experiments show that if special care is taken to
tailor the interaction with the substrate and to provide good contacts to
the leads, semiconducting-like I-V characteristics with currents exceeding
200 nA at 1 V can be achieved even under ambient conditions [855].
It is worth remarking that there are several experiments that do not
fall into any of these three categories. Thus for instance, Fink and
Schönenberger [856] reported ohmic behavior in 16 µm-long DNA molecules
in experiments performed with a field-emission microscope. On the other
hand, Kasumov et al. [857] found the induction of proximity superconduc-
tivity in experiments where a peculiar contacting scheme was used (see
Chapter 3). This observation can only be explained if DNA turns out to
be a very good conductor.
Several experiments have been designed to elucidate the origin of the
discrepancies mentioned above. Thus for instance, both Kasumov et al.
[858] and Heim et al. [859] using a STM or a conducting AFM, respectively,
showed that the interaction between DNA and the underlying substrate
plays a fundamental role in the conduction properties of this molecule. Such
interaction turns out to be in some cases strong enough to deform the DNA
molecule and to induce conformational changes. These changes may be
responsible for the blocking of the current along the molecule, which results
in the insulating behavior observed in many experiments. The importance
of the molecule-substrate interaction have also been emphasized by several
groups in the case of different polymers [504, 860]. For DNA, it is known
that external forces can stabilize several helical conformations [861].
As mentioned above, the transport through DNA is expected to depend
on the exact base sequence. This has been nicely illustrated by Tao’s group
in experiments on short DNA pieces (eight GC base pairs plus a varying
number of AT base pairs) performed with the STM break junction tech-
nique in liquid environment [862]. These authors found qualitative differ-
ences between the transport mechanism for GC base pairs and AT ones, see
Fig. 18.6. The interpretation of this experiment is that coherent transport
would be possible through CG-only DNA, while the AT base pairs act as
tunneling barrier over which the transport takes place via incoherent hop-
ping from site to site. The reported sequence dependence is in agreement
with theoretical predictions [838, 863, 864].
It is interesting to mention at this point a related experiment by Giese
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The hopping regime and transport through DNA molecules 549

(a) (d)

(e)
(b)

(f)
(c)

Fig. 18.6 The left panels show conductance histograms of three DNA duplexes mea-
sured with the STM break-junction technique: (a) 5′ -CGCGCGCG-3′ -thiol linker, (b)
5′ -CGCGATCGCG-3′ -thiol linker, and (c) 5′ -CGCGAATTCGCG-3′ -thiol linker. (d)
Schematic illustration of a single DNA conductance measurement. (e) Natural loga-
rithm of GCGC(AT)m GCGC conductance vs. length (total number of base pairs). The
solid line is a linear fit that reflects the exponential dependence of the conductance on
length. (f) Conductance of (GC)n vs. 1/length (in total base pairs). Reprinted with
permission from [862]. Copyright 2004 American Chemical Society.

et al. [842], where the charge transfer rate in DNA molecules was measured.
Some of these results are shown in Fig. 18.7. As compared to the experi-
ment just described, Giese et al. found a weaker length dependence of the
transfer rates when several AT base pairs were inserted between CG base
pairs. These experiments showed the existence of two different processes for
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550 Molecular Electronics: An Introduction to Theory and Experiment

Fig. 18.7 Sequence dependence of the charge transfer in DNA. Plot of log(PGGG /PG )
(PGGG /PG is proportional to the charge transfer rates) against the number n of the AT
base pairs. Each experiment was performed three times, and their relative errors are
within ±10 − 20%. The steep line corresponds to the coherent superexchange charge
transfer (tunneling). The flat line is drawn in order to make clear the weak distance de-
pendence. The arrows in the depicted DNA strands indicate the superexchange charge
transfer between the guanine radical cation G+ 22 and the GGG sequence for short dis-
tances (n = 2), or the hopping mechanism for long distances (n = 5), where - in addition
adenines act as charge carriers. For clarity, only the double strands with n = 2 and n = 5
are shown. The nucleotides in grey indicate all charge carriers. Reprinted by permission
from MacMillan Publishers Ltd: Nature [842], copyright 2001.

the hole transfer between guanines in DNA: (i) A coherent superexchange

reaction (single-step tunneling), where the bridging adenines are indirectly
affecting the transfer mechanism by mediating the electronic coupling be-
tween the guanines, and (ii) a thermally induced hopping process, where the
guanines oxidize the intervening adenine bases and directly involve them in
charge transport. The efficiency of the tunneling reaction decreases rapidly
with the number of the intervening AT base pairs, whereas the hopping
process is only slightly influenced by the number of the AT base pairs.
The discussion of these two experiments shows that one cannot talk
about a single transport mechanism in DNA. From these experiments it is
expected that CG-DNA could serve as a molecular wire. However, it has
been shown that longer DNA species with a percentage of CG pairs above
approximately 75% undergo a conformational change to the presumably
conductive quartet geometry (nicknamed G4-wires) [865]. Recent experi-
ments on G4 derivatives show an enhanced electrical polarizability, while
dsDNA oligomers appeared electrically “silent” in an equivalent experiment
[866]. This instability makes it difficult to investigate the sequence depen-
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The hopping regime and transport through DNA molecules 551

dence of the transport mechanisms for longer molecules.

On the other hand, a lot of progress has been made in the last years
concerning the relation between electronic properties and the sequence by
using transverse scanning tunneling microscopy and spectroscopy [867, 868]
or theoretical modeling [869]. Thus for instance, Shapir et al. [867] showed
that GC and AT base pairs have very distinct I-V spectra. The authors of
Ref. [868] used this difference for developing a method for determining the
sequence of a DNA strand by measuring their electronic properties.
From the theory side, it has been predicted that a junction with a single
DNA (in B-form) should exhibit a non-monotonic behavior of the electrical
response as a function of its elongation [870]. This non-monotonic behav-
ior originates from a competition between a stretching and a de-twisting
process of the helical structure. To be precise, the elongation of a helix is
predicted to reduce the angle between neighboring base pairs, which results
in an enhancement of the overlap of the conducting orbitals. Simultane-
ously, the stretching enhances the distance and thus reduces the overlap
again.
An important characteristic of the DNA molecule is its remarkable
flexibility. It can be stretched in excess of 1.7 times its B-form length.
Single-molecule stretching experiments have shown that DNA undergoes a
pronounced and abrupt structural transformation to a yet unknown struc-
ture, which is elongated by more than 50% and called S-conformation [871].
This conformation is presumably associated with rotations of specific or-
bitals along the helix axis, in turn influencing the effective orbital overlap
between neighboring base pairs. Theoretical calculations indicate that this
pronounced conformational transition has a strong impact in the conduc-
tion properties of DNA molecules [872–874].
In summary, there are at least three important factors that influence
the conduction properties of single DNA molecules:

• The environment: The presence (or the absence) of a solvent plays a

key role. This is evident in the case of the most conductive form of
DNA, namely the B-conformation, which only exists in solution. On
the other hand, the interaction with an underlying substrate may give
rise to a conformational change, which in turn can modify dramatically
the transport properties of DNA-based junctions [858, 859].
• The contacting method: As usual, the metal-molecule interface plays a
very important role. In this sense, the highest currents through DNA
have been achieved with dithiolated molecules, i.e. with covalent bonds
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552 Molecular Electronics: An Introduction to Theory and Experiment

to gold electrodes at both ends of the molecules [855, 862]. Also, rel-
atively high currents have been measured through dsDNA molecules
covalently attached to single-walled carbon nanotubes (SWNT) [875]
and through duplex DNA coupled to SWNT electrodes via amide link-
ages [876].
• The sequence: The exact base sequence determines finally the level of
the current that can flow through DNA molecules [842, 862, 864]. In
particular, a sequence rich in CG pairs is expected to exhibit a higher
current.

18.4 Exercises

18.1 Length and temperature dependence of the conductance in the

hopping regime: Show that the conductance in the incoherent model of section
18.1 is given by Eq. (18.7). Hint: In order to learn how to solve Eq. (18.3),
consider first the case in which the molecular bridge is composed of only two
sites.
18.2 Activation-like temperature dependence in the resonant tunneling
model: The goal of this exercise is to show that an exponential dependence of the
current on temperature is also possible in the coherent regime. For this purpose,
use the resonant tunneling model of section 13.2 and compute the current as a
function of ǫ0 /kB T for ΓL = ΓR = 0.005ǫ0 for several bias voltages: eV /ǫ0 =
0.1, 0.5, 1.0, 1.5, 2.0. Hint: The solution can be found in Fig. 3(a) of Ref. [471].
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Chapter 19

Beyond electrical conductance: shot

noise and thermal transport

In the previous chapters we have addressed the main transport regimes that
are realized in molecular junctions. In our discussion so far, we have fo-
cused our attention on the analysis of the electrical conductance. However,
there are many other transport properties that provide valuable informa-
tion, which often is not contained in the conductance. A paradigmatic ex-
ample is the current fluctuations or noise. Its investigation has contributed
decisively to our understanding of the transport mechanisms in a great va-
riety of mesoscopic and nanoscale devices [150]. On the other hand, the
charge transport is not the only important aspect in the context of conduc-
tion in molecular junctions. Thermal transport is also a key issue in the field
of molecular electronics from a fundamental as well as a from a practical
viewpoint. Molecular-scale contacts provide a new territory to study heat
conduction in regimes never explored before and, issues like heating will
have to be faced and understood, if molecular electronics wants to become
a viable technology. Obviously, the study of thermoelectric phenomena
in molecular junctions, resulting from the interplay between electrical and
thermal transport, can also give a new insight into the physics of these
nanocircuits.
For these reasons, we shall put aside the electrical conductance for a
while, and in this chapter we shall concentrate on the discussion of other
transport properties. To be precise, in section 19.1 we shall discuss the
basic physics of noise in molecular junctions and describe the first noise
experiments in this field. Then, we shall turn our attention to thermal
transport and in section 19.2 we shall present a detailed discussion of heat-
ing and heat conduction in molecular wires. Finally, section 19.3 is devoted
to the analysis of the thermopower, which is becoming a vital source of
novel information on molecular transport junctions.

553
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554 Molecular Electronics: An Introduction to Theory and Experiment

As in previous chapters, we shall present here both a discussion of the

basic concepts related to these transport properties as well as a review of
the work reported on this subject in recent years. In any case, we shall
concentrate on the analysis of aspects that have been already investigated
experimentally or that are likely to be investigated in the near future. Let
us also say that in some cases we have also included a brief description
of the corresponding phenomena in metallic atomic contacts, since these
systems often paved the way for a later analysis in the context of molecular
junctions.

19.1 Shot noise in atomic and molecular junctions

The electrical current through any conductor exhibits temporal fluctuations

(or noise). As it was already pointed out by Schottky in 1918 [877], when
all sources of spurious noise are eliminated, there remain two types of noise
in the electrical current, namely the thermal noise and the shot noise.
The thermal noise, which is also known as Johnson-Nyquist noise (after
the experimentalist [878] and the theorist [879] who investigated it), is
due to the thermal motion of the electrons and occurs in any conductor.
The nonequilibrium fluctuations known as shot noise are caused by the
discreteness of the charge of the carriers of the electrical current. We have
discussed this transport property within the scattering formalism in section
4.7 and for further details we recommend to the reader the excellent reviews
of Refs. [150, 155].
Noise is characterized by its spectral density or power spectrum P (ω),
which is the Fourier transform at frequency ω of the current-current corre-
lation function,
Z ∞
P (ω) = 2 dt eiωt h∆I(t + t0 )∆I(t0 )i. (19.1)
−∞
Here ∆I(t) denotes the time-dependent fluctuations in the current at a
given voltage V and temperature T. The brackets h· · · i indicate an ensem-
ble average. Both thermal and shot noise have a white power spectrum, i.e.
the noise power does not depend on ω over a very wide frequency range.
Thermal noise (V = 0, T 6= 0) is directly related to the conductance G by
the fluctuation-dissipation theorem [880],
P = 4kB T G, (19.2)
as long as ~ω ≪ kB T . Therefore, the thermal noise of a conductor does
not give any new information as compared to the conductance.
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Beyond electrical conductance: shot noise and thermal transport 555

Shot noise (V 6= 0, T = 0) is more interesting, because it gives infor-

mation on the temporal correlation of the electrons, which is not contained
in the conductance. In devices such as tunnel junctions, Schottky bar-
rier diodes, p-n junctions, and thermionic vacuum diodes, the electrons
are transmitted randomly and independently of each other. The transfer of
electrons can be described by the Poisson statistics, which is used to analyze
events that are uncorrelated in time. For these devices the zero-frequency
shot noise has its maximum value
P = 2eI ≡ PPoisson , (19.3)
which is proportional to the time-averaged current I. Correlations suppress
the low-frequency shot noise below PPoisson . One source of correlations,
operative even for non-interacting electrons, is the Pauli principle, which
forbids multiple occupancy of the same single-particle state. A typical ex-
ample is a ballistic point contact in a metal, where P = 0 because the
stream of electrons is completely correlated by the Pauli principle in the
absence of scattering. In single-channel quantum point contacts, and in the
absence of inelastic scattering, shot noise is predicted to be suppressed by
a factor proportional to τ (1 − τ ), where τ is the transmission probability of
the conduction channel1 [881–883]. This quantum suppression was first ob-
served in point contact devices in a two-dimensional electron gas [884, 97].
For a general multichannel contact in the limit of very low temperatures
the shot noise power is predicted to be [883]
X
P = 2eV G0 τn (1 − τn ), (19.4)
n
2
where G0 = 2e /h is the quantum of conductance. For arbitrary tempera-
ture and voltage the noise is a mixture of thermal noise and shot noise and,
assuming that the transmission coefficients do not depend on energy, it is
given by Eq. (4.75), which we reproduce here2
" µ ¶X #
X
2 eV
P = 2G0 2kB T τn + eV coth τn (1 − τn ) . (19.5)
n
2kB T n

Since the shot noise depends on the sum over the second power of the
transmission coefficients, this quantity is independent of the conductance,
P
G = G0 n τn , and the simultaneous measurement of these two quantities
1 Throughout this chapter we shall denote the transmission as τ in order to avoid con-
fusions with the temperature.
2 Here, we have taken into account the spin degeneracy that will be assumed in our

discussion throughout this section.

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556 Molecular Electronics: An Introduction to Theory and Experiment

1.0

Transmission
0.8

0.3 0.6
0.4
x
Excess noise (2eI)

0.2
0.0
0 1 2 3 4
2
Conductance (2e/h)
0.2

0.1

20%
10%
x=0% 5%
0.0
1 2 3 4
2
Conductance (2e /h)

Fig. 19.1 Noise measurements on Au atomic contacts using the MCBJ technique. The
symbols correspond to the measured excess noise values for 27 contacts at 4.2 K with
a bias current of 0.9 µA. The different lines show the calculations using Eq. (19.5) in
the case of one single partially transmitted channel (full curve) and for various amounts
of contributions of other modes according to the model described in the inset (dashed
curves). In the limit of zero conductance, these curves all converge to full shot noise,
i.e. 2.9 10−25 A2 /Hz. Inset: transmission of modes in the case of x=10% contribution
from neighboring modes. Reprinted with permission from [885]. Copyright 1999 by the
American Physical Society.

should give information about the transmission coefficients of the contact.

The relevant quantity is conveniently expressed in terms of the Fano factor
F , which is the ratio of the shot noise to the noise that the same current
would produce in the classical Schottky limit,
P
P τn (1 − τn )
F = = nP . (19.6)
2eI n τn

Shot noise in atomic-scale contacts was first measured by van den Brom
and J.M. van Ruitenbeek using the MCBJ technique [885]. The measure-
ments were conducted at low temperatures to reduce the thermal noise.
However, in these experiments the noise level of the pre-amplifiers in gen-
eral exceeds the shot noise to be measured. Using two sets of pre-amplifiers
in parallel and measuring the cross-correlation, this undesired noise is re-
duced. By subtracting the zero-bias thermal noise from the current-biased
noise measurements, the pre-amplifier noise, present in both, is further
eliminated. For currents up to 1 µA the shot noise level was found to
have the expected linear dependence on current. For further details on the
measurement technique, we refer to [885].
In Fig. 19.1 we show the results of Ref. [885] for the noise of gold atomic
contacts as a function of the conductance of the junctions. The measured
shot noise is given relative to the classical shot noise value 2eI. All data are
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Beyond electrical conductance: shot noise and thermal transport 557

2.5
0.006 (a) (b)
2.0
0.005
〈S I〉 (pA /Hz)

0.004 1.5

T (K)
2

*
0.003
1.0
0.002
0.5
0.001

0.000 0.0
-2 -1 0 1 2 -3 -2 -1 0 1 2 3
eV/2k B (K) eV/2k B (K)

Fig. 19.2 Noise measurements on Al atomic contacts using microfabricated MCBJs.

Symbols: measured average current noise power density hSI i and noise temperature T ∗ ,
defined as T ∗ = SI /4kB G, as a function of reduced voltage, for a contact in the normal
state at three different temperatures (from bottom to top: 20, 428, 765 mK). The solid
lines are the predictions of Eq. (19.5) for the set of transmissions {0.21,0.20,0.20} mea-
sured independently from the I-V in the superconducting state. (b) Symbols: measured
effective noise temperature T ∗ versus reduced voltage for four different contacts in the
normal state at T = 20 mK. The solid lines are predictions of Eq. (19.5) for the cor-
responding set of transmissions (from top to bottom: {0.21,0.20,0.20}, {0.40,0.27,0.03},
{0.68,0.25,0.22}, {0.996,0.26}. The dashed line is the Poisson limit. Reprinted with
permission from [98]. Copyright 2001 by the American Physical Society.

strongly suppressed compared to the full shot noise value, with minima close
to 1 and 2 times the conductance quantum. For contacts with conductance
below 1 G0 the data are consistent with a single conduction channel having
a transmission probability τ = G/G0 , as expected for this monovalent
metal. For larger contacts there is a tendency for the channels to open
one-by-one, but admixture of additional channels grows rapidly. There
is a very strong suppression, down to F = 0.02, for G = 1 G0 , which
unambiguously shows that the current is carried dominantly by a single
channel. It needs to be stressed that this holds for gold contacts. There is a
fundamental distinction between this monovalent metal and the multivalent
metal aluminum, which shows no systematic suppression of the shot noise
at multiples of the conductance quantum, and the Fano factors lie between
about 0.3 and 0.6 for G close to G0 [886].
Shot noise measurements by Cron et al. [98] have provided a very strin-
gent experimental test of the multichannel character of the electrical con-
duction in Al atomic contacts. In these experiments the set of transmis-
sions τn were first determined independently by the technique of fitting the
subgap structure in the superconducting state, discussed in section 11.4.
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558 Molecular Electronics: An Introduction to Theory and Experiment

The knowledge of the transmission coefficients allows a direct quantitative

comparison of the experimental results on the shot noise with the the-
oretical predictions of Eq. (19.5). The experiments were done using Al
nanofabricated break junctions which exhibit a large mechanical stability.
The superconducting I-V curves for the smallest contacts were measured
below 1 K and then a magnetic field of 50 mT was applied in order to
switch into the normal state. The measured voltage dependence of the in-
trinsic current noise is shown in Fig. 19.2(a) for a typical contact in the
normal state at three different temperatures, together with the predictions
of Eq. (19.5), using the set of transmission coefficients measured indepen-
dently. The noise measured at the lowest temperature for four contacts
having different sets of transmission coefficients is shown in Fig. 19.2(b),
together with the predictions of the theory. This excellent agreement be-
tween theory and experiments provides an unambiguous demonstration of
the presence of several conduction channels in the smallest Al contacts and
serves as a test of the accuracy that can be obtained in the determination
of the τ ’s from the subgap structure in the superconducting I-V curve.
In the last years, van Ruitenbeek’s group has performed shot noise mea-
surements in highly conductive molecular junctions to determine the chan-
nel decomposition of the conductance. A first example was reported by
Djukic and van Ruitenbeek for the hydrogen molecule bridge [568]. In this
case, a Pt-H2 junction was adjusted so as to have a clear vibrational mode
signal, and the shot noise signal was measured for the same junction. An
example of this measurement is shown in Fig. 19.3. Although shot noise
generally does not allow determining the full set of transmission values,
one can obtain information from the property that the noise increases the
more channels are partially transmitted. The result of Fig. 19.3 for a junc-
tion with conductance G = 1.021 G0 was fitted using two channels with
transmissions τ1 = 1.000 and τ2 = 0.021. In principle, the conductance in
this example can be redistributed over more than just two channels. How-
ever, when the transmission τ1 = 1.000 is broken up into more channels this
strongly increases the Fano factor. Thus, the only freedom is to redistribute
the transmission τ2 = 0.021 over two or more channels, that will all have a
very small contribution. Therefore, one can conclude that the conductance
is largely dominated by a single channel with nearly perfect transmission,
which was found to be a very robust result of these measurements. As we
explained in section 14.1.3, this result was decisive to discriminate between
the different possible geometries for the hydrogen bridge which had been
proposed theoretically.
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Beyond electrical conductance: shot noise and thermal transport 559

Fig. 19.3 Shot noise measurements in Pt-D2 break junctions. The left panel shows the
point-contact spectroscopy (PCS) signal for this junction, with a clear vibrational mode
at 76 meV. The right panel shows the excess noise (the white noise in the current above
the thermal noise) as a function of the current. The noise is strongly suppressed below
the full Schottky noise for a tunnel junction. After each measurement of noise at a given
current, the PCS was measured again to verify that the contact had not changed. The
total conductance for this junction is G = 1.021 G0 , and the shot noise can be fitted with
two channels, τ1 = 1.000, and τ2 = 0.021, giving a Fano factor of F = 0.020. Adapted
with permission from [568]. Copyright 2006 American Chemical Society.

More recently, shot noise measurements have also been used to charac-
terize Pt-H2 O-Pt junctions [735] and Pt-benzene-Pt junctions [473]. In the
former case, the noise results indicated that for conductance below 1 G0
there are typically two conduction channels, although one clearly domi-
nates the transport. These results were very important to understand the
crossover between PCS and IETS and to test the so-called 1/2-rule (see
section 16.4). In the case of Pt-benzene junctions (see section 14.1.4),
the analysis of the shot noise results showed that for conductances around
1 G0 (and also well below) several channels contribute significantly to the
transport, while when the conductance is reduced to 0.2 G0 , the number of
channels is eventually reduced to one. As opposed to Pt-H2 O-Pt junctions,
in this case there is no dominant transmission channel when more than a
single channel exists. It was shown theoretically in same work [473] that
the number of channels is roughly determined by the number or carbon
atoms directly coupled to the Pt electrodes.
So far the shot noise measurements in molecular junctions have been
used to extract the channel transmissions in highly conductive junctions,
where the transport is supposed to be coherent. Notice that this applica-
tion is restricted to junctions with a high conductance, let us say above
0.1 G0 . Below that, the quadratic term in the transmission coefficients is
negligible and the shot noise becomes proportional to the conductance (i.e.
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560 Molecular Electronics: An Introduction to Theory and Experiment

linear in the transmission coefficients). Anyway, the shot noise can provide
very important information also in other transport regimes. For instance,
shot noise measurements in the Coulomb blockade regime [887, 888] or in
the Kondo regime [889] have been only reported very recently in the con-
texts of carbon nanotubes and semiconductor quantum dots. In this sense,
weakly coupled molecular junctions (molecular transistors) can be an ideal
playground to further explore the noise in these transport regimes.
On the other hand, the vibrational effects discussed in previous chap-
ters can be further investigated with the help of the noise. For instance,
it has been predicted that the Franck-Condon blockade (see section 17.3.1)
is characterized by remarkably large Fano factors (102 -103 for realistic pa-
rameters), which arise due to avalanche-like transport of electrons [801].
The vibrationally-induced inelastic effects on noise properties of molec-
ular junctions in different transport regimes have been studied using NEGF
techniques by Zhu and Balatsky [784] and by Galperin and coworkers [890].3
Very recently, several theoretical groups have discussed the noise induced
by vibrations in the limit of weak electron-phonon coupling [891–893]. One
of the central issues of these papers was the discussion of the sign of the
inelastic noise as a function of the transmission, which is related to our dis-
cussion of the sign of the inelastic conductance in this regime (see section
16.4). The predictions of these papers could be in principle tested in the
type of experiments discussed above.

19.2 Heating and heat conduction

As mentioned in the introduction, so far we have only discussed the trans-

port of electrical charge in molecular junctions, but heat transport is also
very important for several reasons. From a practical point of view, the
understanding of heat generation in molecular contacts is crucial. When
an electrical current flows through a junction, there is an energy trans-
fer (Joule heating) from the electrons to the vibrations that might cause
a large temperature increase that in turn can affect the stability and in-
tegrity of molecular junctions. From a more fundamental point of view,
it is very exciting to investigate how the heat is conducted through the
tiniest circuits ever built, namely atomic-scale junctions. These structures
3 The approach used in these two references has been criticized in Ref. [893], where it is

claimed that it misses vertex corrections even at the lowest order in the electron-phonon
interaction.
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Beyond electrical conductance: shot noise and thermal transport 561

have dimensions that are much smaller than the inelastic mean free path
for phonons, even at room temperature, and thus they offer the possibility
to study phonon transport (and their contribution to thermal conduction)
in a very special regime.
Heat generation and heat conduction are intimately connected. Indeed,
heat conduction is an essential ingredient in the balance of the processes
that determines local heat generation. For this reason, we have chosen to
organize this section in the following way. After some general comments
about the problem of describing heating and heat conduction in molec-
ular junctions, we shall briefly review the work done so far on thermal
conductance. Then, we shall discuss the issue of heat generation in molec-
ular junctions and, in particular, we shall describe the main experiments
reported to date on this topic.
The subject of thermal properties of molecular junctions is presently
dominated by the theory. The experiments are rather scarce due to the
difficulties of measuring thermal transport at the nanoscale (for a review
on this subject see Ref. [894]). Although it is a very interesting subject,
we shall not discuss here in depth the theoretical techniques to describe
heat transport in molecular junctions, and we shall merely point out the
main ideas and challenges. For a more detailed discussion on the theory,
see section 9 of Ref. [695].

19.2.1 General considerations

In general, both electrons and phonons contribute to the thermal transport
properties. In insulators heat is conducted by atomic vibrations, while in
metals electrons are the dominant carriers, at least at low temperatures.
In molecular junctions, both types of carriers exist and mutually interact.
Therefore, a complete description of the thermal transport in these systems
requires to take into account the energy transport due to both electrons
and phonons, as well as the energy exchange between them due to the
electron-phonon interaction. This problem is quite complicated and so far
no realistic calculations have been performed taking into account all the
ingredients mentioned above.
Even though practical applications can be difficult, a unified description
of both heat generation and heat transport is in principle possible within the
framework of the nonequilibrium Green’s function formalism (NEGF). This
formalism was first applied to thermal transport by Datta and coworkers
[895, 896] and it has been extended by several groups to treat different
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562 Molecular Electronics: An Introduction to Theory and Experiment

aspects of this problem [897–900].

The analysis of heat transport can be greatly simplified, in particu-
lar, when electrons and phonons can be considered separately (e.g. when
the electron-phonon interaction is negligible). For instance, in highly con-
ductive molecular junctions the low-temperature thermal conductance is
expected to be dominated by electrons. Assuming that the transport is co-
herent, the contribution of electrons can be computed within the scattering
formalism, as we have shown in section 4.8. In this case, the heat cur-
rent is simply determined by the (electronic) transmission coefficient and
it is given by Eq. (4.84). In this sense, it can be computed from the usual
methods for coherent transport like DFT-based ones.
In the case of junctions with a low electrical conductance, the dominant
contribution to thermal transport comes from phonons (or vibrations). Ig-
noring anharmonic effects and the electron-phonon interaction, the heat
current can be expressed in terms of a Landauer-like formula [901–903],
where the phonon transmission can be determined using Green’s function
techniques analogous to those of the corresponding electronic problem, see
e.g. Ref. [899].

19.2.2 Thermal conductance

We consider now the heat conduction through a molecular wire suspended
between two reservoirs characterized by different temperatures. In particu-
lar, we shall focus here on situations where the heat transfer is dominated by
phonons. The theoretical analysis of the transport of phonons and the cor-
responding thermal transport goes back to Peierls’ early work [904]. In the
recent years, it has become clear that the thermal properties of nanowires
can be very different from the corresponding bulk properties. For example,
Rego and Kirczenow [901] have shown theoretically that in the low temper-
ature ballistic regime, the phonon thermal conductance of one-dimensional
(1D) quantum wires is quantized in units of π 2 kB 2
T /3h, where T is the
temperature. This prediction was confirmed experimentally by Schwab et
al. [905] in a nanofabricated 1D structure, which behaves essentially like a
phonon waveguide.
An aspect that has attracted a lot of attention in the last decades is the
validity of the macroscopic Fourier law of heat conduction in 1D systems
[906–912]. The Fourier law is a relationship between the heat current J per
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Beyond electrical conductance: shot noise and thermal transport 563

unit area A and the temperature gradient ∇T

J/A = −K̃∇T, (19.7)

where A is the cross-section area normal to the direction of heat propagation

and K̃ is the thermal conductivity (the thermal conductance K is defined
as K = J/∇T ). In spite of all the work on this subject, there is yet
no convincing and conclusive result about the validity of this law in 1D
systems.
Another aspect that has been the subject of recent discussions is the
possibility of having an asymmetry in the directionality of heat transfer.
Several authors have proposed both classical and quantum-mechanical mod-
els that exhibit heat rectification [913–918]. In these models, rectification
is usually associated with a non-linear (anharmonic) response.
From the experimental point of view, remarkable progress has been
made in the last decade in nanoscale thermometry, and measurements on
the scale of the mean free path of phonons and electrons are now possible.
Using scanning thermal microscopy methods one can obtain the spatial tem-
perature distribution of the sample surface, study local thermal properties
of materials, and perform calorimetry at nanometric scale [894, 919, 920].
These advances have allowed, for instance, studying the thermal transport
in single carbon nanotubes (see e.g. [921] and references therein) and es-
tablishing a quantitative comparison with the theory (see e.g. [922] and
references therein).
Experimental work on thermal transport in molecular junctions is how-
ever very limited. The first thermal conductance measurements that we
are aware of were reported by Wang et al. [923]. These authors studied
solid-solid junctions with an interfacial self-assembled monolayer (SAM). To
be precise, Au-SAM-GaAs junctions were made using alkanedithiol SAMs
and fabricated by nanotransfer printing. Measurements of thermal conduc-
tance were very robust and no thermal conductance dependence on alkane
chain length was observed. The thermal conductances using octanedithiol,
nonanedithiol, and decanedithiol SAMs at room temperature were found
to be 27.6 ± 2.9, 28.2 ± 1.8, and 25.6 ± 2.4 MW m−2 K−1 , respectively.
The thermal conductance of an alkanedithiol SAM anchored to a gold
substrate was studied by ultrafast heating of the gold with a femtosecond
laser pulse in Ref. [924]. It was found that when the heat reached the methyl
groups at the chain ends, a nonlinear coherent vibrational spectroscopy
technique detected the resulting thermally induced disorder. The flow of
heat into the chains was limited by the interface conductance. The leading
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564 Molecular Electronics: An Introduction to Theory and Experiment

-11
x 10
20

15
K (W/K)

10

0
5 10 15 20
N
Fig. 19.4 Theoretical results for the heat transport coefficient (heat flux per unit T
difference between hot and cold bath) displayed as a function of alkane bridge length,
for a particular model of molecule-heat bath coupling at 50 K (full line), 300 K (dotted
line) and 1000 K (dashed line). Reprinted with permission from [903]. Copyright 2003,
American Institute of Physics.

edge of the heat burst traveled ballistically along the chains at a velocity
of 1 kilometer per second. The molecular conductance per chain was 50
pW/K.
The thermal conductance of alkane-based junctions was indeed ad-
dressed theoretically by Segal et al. [903] a few years before the realization
of the experiments mentioned above. These authors computed the phonon
contribution to the heat current, which should be the dominant one in these
low transmissive junctions. To be precise, they computed the heat flux for
a harmonic molecule characterized by a set of normal modes and coupled
through its end atoms to harmonic heat reservoirs. They have also per-
formed classical mechanics simulations in order to assess the role played by
anharmonicity. The general conclusions of this work are: (i) At room tem-
perature and below, molecular anharmonicity is not an important factor
in the heat transport properties of alkanes of length up to several tens of
carbon atoms. (ii) At room temperature, the efficiency of heat transport by
alkane chains decreases with chain size above 3-4 carbons, then saturates
and becomes length independent for moderate sizes of up to a few tens
of carbon atoms (this prediction agrees with the observations of Ref. [923]
mentioned above). (iii) At low temperature, the heat transport efficiency
increases with chain length. This is a quantum effect: at low temperatures
only low frequency modes can be populated and contribute to phonon trans-
port, however such modes are not supported by short molecules and become
available only in longer ones. In Fig. 19.4 we reproduce the results for the
thermal conductance of Segal et al. [903] that illustrate these conclusions.
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Beyond electrical conductance: shot noise and thermal transport 565

To conclude this discussion, let us simply say that the investigation of

heat transport in molecular junctions is still in its infancy both theoretically
and experimentally and this is one of the issues in which a lot of progress
is expected (and desired) in the next years.

19.2.3 Heating and junction temperature

In the context of thermal properties of molecular junctions, heating or heat
generation is the most studied aspect from the experimental point of view.
When an electrical current is driven through an atomic or molecular junc-
tion, there is a continuous energy transfer from the conduction electrons to
the vibrational degrees of freedom that, loosely speaking, tends to increase
the local temperature inside junction. This heating effect is partially allevi-
ated by the conduction out of the junction via phonon thermal conduction.
The balance between these two mechanisms determines the excess energy
that is deposited in the phonon subsystem.
This energy transfer is usually described in terms of an effective local
temperature. This is, of course, questionable since the system is out of
equilibrium and the phonon distribution can differ significantly from that
described by the Bose function. From a theoretical point of view, the ef-
fective temperature is sometimes defined by forcing the phonon occupation
to adopt the form of the Bose function with an effective temperature, Teff .
Other definitions have been introduced and for a detailed discussion of this
issue we refer to Refs. [894, 899].
From the experimental point of view, indirect information about the
local temperature is obtained by measuring temperature-dependent prop-
erties like the switching rate between two different configurations in atomic
sized contacts [925, 926], the fracture rate in atomic chains [927], the force
required to break a molecule-electrode bond [928] or the distance over which
molecular junctions can be stretched before breakdown [929]. The exper-
imental data on the current-induced local heating are typically analyzed
with the help of the theory of Todorov and coworkers [930, 267, 931]. This
theory provides a simple estimate for the voltage dependence of the local
effective temperature. In particular, it predicts that the temperature in the
center of a general ballistic nanoscale junction of length L at a voltage V
is given by

Tef f = (T04 + TV4 )1/4 , (19.8)

p
where T0 is the ambient temperature and TV = γ L|V |, where γ is a
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566 Molecular Electronics: An Introduction to Theory and Experiment

material-dependent parameter. For a typical metal contact, γ = 60 K

V−1/2 nm−1/2 . This means that with L ≈ 2 nm and V = 1.5 V, TV ≈ 100
K. Three factors are taken into account in this estimate: (i) heating by
the electrons due to creation of phonons in the junction, (ii) cooling by the
electrons due to absorption of phonons, and (iii) cooling by the thermal
transport of energy away from the contact into the metal reservoirs.
Eq. (19.8) was experimentally tested by Smit et al. [927]. These authors
investigated the breaking mechanism of Au and Pt atomic chains as a func-
tion of the bias voltage. The chain breaking is a thermally activated process
and the fracture rate contains information about the bias-dependent local
temperature. An analysis of the data showed a reasonable agreement with
the predictions based on Eq. (19.8). From this analysis, the authors could
estimate the effective (lattice) temperature inside the atomic wire. It rises
in proportion to the square root of the bias voltage for sufficiently high bias,
and for a monatomic gold chain of length L = 1 nm at V = 1 V it reaches
a temperature of 60 K, which is well above the bath temperature of 4.2 K
in the experiments.
The derivation of Eq. (19.8) assumes a bulk T 3 law for heat capacity
of the contact, which is an approximation only valid for temperatures well
below the Debye temperature. Therefore, it should not be surprising to
find deviations from the square-root dependence of Eq. (19.8) at elevated
temperatures. In this respect, Tsutsui et al. [926] found the effective tem-
perature √of Zn atomic-sized contacts at 77 K rises more rapidly with bias
than the V dependence of Eq. (19.8).
In the case of molecular junctions, Tao’s group measured the local ef-
fective temperature. In this case, the authors studied the force required
to break Au-octanedithiol-Au junctions under finite bias. The breakdown
process is thermally activated, which can be used to extract the effective
temperature. The data could be roughly fitted with Eq. (19.8). It was
found that at a bias voltage of 1 V, the temperature of the junction is
raised ∼ 30 K above the ambient room temperature. Above this bias, the
molecular junctions become increasingly unstable.
In another work of Tao’s group, the effective temperature of single-
molecule (n-alkanedithiol) junctions due to current-induced local heating
was measured as a function of molecular length and applied bias voltage. In
this case the method was based on analyzing the average stretching length
over which a molecular junction can be stretched before breakdown, using
the STM break junction approach. By measuring the stretching length
as a function of stretching rate and temperature, the authors showed that
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Beyond electrical conductance: shot noise and thermal transport 567

the breakdown of the molecular junctions is thermally activated and the

dependence of the stretching length on temperature was used to extract
the effective temperature of single-molecule junctions. They reported the
following two notable findings. First, at a given bias, the local ionic heating
increases with decreasing molecular length, in agreement with the theoret-
ical predictions of Ref. [766] (see discussion below). Second, for a given
molecule, the effective local temperature first increases with bias, and then
decreases after reaching a maximum value at ∼ 0.8 V. This is in agreement
with a transport theory based on a hydrodynamic approach [933], which
predicts that effective cooling of the temperature at high biases can occur
due to electron-electron interaction with consequent local electron heating
at the junction.
In attempts to go beyond the simple estimate of Eq. (19.8), Di Ventra
and co-workers have reported quantitative calculations of the temperature
rise in realistic models of atomic and molecular junctions [766, 932, 934].
These calculations are based on a microscopic description of the heat gen-
eration, while the heat conduction is estimated via a simplified approach.
The following important observations based on these calculations have been
made: (i) For the same voltage, the temperature rise in a benzenedithiol
junction is considerably smaller than that of a gold wire of similar size
because of the larger conduction (therefore higher current) in the latter.
In absolute terms, the temperature rise is predicted to be about 15 and
130 K above ambient temperatures at a voltage bias of ∼ 1 V [932]. (ii)
In dithiolate alkane chains, the estimated temperature rise is a few tens
degrees at 0.5 V and depends on the chain length, see Fig. 19.5. The tem-
perature rise is smaller in longer chains characterized by smaller electrical
conduction [766], which is in agreement with the experimental results of
Ref. [929]. In this case, decreasing conduction with molecular lengths over-
shadows the less efficient heat dissipation in these systems. (iii) In contrast
to alkanes, in Al wires the temperature rise in current carrying wires is
more pronounced for longer chains [934]. In these good conductors the
balance between the length effects on conduction and heat dissipation is
tipped the opposite way from their molecular counterparts, because length
dependence of conduction is relatively weak.
The main difficulty in the calculation of the effective temperature is
the description of the energy transfer between the local vibrations and the
phonons in the reservoirs. Some progress has been made in the last years,
see e.g. Refs. [777, 935, 936], but the description of the phonon transport in
molecular junctions has not yet reached the level of sophistication achieved
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568 Molecular Electronics: An Introduction to Theory and Experiment

60
50 C8
40 C10
T (K)

30 C12
20
10 C18
0
0 100 200 300 400 500
Bias (mV)
Fig. 19.5 Estimated junction temperature as a function of bias in alkane-dithiol junc-
tions of various chain lengths. Reprinted with permission from [766]. Copyright 2005
American Chemical Society.

for the electron transport problem.

Recently, new experimental methods have been introduced to study the
mechanisms of heating and heat dissipation induced by the flow of current
across a single molecule. For instance, in the context of STM experiments,
Schulze et al. [937] have used a method based on detecting the maximum
power that one molecule can sustain. In particular, these authors used
a low temperature STM to control the flow of electrons through a single
C60 molecule at an increasing rate until the molecule decomposes. By
comparing the power applied for decomposition of the molecule (Pdec ) in
the tunneling regime and in contact with the STM tip, they found that
it depends significantly on two factors: (i) Pdec decreases when molecular
resonances participate in the transport, evidencing that they enhance the
heating; (ii) Pdec increases as the molecule is contacted to the source and
drain electrodes, revealing the heat dissipation by phonon coupling to the
leads. A good contact between the single-molecule device and the leads
is hence an important requirement for its operation under large current
densities.
Probably the most direct method to investigate local heating in molec-
ular junctions has been reported by Ioffe et al. [938]. These authors have
shown that the effective temperature of current-carrying junctions can be
monitored with surface-enhanced Raman spectroscopy (SERS) that in-
volves measuring both the Stokes and anti-Stokes components of the Ra-
man scattering. The ratio of these two components for each Raman active
vibrational mode gives direct information about its steady-state nonequi-
librium population. This ratio can be translated into a mode-specific effec-
tive temperature [939]. In Ref. [938], Ag-SAMBPDT-Ag junctions were
studied, where SAMBPDT stands for self-assembled monolayer of 4,4′ -
biphenyldithiol. In Fig. 19.6 we reproduce the results of this work for
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Beyond electrical conductance: shot noise and thermal transport 569

(a) (b)
Temperature (K)

Temperature (K)
Bias voltage (V) Bias voltage (V)

Fig. 19.6 Measurements using surface enhanced Raman spectroscopy of the mode-
specific effective temperature [Teff (ν)] as a function of bias for two representative Ag-
SAMBPDT-Ag junctions. (a) Plot of Teff (ν) as a function of bias voltage for a mode
with 1,585 cm−1 (triangles) and 1,083 cm−1 (squares) modes (532 nm laser). (b) Plot of
Teff (ν) as a function of bias voltage for a mode with 1,585 cm−1 (triangles), 1,280 cm−1
(circles) and 1,083 cm−1 (squares) modes (671 nm laser). Reprinted by permission from
Macmillan Publishers Ltd: Nature Nanotechnology [938], copyright 2008.

the effective temperature of several modes as a function of the voltage for

two representative junctions. As one can see, the apparent dependence of
the effective temperature on the applied bias reveals two types of behavior:
(i) Between 0 and ∼ |0.2 V| in both polarities an apparent cooling process
is observed and (ii) at bias values higher than |0.2 V|, heating of the vibra-
tional modes takes place. As explained by the authors, these experimental
results reveal a rich heating/cooling behavior that is inexplicable using ex-
isting models. The calculations of the bias dependence of heat dissipation
in molecular junctions described above do not include the additional in-
termolecular dissipation channel prevailing in these monolayer junctions.
The explanation of these experimental results constitutes at the moment
an interesting open problem.

19.3 Thermoelectricity in molecular junctions

A property closely related to heat transport, namely the thermopower, has

recently received considerable attention and it deserves a separate discus-
sion. The thermopower (also called the Seebeck coefficient) of a material
is a measure of the magnitude of an induced thermoelectric voltage in re-
sponse to a temperature difference across that material.4 Classically, an
4A brief discussion of thermoelectric phenomena in nanocontacts, including the Seebeck
effect, can be found in section 4.8.
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570 Molecular Electronics: An Introduction to Theory and Experiment

applied temperature difference causes charged carriers in the material to

diffuse from the hot side to the cold side, similar to a gas that expands
when heated.
Mobile charged carriers migrating to the cold side leave behind their
oppositely charged and immobile nuclei at the hot side thus giving rise to
a thermoelectric voltage.5 Since a separation of charges also creates an
electric field, the build-up of charged carriers on the cold side eventually
ceases at some maximum value since there exists an equal amount of charge
carriers drifting back to the hot side as a result of the electric field at
equilibrium.
The thermopower of a (bulk) material, represented as S, depends on the
material temperature and crystal structure. Typically, metals have small
thermopower because most have half-filled bands. Electrons and holes both
contribute to the induced thermoelectric voltage thus canceling each other’s
contribution to that voltage and making it small. In contrast, semiconduc-
tors can be doped with an excess amount of electrons or holes and thus
can have large negative (for n-type materials) or positive values (for p-type
materials) of the thermopower depending on the charge of the excess carri-
ers. The sign of the thermopower can thus determine which charge carriers
dominate the electric transport in both metals and semiconductors. This is
one of the key ideas that makes the thermopower interesting for molecular
electronics.
If the temperature difference ∆T between the terminals of a junction
(or the two ends of a material) is small, then the thermopower of a material
is conventionally defined as6
∆V
S=− , (19.9)
∆T
where ∆V is the thermoelectric voltage seen at the terminals. In general,
there are two main contributions to the thermopower, namely an electronic
one and the contribution of phonons, the so-called phonon drag7 [940]. It
has been argued that for point contacts (and in general for nanoconstric-
5 Thermoelectric refers to the fact that the voltage is created by a temperature differ-

ence.
6 Strictly speaking, this expression is only approximate. The numerator should be the

difference in electrochemical potential divided by −e, not the electric potential, see
Eq. (4.79). However, the chemical potential is often relatively constant as a function of
temperature, so using electric potential alone is in these cases a very good approximation.
7 Any thermal gradient gives rise to the transport of heat by the phonons, while an

electric current, though carried by the electrons, cannot fail to transfer some of its
momentum to the lattice vibration, and drag them along with it.
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Beyond electrical conductance: shot noise and thermal transport 571

tions), the phonon drag contribution to the thermopower becomes neg-

ligible [941], which simplifies enormously the theoretical analysis of this
property. As we have shown in section 4.8, if the transport is assumed to
be phase-coherent (no inelastic scattering), the electronic contribution to
the thermopower can be expressed in terms of the zero-bias transmission
function τ (E) as [160]
R∞
1 −∞ (E − µ) τ (E) [∂f (E, T )/∂E] dE
S= R∞ , (19.10)
eT −∞
τ (E) [∂f (E, T )/∂E] dE
−1
where f (E, T ) = {exp [(E − µ) /kB T ] + 1} is the Fermi function and µ
the chemical potential with µ ≈ EF . From the numerator of Eq. (19.10), it
is evident that a non-vanishing thermopower requires a certain electron-hole
asymmetry in the transmission function. This asymmetry also determines
the sign of this transport property.
At low temperatures, the leading-order term in the Sommerfeld expan-
sion for the thermopower yields
π 2 kB
2
T τ ′ (EF )
S=− , (19.11)
3e τ (EF )
where the prime denotes derivative with respect to energy. Let us remind
that the linear conductance in this limit is given by G = G0 τ (EF ).
As the in the case of shot noise, experiments in metallic atomic-sized
contacts paved the way for the analysis of thermopower in molecular junc-
tions. In 1999 Ludoph and van Ruitenbeek reported the first thermopower
measurements in gold atomic contacts [942]. The principle of the measure-
ment is illustrated in the left panel of Fig. 19.7. By applying a constant
temperature difference over the contacts, the thermally induced potential
could be measured simultaneously with the conductance. In this experi-
ment, large thermopower values were obtained, which jump to new values
simultaneously with the jumps in the conductance. The values are ran-
domly distributed around zero with a roughly bell-shaped distribution, as
one can see in the right panel of Fig. 19.7. Negative values of the ther-
mopower are not expected in simple adiabatic models for point contacts
[161, 163, 943]. Ludoph and van Ruitenbeek proposed a convincing inter-
pretation in terms of coherent backscattering of the electrons with impu-
rities near the contact [942]. As a result of the interference of waves with
different path length, the transmission of the contact shows fluctuations as
a function of energy, which according to Eq. (19.10) lead to a finite ther-
mopower with a sign that can be either positive or negative. So in other
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572 Molecular Electronics: An Introduction to Theory and Experiment

Fig. 19.7 Left panel: Schematic diagram of the modified MCBJ configuration, used
for the simultaneous measurement of conductance and thermopower of metallic atomic
contacts. Right panel: Density plot of thermopower of gold atomic contacts against con-
ductance constructed from 220 breaking curves for for two samples. Black represents no
data points and white more than 100. Reprinted with permission from [942]. Copyright
1999 by the American Physical Society.

words, the thermopower signal was shown to be dominantly of the same

origin as the conductance fluctuations discussed in Chapter 11.
In the context of molecular junctions, Paulsson and Data stressed in
a theoretical study the importance of measuring the thermopower [944].
They showed that in molecular contacts this transport property is large
enough to be measured, it is rather insensitive to the detailed coupling
to the contacts and it provides valuable information about the position of
the Fermi energy relative to the molecular levels. Let us illustrate these
ideas with a simple model. Following Ref. [944], let us assume that the
transmission function exhibits a double-peak structure described by two
independent Lorentzian:
2
X 4ΓL ΓR
τ (E) = , (19.12)
i=1
(E − ǫi + (ΓL + ΓR )2
)2
were ǫi is the energy of the two levels, and ΓL and ΓR the broadenings
by contacts L and R. For simplicity, we assume here that the broadenings
are the same for both levels. Eq. (19.12) describes a typical situation that
is realized in many organic molecules where the two levels typically cor-
respond to the HOMO and LUMO of the molecule and the Fermi energy
lies somewhere between them. The transmission function of Eq. (19.12) is
illustrated in Fig. 19.8(a) for a case in which ǫ1 = −7 eV, ǫ2 = −3 eV and
two values of the broadenings, 100 and 30 meV, which are assumed to be
equal for both contacts. The exact value of the linear conductance depends
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Beyond electrical conductance: shot noise and thermal transport 573

HOMO LUMO
1 60
(a) 40 (b)

S (µV/K)
0.1 20
τ(E)

0
0.01 -20
-40
0.001 -60
-8 -7 -6 -5 -4 -3 -2 -8 -7 -6 -5 -4 -3 -2
EF (eV) EF (eV)
Fig. 19.8 (a) Transmission as a function of the Fermi energy computed from Eq. (19.12).
The values of the parameter are: ǫ1 = −7 eV, ǫ2 = −3 eV and ΓL = ΓR = 100 meV
(solid line) and ΓL = ΓR = 30 meV (dashed line). (b) Corresponding thermopower as a
function of the Fermi energy calculated from Eqs. (19.12) and (19.11).

on the position of the Fermi level, which we leave as a free parameter. Us-
ing Eq. (19.11) one can compute the thermopower and the result is shown
in Fig. 19.8(b). Notice that depending on the position of the Fermi energy
with respect to molecular levels, the thermopower can be either positive or
negative. If EF is closer to the HOMO, the sign is positive and one talks
about hole-dominated transport. If at the contrary, the LUMO is closer
to EF , then the thermopower is negative and one has electron-dominated
transport. Notice also that the thermopower is in the range of µV /K (or
larger), like in the case of atomic contacts, and therefore it should be mea-
surable. Finally, when EF is not too close to one of the frontier orbitals,
the thermopower is very similar for the two cases shown in Fig. 19.8(b),
although the broadenings differ by a factor of three. Indeed, if the Fermi
energy is located between the HOMO and LUMO and far away from them,
it is easy to show from Eqs. (19.12) and (19.11) that, to first order, the
thermopower is independent of the metal-molecule coupling [944].
The first experiment measuring the thermopower in single-molecule
junctions was reported by Reddy et al. [101]. These authors used STM
break junctions to trap molecules between two gold electrodes with a tem-
perature difference across them. In this way they were able to measure
the thermopower (or Seebeck coefficient) of 1,4-benzenedithiol (BDT), 4,4′ -
dibenzenedithiol (DBDT), and 4,4′′ -tribenzenedithiol (TBDT) in contact
with gold at room temperature and found the values +8.7 ± 2.1 µV/K,
+12.9 ± 2.2 µV/K, and +14.2 ± 3.2 µV/K, respectively. As explained
above, the positive sign indicates p-type (hole) conduction in these hetero-
junctions, i.e. the transport is dominated by the HOMO of the molecules.
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574 Molecular Electronics: An Introduction to Theory and Experiment

Fig. 19.9 Histograms obtained by analyzing approximately 1000 consecutive thermo-

electric voltage curves obtained in measurements of Au-BDT-Au junctions with tip-
substrate temperature differential (A) ∆T = 10 K, (B) ∆T = 20 K, and (C) ∆T = 30
K. a.u., arbitrary units. (D) Plot of the peak values of the thermoelectric voltage in his-
tograms as a function of the temperature differential. The error bars represent FWHM
of the corresponding histograms. It can be seen that the measured voltage varies linearly
with the temperature differential, as expected. (E) Plot of measured junction Seebeck
coefficient as a function of molecular length for BDT, DBDT, and TBDT. From [101].
Reprinted with permission from AAAS.

It was also observed that S grows roughly linearly with the number N of
the phenyl rings in the molecule. These results are illustrated in Fig. 19.9.
This pioneering experiment motivated new theoretical work on this sub-
ject. Thus for instance, Pauly et al. [546] presented an ab initio (DFT-
based) study of the thermopower in metal-molecule-metal junctions made
up of dithiolated oligophenylenes contacted to gold electrodes. It was found
that, in agreement with the experiment, the transport is dominated by
the HOMO of these molecules. Moreover, it was shown that while the
conductance decays exponentially with increasing molecular length, the
thermopower increases linearly as in the experiments of Ref. [101]. This
is illustrated in Fig. 19.10, where the conductance and thermopower for
oligophenylenes with up to 4 phenyl rings are shown in panel (c) and (d),
respectively. Notice that the transmission functions for these molecules, see
Fig. 19.10(a), resemble those obtained with the simple model of Eq. (19.12).
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Beyond electrical conductance: shot noise and thermal transport 575

R1 R1
EF R2 R2

-τ’(E) / τ(E) (1/eV)

0
10 R3 10 R3
R4 R4
-1
10 0
τ(E)

-2
10
-10 EF
-3
10 (a) (b)
-4 -20
10 -7 -6 -5 -4 -3 -2 -1 -6 -5 -4 -3 -2
E (eV) E (eV)
0
10 40
-1
(c) R, exp (d)
10 30 R
S (µV/K)
G / G0

-2
10 20
-3
10 10
-4
10 0 0
1 2 3 4 0 1 2 3 4
N N

Fig. 19.10 Ab initio calculations for the conductance and thermopower of dithiolated
oligophenylenes contacted to gold electrodes. N is the number of phenyl rings in the
molecules. (a,b) Transmission function and the negative of its logarithmic derivative.
(c,d) The corresponding conductance (G) and thermopower (S). The experimental data
in (d) are form Ref. [101]. The straight lines are the best fits to the numerical results.
Adapted with permission from [546]. Copyright 2008 by the American Physical Society.

As one can see in Fig. 19.10(b), there is a qualitative agreement with the ex-
perimental results which, taking into account the usual theory-experiment
disagreement for the conductance, is certainly encouraging.
Pauly et al. also explained in simple terms the origin of the linear
increase of the thermopower with the length of the molecules (see also
Ref. [945]). As mentioned above, the transport in oligophenylenes proceeds
through the tail of the HOMO and the off-resonant tunneling is reflected
in the typical exponential decay of the linear conductance: G/G0 ∼ e−βN ,
where N is the number of phenyl rings. This off-resonant transport is
the origin of the linear increase in the thermopower. The idea goes as
follows. Assuming that the transmission around E = EF is of the form
τ (E) = α(E)e−β(E)N , then Eq. (19.11) yields S = SC + βS N , where
π 2 kB
2
T π 2 kB
2
T ′
SC = − [ln α(EF )]′ and βS = β (EF ). (19.13)
3e 3e
It is important to notice that, while SC depends on the prefactor α(E), βS
does not. Since α(E) contains the most significant uncertainties related to
the contact geometries, one expects βS to be described at a higher level
of confidence than SC . This linear dependence of the thermopower on
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576 Molecular Electronics: An Introduction to Theory and Experiment

molecular length has also been obtained in model calculations [222].

Another interesting suggestion of the work of Pauly et al. [546] is the
idea that the thermopower can be tuned to a large extend by modifying
the molecules with the inclusion of appropriate side-groups. In particu-
lar, in that work the introduction of methyl groups in the oligophenylenes
molecules was shown to have a two-fold effect: (i) the substituents push the
energies of the π electrons up as a result of their electron-donating behav-
ior and (ii) they increase the tilt angles between the phenyl rings through
steric repulsion. The latter effect tends to decrease both G and S due to
a reduction of the degree of the π-electron delocalization, while the former
opposes this tendency by bringing the HOMO closer to EF , see Ref. [546]
for further details.
Indeed, thermopower measurements were used by Majumdar’s group
[102] to elucidate the role of side-groups on the electronic structure and
charge transport in molecular junctions. Again, this group used a STM
break junction technique to study the thermopower of several benzene
derivatives. To be precise, 1,4-benzenedithiol (BDT) was modified by the
addition of electron-withdrawing or -donating groups such as fluorine, chlo-
rine, and methyl on the benzene ring. Moreover, the thiol end groups
on BDT were replaced by cyanide end groups. It was observed that the
thermopower of the molecular junction decreases for electron-withdrawing
substituents (fluorine and chlorine) and increases for electron-donating
substituents (methyl). The authors interpreted these results as follows.
Electron-withdrawing groups remove electron density from the σ-orbital
of the benzene ring allowing the rings high energy π-system to stabilize.
Because the HOMO has a largely π-character, its energy is therefore de-
creased, shifting it further away from EF . According to the simple model
discussed above, such a shift results obviously in a decrease of the ther-
mopower. Alternatively, the addition of electron-donating groups increases
the σ-orbital electron density in the benzene ring, leading to an increase
in the energy of the π-system and thereby shifting the HOMO closer to
EF . This shift causes in this case the enhancement of the thermopower.
Finally, let us say that cyanide end groups were found to radically change
transport relative to BDT. The thermopower in this case was found to be
negative, which indicates that transport in 1,4-benzenedicyanide is domi-
nated by the LUMO. For a recent theoretical study of thermopower of some
of these molecules, see Ref. [946].
In yet another experiment of Majumdar’s group, the alignment and cou-
pling of the molecular orbitals with the states in the metal contacts were
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Beyond electrical conductance: shot noise and thermal transport 577

Fig. 19.11 Measurements of the Seebeck coefficient vs. molecular length for N -unit
phenylenedithiols (N = 1, 2, 3), phenylenediamines (N = 1, 2, 3), and alkanedithiols
(N = 2, 3, 4, 5, 6, 8). Fit lines to the data indicate that thermopower increases with
length at a similar rate (βS ) for phenylenediamines and phenylenedithiols but decreases
with length for alkanedithiols. Reprinted with permission from [947]. Copyright 2009
American Chemical Society.

investigated [947]. For this purpose, thermopower measurements were con-

ducted for a series of phenylenes and alkanes with varying binding groups.
As shown in Fig. 19.11, the thermopower increases linearly with length for
phenylenediames and phenylenedithiols while it decreases linearly in alka-
nedithiols. The comparison between the two phenylenes series suggests that
the molecular backbone determines the length dependence of S, while the
binding group determines the zero length or contact S. Notice that for
both thiol and amine end groups, the transport in phenylenes is dominated
by the HOMO. Analyzing the data in terms of the model of Eq. (19.12),
the authors concluded that for phenylenes the HOMO aligns closer to the
Fermi energy of the contacts as L−1 , but becomes more decoupled from
them as e−L . Notice that this approximate behavior is reproduced by the
ab initio results for the phenylenedithiols shown in Fig. 19.10(a). There,
one can see that the HOMO shifts progressively towards the Fermi energy,
while the corresponding resonance becomes narrower. The shift of the level
can be traced back to the electronic structure of the isolated molecules. As
shown in Fig. 2 of Ref. [546], the HOMO of the molecules moves closer to
the gold Fermi energy as the number of phenyl rings increases.
The case of the alkanedithiols is more complicated to understand. As
shown in Eq. (19.13), the linear coefficient βS is determined by β ′ (EF ),
i.e. the derivative with respect to energy of the attenuation factor β at
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578 Molecular Electronics: An Introduction to Theory and Experiment

the Fermi energy. As shown in section 13.4, one can use simple bridge
models to obtain an expression of the energy dependence of the β-factor.
Assuming off-resonant tunneling, we obtained in section 13.4 the expres-
sion of Eq. (13.15) for β. Such a simple model predicts that a positive
thermopower is accompanied by an increase with length. This explains the
trend for the phenylenes, but not for the alkanes.8 The authors of Ref. [947]
suggested an explanation of this peculiar behavior in terms of gold-sulfur
metal induced gap states residing between the HOMO and the LUMO. As
we mentioned in section 14.1.2, several theoretical groups have concluded
that the transport in alkanes can be influenced by states that originate from
the hybridization of gold and sulfur orbitals with localized orbitals of the
alkane chain [948–950]. Indeed, evidence of the existence of these states has
been reported in STM experiments [951]. These hybrid states are localized
at the interfaces and therefore, they are expected to have a major impact
on the conductance for short molecules, while for long ones the transport is
expected to be dominated by the HOMO of the alkane chains. The different
length dependence of the metal induced gap states and the HOMO of the
chains could be the origin of the decreasing thermopower [947].
As we already discussed in section 13.6, the transmission function of
a molecular contact can exhibit lineshapes that completely differ from the
double-peak structure shown in Fig. 19.8(a). One way to increase the ther-
mopower is by “engineering” a much more pronounced energy dependence
of the transmission function. As shown recently by Finch et al. [487], some
molecules can exhibit sharp Fano resonances very close to the Fermi energy
that in turn can lead to a huge thermopower in molecular junctions.
In the discussion so far we have focused on the thermopower in the co-
herent transport regime. However, this transport property can also provide
very valuable information in many other transport regimes. For instance,
Koch et al. [800] have shown theoretically that the thermopower of weakly
coupled molecular junctions can give access to the electronic and vibrational
excitation spectrum of the molecule even in a linear-response measurement.
To summarize, we have shown in this section that thermopower mea-
surements in molecular junctions provide very important information not
contained in the conductance. In this sense, we believe that measurements
of this thermoelectric property will play a crucial role in the immediate
future of molecular electronics.
8 The simple bridge model of section 13.4 suggests that a decreasing S with length can

only be obtained when the transport is dominated by the LUMO and therefore, the
thermopower is negative.
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Chapter 20

Optical properties of current-carrying

molecular junctions

We have discussed so far different ways of controlling the current through

a molecular junction such as gating or appropriate chemical synthesis. An-
other possibility is the use of an external electromagnetic field, which has
been widely explored in larger mesoscopic structures [214]. In addition to
controlling transport with external radiation, many other issues related to
the optical properties of molecular junctions are of interest and some of
them have been recently studied [211, 952]. In this sense, the goal of this
chapter is to discuss the physical phenomena that emerge as a result of the
interplay between current-carrying molecular junctions and an electromag-
netic field.
The optical properties of molecular transport junctions involve many
different aspects and it is certainly impossible to address all of them. Here,
we shall focus our attention on the topics related to the following funda-
mental questions:

(1) Is it possible to use conventional optical spectroscopies to characterize

molecular transport junctions?
(2) What is the effect of an electromagnetic radiation on molecular con-
duction?
(3) Can we use an external electromagnetic field to control the current or to
learn something about the electronic structure of molecular contacts?
(4) What are the new transport phenomena than can be expected in ac
driven molecular junctions?
(5) How does a molecular transport junction radiate?
(6) Different molecules have very peculiar and interesting optical proper-
ties. Can those molecules be used to design novel optoelectronic de-
vices?

579
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580 Molecular Electronics: An Introduction to Theory and Experiment

With those questions in mind, we have organized the rest of this chapter
as follows. First, we shall describe recent experiments in which molecu-
lar junctions have been characterized using surface-enhanced Raman spec-
troscopy. Then, we shall discuss the physical mechanisms that are expected
to play a major role in irradiated atomic and molecular junctions. In par-
ticular, we shall pay special attention to the so-called photon-assisted tun-
neling or current rectification.1 Section 20.4 presents a description of some
recent experimental results on the electronic transport through irradiated
atomic and molecular contacts. In section 20.5 we shall briefly discuss the
phenomenon of current amplification and other novel transport phenom-
ena that have been predicted to appear in ac driven molecular junctions.
Section 20.6 is devoted to the analysis of fluorescence of current carrying
junctions. Finally, in section 20.7 we shall review some of the experiments
in which the optical properties of certain molecules have been exploited to
design primitive molecular optoelectronic devices.

20.1 Surface-enhanced Raman spectroscopy of molecular

junctions

With respect to the first question posed above, it is obvious that combined
optical and transport experiments on molecular transport junctions could
reveal a wealth of additional information beyond that available from purely
electronic measurements. It is, however, very challenging to use conven-
tional optical spectroscopies to obtain local information about molecular
junctions. First of all, it is not easy to inject light into slits of molecu-
lar size between two metal leads and second, the molecular emission may
be strongly damped because of the proximity to a metal surface. Fortu-
nately, recent work has shown that surface-enhanced Raman spectroscopy
(SERS) can offer a way out of these problems [953–956]. The idea is based
on the fact that metallic nanostructures, similar to those used to form the
electrodes of molecular junctions, can act as effective plasmonic antennas,
leading to a dramatic enhancement of the electric field locally at the junc-
tion region (see e.g. Ref. [957]). This enhanced field can then be used to
perform Raman spectroscopy of objects placed in these nanogaps (for a
review on SERS, see e.g. Ref. [958]). This idea has been explored recently
in the context of molecular electronics, in particular, by Natelson’s group
1 Thesetwo terms are sometimes believed to refer to two different physical mechanisms.
However, we shall show in section 20.3 that they are indeed identical.
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Optical properties of current-carrying molecular junctions 581

in two works that we now proceed to describe [955, 956].2

This group has performed a series of optical experiments in Au nanogap
structures prepared with the electromigration technique. The measure-
ments were made with a confocal Raman microscope with a 785 nm
diode laser at room temperature in air. The initial experiments of this
group examined nanogaps as a potential SERS substrate [955], with para-
mercaptoaniline (pMA) as the molecule of interest. Following electromi-
gration, the authors observed a SERS response strongly localized to the re-
sulting gaps. Successive spectra measured directly over the SERS hot-spot
revealed “blinking” and spectral diffusion, phenomena often associated with
single-or few-molecule Raman sensitivity. Blinking was observed to occur
when the Raman spectrum rapidly changed on the second timescale with
the amplitudes of different modes changing independently of one another.
Spectral shifts as large as ±20 cm−1 were observed, making it difficult
to directly compare SERS spectra with other published results. Blinking
and spectral shifts are attributed to movement or rearrangement of the
molecule relative to the metallic substrate. It is unlikely that an ensemble
of molecules would experience the same rearrangements synchronously and
thus blinking and wandering are expected to be observed only in situations
where a few molecules are probed.
In a second experiment, the same group performed simultaneous SERS
and transport measurements [956], including Raman microscope observa-
tions over the center of nanogap devices during electromigration. The
molecules of interest, pMA or a fluorinated oligomer (FOPE), were as-
sembled on the Au surface prior to electromigration. It was observed that
once the resistance exceeds approximately 1 kΩ, SERS can be seen. This
indicates that localized plasmon modes responsible for the large SERS en-
hancements may now be excited. As the gap further migrates the SERS
response was seen to scale logarithmically with the device resistance until
the resistance reaches approximately 1 MΩ. In most samples the Raman re-
sponse and conduction of the nanogap became decoupled at this point with
the conduction typically changing little while uncorrelated Raman blinking
occurred.
In some devices, however, the Raman response and conduction showed
very strong temporal correlations. A typical correlated SERS time spec-
trum and conductance measurement for a FOPE device are presented in
Fig. 20.1. The temporal correlations between SERS and conduction are

2 These experiments have been reviewed by the authors in Ref. [607].

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582 Molecular Electronics: An Introduction to Theory and Experiment

Fig. 20.1 The upper panel shows the Raman spectrum (1 s integration) for a single
FOPE molecule in a gold junction formed by electromigration. The lower panel shows
the correlated measurement of the conductance of this junction. The Raman mode ob-
served between 1950 and 2122 cm−1 is believed to be for the same 2122 cm−1 mode
associated with the C≡C stretch of the FOPE molecule. The large spectral shifts ob-
served for this mode are attributed to interactions between the molecule and its nanogap
environment. Clear correlations between the Raman structure and conductance can be
seen. In particular in region B and for part of region E the Raman spectrum is observed
to disappear while the conductance drops to zero. Reprinted with permission from [607].
Copyright 2008 IOP Publishing Ltd.

evident. Since the conduction in nanogaps is dominated by approximately

a single molecular volume, the observed correlations between conductance
and Raman measurements strongly indicate that the nanogaps have single-
molecule Raman sensitivity. It is then possible to confirm that electronic
transport is taking place through the molecule of interest, via the char-
acteristic Raman spectrum. Data sets such as those of Fig. 20.1 contain
implicitly an enormous amount of information about the configuration of
the molecule in the junction.
Let us mention that Tian et al. [954] have used the MCBJ technique
to study the intensity of the surface-enhanced Raman signal of molecular
junctions. They showed that this signal depends critically on the separation
of the electrodes and the incident light polarization. In particular, it was
shown that when the incident laser polarization is along the two electrodes,
the field in the nanogap is the strongest because of the coupling to the
localized surface plasmon resonance of two gold electrodes [957].
It would be highly desirable to perform simultaneous measurements of
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Optical properties of current-carrying molecular junctions 583

Raman and IET spectra, which, unfortunately, was not yet possible in the
experiments just described. As we explained in the previous chapter, IETS
requires cryogenic temperatures, at which it is not easy to operate a Raman
microscope. However, in the experiment of Ioffe et al. [938], which was dis-
cussed in section 19.2.3, a comparison between vibrational modes revealed
by Raman scattering and IETS could be established, although these two
type of measurements were not performed at the same time. Let us re-
mind that these authors reported SERS measurements of junctions based
on biphenyldithiol SAMs3 . In this experiment the Raman spectra were
acquired at room temperature, while the IET spectra were obtained in
transport measurements at 4 K. Interestingly, all Raman active vibrational
modes were revealed in the IETS measurements, in spite of the fact that the
selection/propensity rules are different in these two types of spectroscopies.
Let us also recall that the main goal of this work was to measure the voltage
dependence of the effective temperature of these current-carrying junctions.
This was achieved by measuring both the Stokes and anti-Stokes compo-
nents of the Raman scattering. Then, the effective temperature Tef f (ν) for
each mode was calculated at each bias using the following expression4
IAS (νL + νν )4 σAS A2AS
= exp (−hνν /kB Tef f (ν)) , (20.1)
IS (νL − νν )4 σS A2S
where IAS(S) is the intensity of the anti-Stokes (Stokes) Raman mode,
νL(ν) is the frequency of the laser (Raman mode), σAS(S) is the anti-Stokes
(Stokes) scattering cross-section of the adsorbed molecules and AAS(S) is
the average local field enhancement at the molecules at the anti-Stokes
(Stokes) frequency. Strictly speaking, this expression is only valid in ther-
modynamical equilibrium and one may wonder whether this relation still
holds at a finite bias voltage. For a discussion of this issue we refer the
reader to Refs. [959, 960], where a detailed theoretical study of Raman
scattering in current-carrying molecular junctions is presented.

20.2 Transport mechanisms in irradiated molecular junc-

tions

A prerequisite to answer questions 2-4 in the list presented in the introduc-

tion, i.e. to understand how an electromagnetic field alters the electrical
current of a molecular junction, is to identify the physical mechanisms that
3 The use of SAMs facilitates the acquisition of Raman spectra.
4 It was assumed that σAS A2AS = σS A2S .
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584 Molecular Electronics: An Introduction to Theory and Experiment

can play a role in this problem, which is by no means a trivial task. The
theoretical and experimental work reported so far on this subject suggests
that the main “suspects” are the following:

• Current rectification or photon-assisted tunneling: When an electrical

contact is irradiated, the ac field may induce an alternating voltage in
the junction with a frequency equal to that of the field. This ac bias
in turn gives rise not only to an ac current, but also modifies the dc
component. This phenomenon, in which an ac signal is converted to
a dc current, is known as current rectification. The phenomenon is
also known as photon-assisted tunneling (PAT) due to the nature of
the inelastic tunneling processes that govern the electrical conduction
in the presence of an ac bias (see discussion below).5
• Internal molecular transitions: While electronic transitions in the pre-
vious mechanism take place at the electrode-molecule interfaces, the
radiation can also induce the standard optical transitions inside the
molecules. This requires radiation frequencies comparable to the en-
ergy of the electronic excitations of the molecules, which are typically
in the optical range. The induction of such transitions can in principle
lead to phenomena like the resonant current amplification that will be
discussed in section 20.5.
• Hot electrons: If the radiation frequency is close to the plasma fre-
quency of the metal electrodes, the field can penetrate in the leads and
excite the conduction electrons to high energy (hot electrons) . If these
electrons are sufficiently close to the junction (closer than the inelas-
tic mean free path at the corresponding energy), they can contribute
significantly to the transport characteristics [961, 962].
• Heating: At optical frequencies a metal does not completely reflect the
radiation and part of it can be absorbed. This absorption is usually
accompanied by heating, which has several important consequences.
First of all, heating can result in thermal expansion of the samples,
which can be reflected in the junction current. This is well documented
in the STM context [215], where a change in tip-sample distance due
to thermal excitation has a dramatic effect on the tunneling current.
On the other hand, heating can also create a temperature gradient
5 Photon-assisted tunneling is maybe not a good name for this phenomenon since no real

photons are emitted or absorbed in the tunneling processes. However, it is commonly

used in the mesoscopic physics community, in which current rectification is rarely used.
We shall use here both terms, but it must be clear that they refer to the same physical
mechanism.
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Optical properties of current-carrying molecular junctions 585

or a temperature difference between the electrodes. In both cases,

thermoelectrical currents can appear in the junctions [215].

This list is not complete, and other effects can also play an important
role. In particular, some readers might miss surface plasmons in this list.
In this respect, we would like to say that for optical frequencies there is no
doubt that plasmons play a key role. Surface plasmons are responsible for
the local field distribution at the junction and, in particular, for its enhance-
ment with respect to the incident field. In this sense, we can consider that
plasmons determine the effective amplitude (and frequency dependence) of
the ac potential induced in the junction, but the transport mechanism is
still PAT or current rectification. In other words, we rather prefer to say
that plasmons play an important role in the PAT mechanism than to say
they constitute a different mechanism. After all, an ac field would also
appear in the absence of plasmons and their role is only to modify the field
distribution.
As we have already mentioned above, the importance of the different
mechanisms depends primarily on the radiation frequency. For instance, in
the microwave range PAT (or current rectification) largely dominates the
transport. This has been firmly established in a great variety of mesoscopic
structures [214] and more recently in atomic and molecular junctions (see
discussion below). In the optical range, however, the other three mecha-
nisms can also be very important.

20.3 Theory of photon-assisted tunneling

In the this section we shall present a description of the PAT theory for
several reasons. First, this mechanism is likely to operate in almost any
situation since when a junction is illuminated most of the radiation indeed
impinges on the electrodes. Second, it is believed to be the dominant one
at low frequencies and finally, recent experiments in atomic and molecular
contacts seem to suggest that this mechanism is the dominant one even at
optical frequencies. In what follows, we shall first present the basic theory
of PAT and then, we shall discuss the basic predictions of this theory for
atomic and molecular junctions.
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586 Molecular Electronics: An Introduction to Theory and Experiment

20.3.1 Basic theory

In order to explain the steps in the current-voltage characteristics of
microwave-irradiated superconductor-insulator-superconductor junctions
[963], Tien and Gordon [213] proposed a heuristic theoretical treatment
of electron tunneling in the presence of an ac field, which is of appealing
simplicity. The central idea goes as follows.6 First of all, the presence of
the ac field is represented by a time-dependent voltage applied across the
junction in addition to the dc bias7
V (t) = V + Vac cos(ωt). (20.2)
Indeed, Tien and Gordon assumed that this ac voltage is applied to one
of the electrodes, while the other remains grounded. This applied voltage
is assumed to modulate adiabatically the potential energy for each quasi-
particle level on the ungrounded side of the barrier. This assumption is
expected to be valid below the plasma frequencies of the two electrodes,
typically well into the ultraviolet. The time dependence of the wave func-
tion for every single-electron state in the ungrounded electrode is therefore
modified according to
i t ′
· Z ¸
′
ψi (x, t) = ψi (x) exp − dt [Ei + eV (t )] (20.3)
~
X∞
= ψi (x) exp [−i(Ei + eV )t/~] Jn (α)e−inωt ,
n=−∞
where Ei is the unperturbed energy of the single-electron state, Jn is the
Bessel function of the first kind (of order n) and α ≡ eVac /~ω. The adia-
batic modulation of the Fermi sea on this side of the junction can be thus
viewed in terms of a probability amplitude Jn (α) for each quasiparticle
level to be displaced in energy by n~ω. This interpretation is illustrated
schematically in Fig. 20.2. Since all electron states are modulated together,
these displacements in energy are equivalent to dc voltages (V + n~ω/e)
applied across the junction with a probability Jn2 (α) that depends upon the
ac signal amplitude. The resulting dc tunneling current is, therefore, given
by the expression
∞
X
I(V ; α, ω) = Jn2 (α)I0 (V + n~ω/e), (20.4)
n=−∞
6 We present here an extension of the original Tien-Gordon argument due to Tucker
[212].
7 The dc part of the voltage will be simply denoted as V to follow the notation used

so far. The total time-dependent voltage will always be denoted as V (t), i.e. including
explicitly the time argument.
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Optical properties of current-carrying molecular junctions 587

..
.
J2 (α) exp(+2i ω t)

Ei(t) = E i + e(V+Vac cos ωt) J1 (α) exp(+i ω t)

hω
J0 (α)
eV
J−1(α) exp(−i ω t)

J−2(α) exp(−2i ω t)
..
.
Fig. 20.2 Virtual energy levels generated according to Eq. (20.3) by adiabatic modula-
tion of the energy Ei (t) for each quasiparticle state within the ungrounded electrode of
a junction in the presence of an ac field.

where I0 (V ) represents the current in the absence of radiation. This is a

remarkable result that tell us that the I-V characteristics in the presence
of radiation can be understood in terms of the I-V curves in the absence of
external ac driving.
Notice that in this Tien-Gordon picture electrons undergo virtual tran-
sitions in the ungrounded electrode by “absorbing” or “emitting” an integer
number of electromagnetic energy quanta (n~ω) and then they tunnel elas-
tically through the junction. In this sense, one can interpret the tunneling
processes as photon-assisted events and this is the reason why the asso-
ciated phenomenon is known as photon-assisted tunneling (PAT). Notice,
however, that no “real” photons are involved in these processes and, in
particular, the description of this phenomenon does not involve the quan-
tization of the electromagnetic field. In this sense, the name PAT is maybe
not very accurate, but since it is so commonly used, we shall also employ
it here.
On the other hand, notice that Eq. (20.4) expresses how the dc current
is modified by the radiation, i.e. it tells us how an ac signal results in a dc
current. As we already mentioned above, this conversion process is known
as current rectification and in this context, PAT and current rectification
will be considered as synonyms.
A rigorous treatment of the electronic transport through an arbitrary
junction subjected to an ac field can be done in the framework of differ-
ent approaches such as the scattering formalism [165], the Floquet the-
ory [211, 214] and the nonequilibrium Green’s function formalism (NEGF)
[214, 216–221]. This latter approach has been explained in detail in Chap-
ter 8. In particular, we have shown there that the current through a junction
in the presence of an ac voltage can be written in the form of the Tien-
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588 Molecular Electronics: An Introduction to Theory and Experiment

Gordon formula of Eq. (20.4) under the following assumptions: (i) The
energy dependence of the lead density of states is negligible and (ii) the
ac potential does not vary spatially along the central part of the junction
(i.e. across the molecule in the case of molecular junctions). While the first
assumption is often justified, the second one may seem rather restrictive.
However, Viljas et al. [221, 222] have shown that if the amplitude of the
ac voltage is not too large, the precise shape of the profile does not play a
crucial role.
In the derivation of Eq. (20.4) it was assumed that the ac voltage is
applied to one of the electrodes, while the other is grounded. If we consider
that the ac voltage is applied symmetrically, i.e. it drops equally at both
interfaces, the current can be then written as
X∞ h ³ α ´i2
I(V ; α, ω) = Jn I0 (V + 2n~ω/e). (20.5)
n=−∞
2

Here, assuming that the transport in the absence of ac drive is elastic, the
current I0 is given by the standard Landauer formula. At low temperatures
and in the linear response regime (vanishing dc bias), the conductance takes
the particularly simple form
X∞ h ³ α ´i2
G(V = 0; α, ω) = G0 Jn τ (EF + n~ω), (20.6)
n=−∞
2

where τ (E) is the zero-bias equilibrium transmission and EF is the Fermi

energy. This quantity will be of special interest in our discussion below,
and it will be referred to as photoconductance. Note that if the transmission
does not depend on energy in the range probed by the inelastic processes,
the conductance reduces to the conductance in the absence of drive, i.e.
G(V = 0; α, ω) = G0 τ (EF ).8
It is interesting to consider the limit of small ac amplitudes. Using
J0 (x) ≈ x2 /4 and J±n (x) ≈ (±x/2)n /n! in Eq. (20.5) and retaining only
the lowest-order terms in the ac potential Vac , the correction to the current
in this limit can be expressed as
∆Idc (V ; α, ω) ≡ I(V ; α, ω) − I0 (V ) (20.7)
· ¸
1 2 I0 (V + 2~ω/e) − 2I0 (V ) + I0 (V − 2~ω/e)
= Vac .
4 (2~ω/e)2
The quantity in large parentheses is a finite second difference of the I-
V characteristics in the absence of radiation that reflects the emission or
8 This Jn2 (x) = 1.
P
can be easily shown using the relation n
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Optical properties of current-carrying molecular junctions 589

absorption of a single quantum during the tunneling. All higher-order pro-

cesses n = 2, 3, . . . , contributing to the dc current may be neglected in the
limit of small ac amplitude. When the photon energy ~ω/e is smaller than
the voltage scale of the dc nonlinearity, the finite difference can be replaced
by the second derivative of the current, i.e.

d2 I0
µ ¶
1 2
∆Idc (V ; α, ω) ≈ Vac . (20.8)
4 dV 2

This expression reproduces the classical result for rectification [212] and
illustrates the connection between PAT and current rectification. This is a
very important relation since it provides a direct way to test whether the
dominant transport mechanism is indeed PAT/current rectification. Such
test requires to measure independently the induced dc current and the
second derivative of the current with respect to the bias in the absence
of radiation. Moreover, according to Eq. (20.8), the ratio of these two
quantities gives the amplitude of the ac bias, which is typically unknown.
This amplitude gives information about the field enhancement locally in
the junction region.
Notice that Eq. (20.8) suggests that if the I-V characteristics in the
absence of radiation exhibit an asymmetry at vanishingly bias voltage due
to material and/or geometrical asymmetries (i.e. if d2 I0 /dV 2 6= 0 at V = 0),
a radiation-induced current can flow in the system even in the absence
of any dc bias voltage. This phenomenon of rectification at zero dc bias
voltage was predicted by Cutler et al. in 1987 [964] and it was first reported
by Walther’s group in 1991 in laser-driven STM experiments on graphite
surfaces [965] (for a detailed discussion of this phenomenon, see the review
of Ref. [215]). The current generated by the ac field in the absence of dc
bias is often referred to as photocurrent.9
The classical expression of Eq. (20.8) is probably valid in a wide range
of molecular junctions for microwave frequencies, while in the optical range
significant deviations from this expression are likely to appear and it has
to be replaced by its quantum version of Eq. (20.7) [see Exercise 20.2(ii)].
On the other hand, it is interesting to derive similar expressions for the
linear conductance. Defining the induced linear conductance correction as
∆Gdc (α, ω) ≡ G(V = 0; α, ω)−G(ω = 0), where G(ω = 0) = G = G0 τ (EF ),

9 Laterin this chapter we shall discuss the so-called ratchet effect in molecular junctions,
which is just another name for rectification at zero bias.
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590 Molecular Electronics: An Introduction to Theory and Experiment

the relative correction for small ac amplitudes becomes

(eVac )2 τ (EF + ~ω) − 2τ (EF ) + τ (EF − ~ω)
· ¸
∆Gdc (α, ω)
=
G 16τ (EF ) (~ω)2
2 ′′
(eVac ) τ (EF )
≈ , (20.9)
16 τ (EF )
where the last expression has been obtained assuming that ~ω/e is smaller
than the energy scale over which the transmission varies significantly. We
thus see that in this limit ∆Gdc gives experimental access to the second
derivative of the transmission function at the Fermi energy.
In the next section we shall review recent transport experiments per-
formed with irradiated atomic and molecular junctions. In this sense, it is
interesting to know the basic predictions of PAT theory for these systems.
This is addressed in the next two subsections.

20.3.2 Theory of PAT in atomic contacts

In metallic atomic-sized contacts, the I-V curves in the absence of radia-
tion are rather linear up to voltages of the order of 0.5-1.0 V. This is a
consequence of the fact that the transmission does not change significantly
around the Fermi level in an energy window of a few tenths of eV. According
to Eqs. (20.8) and (20.9), this suggests that no significant changes in the
transport characteristics are expected under irradiation up to frequencies
close to the optical range.
As a side remark, let us say that in the superconducting state, atomic
contacts are very sensitive to microwave frequencies. The reason is the pres-
ence of a gap in the spectrum, which ranges from 0.1 to 1 meV depending
on the material. Recently, the subgap transport in superconducting atomic
contacts under microwave irradiation was studied experimentally [966]. It
was found that the subharmonic gap structure in the dc current is strongly
modified in quantitative agreement with the theory of photon-assisted mul-
tiple Andreev reflections [967]. The importance of these results for our
discussion here is that they provide firm support for the PAT mechanism
in the microwave range.
A detailed theoretical study of PAT in atomic contacts has been re-
ported by Viljas and one of the authors [221]. In this work the NEGF
formalism of section 8.3 was used to compute the photoconductance as a
function of frequency in one-atom thick contacts of several metals (Au, Pt
and Al). In Fig. 20.3 we show an example of the results for a dimer con-
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Optical properties of current-carrying molecular junctions 591

3 3
(a) Au contact τtotal (c) Pt contact τtotal
2.5 τ1 2.5 τ1
Transmission

Transmission
τ2 τ2
2 2
τ3 τ3
1.5 τ4 1.5 τ4
1 1
0.5 0.5
0 0
-5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5
E-EF (eV) E-EF (eV)
λ (nm) 800 400 λ (nm) 800 400

1.2 α=0.25 2.2

α=0.5
α=0.75
Gdc(ω) / G0

Gdc(ω) / G0

1.1 α=1.0 2 α=0.25

α=1.25 α=0.5
1 α=1.5 α=0.75
1.8 α=1.0
0.9 (b) (d) α=1.25
α=1.5
0.8 1.6
0 0.5 1 1.5
_
2 2.5 3 0 0.5 1 1.5
_
2 2.5 3
hω (eV) hω (eV)

Fig. 20.3 Theoretical results for the photoconductance of Au and Pt atomic contacts.
(a) Equilibrium transmission τtotal and its decomposition into conduction channels
τ1,2,3,4 for an Au dimer contact. (b) Zero-temperature photoconductance for several
values of α as a function of frequency ω computed using Eq. (20.6). In (b) the wave-
lengths λ with a tick spacing of 400 nm are shown. The range of visible light is indicated
by vertical dotted lines. (c-d) The same as in panels (a-b) but for a Pt contact. Adapted
with permission from [221]. Copyright 2007 by the American Physical Society.

tact10 of Au and Pt. Here, we just reproduce the results obtained with the
simple approximation of Eq. (20.6), which was usually found to reproduce
qualitatively the more rigorous results obtained with the NEGF formalism
[221]. In the case of Au, as can seen in Fig. 20.3(a), the conductance for
ω = 0 is equal to 1 G0 with a single open channel arising from the contri-
bution of the 6s orbitals. Moreover, notice that the transmission around
EF is very flat. Due to this flatness, for frequencies up to ~ω ≈ 1.5 eV
(λ ≈ 827 nm) the effect of radiation is practically negligible. In the red
part of the visible range (~ω . 2 eV) ∆Gdc > 0 and it can reach up to 20%
depending on the value of α.11 This increase in the conductance is due to
the contribution of the 5d bands located 2 eV below EF , where the number
of open transmission channels is higher than at EF .
10 The exact geometry of this dimer contact can be seen in Fig. 1 of Ref. [221]. This

type of geometry is typically responsible for the last conductance plateau in the breaking
process of an atomic contact.
11 It is important to remark that for the case of Au it was found that the results were

quite sensitive to the exact profile of the ac voltage. For the other metals the profile did
not play a major role.
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592 Molecular Electronics: An Introduction to Theory and Experiment

Fig 20.3(c-d) show the corresponding results for Pt. In the absence
of radiation the conductance is close to 2.1 G0 due to the contributions
of mainly three conduction channels, which originate from the 6s and 5d
orbitals. In this case, and in general in the contact regime for Pt, the effect
of the radiation is always a significant reduction in conductance. This
is understandable, since EF lies at the edge of the d band, and photon
absorption leads to an energy region where less open transmission channels
are available and τtotal is smaller.
To conclude, the key message is that the photoconductance simply re-
flects the energy dependence of the transmission of the contacts. The sign
of the induced correction can be both positive (like for Au) and negative
(like for Pt) depending on the material. With respect to the order of mag-
nitude of the correction, it can reach up to 50%-100% in some special cases
depending on the geometry, frequency and power of the radiation, but it is
usually below those values.

20.3.3 Theory of PAT in molecular junctions

From the discussion above, it is obvious that irradiation can lead to more
dramatic effects in the case of molecular junctions. These junctions typi-
cally exhibit a much more pronounced energy dependence of the transmis-
sion function, which can lead to much larger modifications of the trans-
port characteristics than in atomic contacts. In section 8.3.1 we have used
the resonant tunneling model to illustrate some of the effects that may be
expected from PAT in molecular junctions. The most prominent one is
the resonant enhancement of the photoconductance. The idea is the fol-
lowing. The transmission function of most molecular junctions exhibit a
deep pseudo-gap in the energy region between the HOMO and LUMO of
the molecule, while the Fermi level lies somewhere in between. Then, if
the photon energy is equal to the distance between the Fermi energy of
the closest frontier orbital, the low-bias conductance can be greatly en-
hanced. This fact together with other interesting predictions are further
illustrated in Exercise 20.2 at the end of the chapter with the use of the
double-Lorentzian transmission function that we employed in section 19.3
to understand the thermopower of molecular junctions.
More realistic models of PAT in molecular junctions confirm these con-
clusions [968, 969]. Thus for instance, Viljas et al. [970] have reported a
study of the photoconductance in organic single-molecule contacts. This
study is based on Eq. (20.6), whereas the equilibrium transmission was
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Optical properties of current-carrying molecular junctions 593

computed using a DFT-based method. It was found that the radiation

can indeed lead to large enhancements of the conductance of such contacts
by bringing off-resonant levels into resonance through photon-assisted pro-
cesses. The conductance enhancement was demonstrated for oligophenylene
molecules between gold electrodes. It was shown that the exponential decay
of the conductance with the length of the molecule can be replaced by a
length-independent value in the presence of radiation. In other words, the
photon-assisted processes turn the off-resonant tunneling into on-resonance
transport. Results of this work are reproduced in Fig. 20.4. Notice first
that the transmission in the absence of radiation exhibits a pseudo-gap
in the region between the HOMO and LUMO (see panel (a) in the right
figure). The HOMO lies closer to the Fermi energy in this case (∼1 eV
away). Second, in all of cases the low-bias conductance is greatly enhanced
(for still reasonable values of α) and the onset of the enhancements is well
inside the infrared region of the electromagnetic spectrum. Finally, panel
(f) of the right figure shows how the typical exponential decay of the con-
ductance with length is replaced by constant conductance in the presence
of the radiation.
The fact that the conductance enhancement takes place in this case in
the infrared region has important consequences. First, at these frequencies
no internal transitions inside the molecules are possible. Second, metals
do not absorb in the infrared and thus the associated heating effects are
minimal. Therefore, this mechanism for enhanced photoconductance should
be quite robust and, in principle, it could take place for a great variety of
molecules since the only requirement is the existence of a pronounced gap
in the transmission function. In this sense, molecular junctions can behave
similarly to superconductor-insulator-superconductor systems, which have
been used as microwave detectors in a variety of applications [212].
To conclude this section, let us also say that Viljas et al. [222] have
studied the photon-assisted tunneling in more detail using simplified mod-
els and they have made further predictions that can be used as fingerprints
of the PAT mechanism. First, in off-resonant situations, where the con-
ductance in the absence of radiation decays exponentially with the length
of the molecule, the correction to the dc linear conductance grows as the
length square, i.e. the conductance enhancement is more pronounced for
longer molecules. Second, at low frequencies additional steps can appear in
I-V characteristics. Their separation, in the case of a symmetric junction,
is roughly 2~ω. For a discussion of the origin of these steps see section 8.3.1
or Exercise 20.2(iv) at the end of the chapter.
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594 Molecular Electronics: An Introduction to Theory and Experiment

1.5 1
(a) R1
(a) EF 0.1

Gdc / G0
1 G2

T(E)
0.01
D = 0.97 nm
0.5 0.001 (f) G1
R2
0 0.0001
-8 -6 -4 -2 1 2 3 4
E (eV) n
1
(b) α = 1.8
Gdc(ω) / G0
D = 1.40 nm (c)
R3 0.1

0.01 α = 0.2 R1 R2
D = 1.83 nm
R4 1
Gdc(ω) / G0

0.1 (d) (e)

0.01
D = 2.27 nm R3 R4
(b) 0.001
V ac
0 0.5_ 1 1.5 2 0 0.5_ 1 1.5 2
h ω (eV) h ω (eV)

Fig. 20.4 Calculations of the photoconductance of oligophenylene-based single-molecule

junctions. Left figure: (a) The four studied molecular contacts R1–R4, containing
oligophenylenes with one to four phenyl rings and coupled to Au [111] pyramids through
sulfur atoms. (b) In the calculation it was assumed the induced ac voltage Vac to drop
in a double-step manner. Right figure: (a) Transmission versus energy [T (E)] for the
contacts R1–R4 (dash-dot-dotted, dash-dotted, dashed, and solid lines, respectively).
(b)–(e) The photoconductance versus external frequency ω for the contacts R1–R4, re-
spectively. For each case the results for the following values of α are shown: 0.2, 0.6,
1.0., 1.4, and 1.8, in order of increasing conductance. (f) The dc conductances in the
absence (G1 , dots) and presence (G2 , crosses) of radiation with ~ω = 1.5 eV and α = 1.8
for an increasing number n of phenyl rings. The gray line is a fit of the G1 results to an
exponential law. Reprinted with permission from [970]. Copyright 2007 by the American
Physical Society.

20.4 Experiments on radiation-induced transport in atomic

and molecular junctions

The experimental study of the electronic transport in irradiated atomic-

scale junctions started around 20 years ago in the context of STM. In
particular, in the early 1990’s there was an intense activity related to the
study of rectification (for a review see Ref. [215]). Although the observation
of radiation-induced dc currents in STM experiments has been reported
by many groups, it has always been difficult to show unambiguously that
those currents were due to rectification and not to other mechanisms like,
for instance, the generation of thermocurrents.
A convincing evidence of atomic-scale rectification was reported by Ho’s
group in 2006 [971]. This group presented STM experiments in which
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Optical properties of current-carrying molecular junctions 595

microwave-induced dc currents were measured for Mn atoms and MnCO

molecules adsorbed on NiAl(110). The frequency of the microwave signal
was 800 MHz. In Fig. 20.5(a) one can see results from this work for the
differential conductance measured through a single Mn atom as a function
of the bias voltage. In this figure VB denotes the dc voltage, VJ is the local
amplitude of the ac voltage and VIN denotes the ac signal far away from
the contact. In the absence of microwave (VIN = 0), the differential con-
ductance has a narrow peak at 2.0 V and a broad peak at 1.3 V, associated
with the spin splitting of Mn sp states from magnetic interaction with its d
electrons [971]. As VIN is increased, the peak at 2.0 V becomes broader and
eventually splits into two. In order to find out whether this modification of
the dc current was due to rectification, the authors fitted the results with
a classical rectification formula where the dc current is given by [971]

ω
Z 2π/ω h √ i
I(VB , VJ ) = I VB + 2VJ cos(ωt) dt. (20.10)
2π 0
Here, the function I(V ) in the integrand has the same form as the static
I-V characteristics measured in the absence of microwave, but with a time
dependent argument. An example of the fits is shown in Fig. 20.5(b) for
VIN = 5 mV. Notice the high accuracy of the fit, which provides a strong
support for the interpretation of the results in terms of rectification. More-
over, from the fits the value of VJ could be extracted. A plot of VJ versus
VIN is shown in Fig. 20.5(c), yielding a slope 45.5 from the best linear fit.
This slope gives a direct information about the field enhancement at the
contact.
On the other hand, it was also shown that the induced dc current as
a function of voltage for a single Mn atom follows closely the d2 I/dV 2
spectrum in the absence of microwaves (see Fig. 2 of Ref. [971])12 . This
can be understood from Eq. (20.8), which tells us that the correction to
the dc current is proportional to the d2 I/dV 2 spectrum without radiation.
Notice that such relation can also be derived from Eq. (20.10) in the limit
of small VJ by expanding the integrand up to second order in VJ . The close
relation between the induced dc current and the d2 I/dV 2 spectra was also
found in the case of transport through individual MnCO molecules, which
constitutes a convincing proof of the fact that the rectification mechanism
dominates the transport in irradiated atomic-scale junctions at microwave
frequencies.
12 The main difference between this experiment and previously reported ones was the use
of low temperatures (∼ 18 K) that made possible to measure directly d2 I/dV 2 spectra.
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596 Molecular Electronics: An Introduction to Theory and Experiment

Fig. 20.5 Differential conductance spectra of a single Mn atom adsorbed on NiAl(110)

surface with constant microwave input (no amplitude modulation). (b) The spectrum
with VIN = 5 mV was fitted numerically (line) to extract VJ , the microwave amplitude
across the STM junction. (c) Plot of extracted VJ vs VIN . The line is a linear fit with the
constraint of zero intercept, which yields VJ = 45.5 × VIN . Reprinted with permission
from [971]. Copyright 2006, American Institute of Physics.

Let us emphasize that the occurrence of PAT (or rectification) in the

microwave regime has been firmly established in a variety of nanostructures
[214], including carbon nanotube quantum dots [972], which are very closely
related to our systems of interest.
Let us turn now to experiments in the optical regime. Recently, two
different experiments have explored the influence of laser light on the trans-
port through gold atomic contacts. The first one has been performed by
the group of one of the authors [973]. In this case, the microfabricated ver-
sion of the MCBJ technique was employed to fabricate gold atomic-sized
contacts at room temperature. As a light source an argon-krypton cw laser
was used, which allows to select a wavelength in the range between 480
nm and 650 nm. Moreover, pulsed light was used (with pulse durations of
∼ 700 µs) to avoid irreversible deformations of the atomic junctions.13 The
conductance with and without light was measured simultaneously during
the opening and closing of the atomic bridges. Fig. 20.6 shows an example
of the results obtained for green light with λ = 515 nm. Notice that the

13 Itwas found that continuous irradiation of the devices with λ = 488 nm for several
seconds with a power of a few mW results in irreversible conductance changes.
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Optical properties of current-carrying molecular junctions 597

16
150
14
P = 1.9 mW
laser

= 515 nm

ctance
12 sample condu

LIS

10 100

8 970 980 990 1000

G/G [%]
G0]

8 150
G [

i
i

6
6 100 50
4 4
50
2 2
0 0
0 0
880 900 920 940 960 980 1000
Time [s]

Fig. 20.6 Conductance and light-induced relative conductance change ∆G/Gi versus
time when opening the break junction continuously. Inset: close-up of the few-atom
region with Gi < 9 G0 . Reprinted with permission from [973]. Copyright 2007 by the
American Physical Society.

relative change14 in the conductance induced by the light is positive and it

can reach up to more than 100% for G ≈ 2 G0 .
A statistical analysis revealed the following important findings: (i) Illu-
mination always results in an enhancement of the conductance (∆G > 0),
(ii) ∆G is usually smallest in the tunnel regime (when the contacts are bro-
ken) and (iii) ∆G depends very much on the wavelength of the light, the
size and geometry of the contact and even on the exact spot in which laser
light is focused on. In particular, the largest enhancements were found for
λ = 488 nm with ∆G reaching up to 200%. However, for longer wavelengths
the enhancements were typically below 20-30%. Findings (i) and (ii) rule
out thermal expansion as a dominant mechanism in these experiments. On
the other hand, the fact that ∆G > 0 and the order of magnitude of the
light-induced correction are compatible with the PAT mechanism explained
in the previous section [221]. However, a quantitative comparison was not
possible because the theoretical analysis of Ref. [221] focused on the case of
single-atom contacts, which are not easy to stabilize at room temperature
with the MCBJ technique.
Such a quantitative comparison with the theory has been done by Ittah
et al. [974]. This group has developed a new method to form atomic contacts
14 This relative change is defined as ∆G ≡ (Gf − Gi )/Gi , where Gf is the conductance
under illumination and Gi the conductance without light.
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598 Molecular Electronics: An Introduction to Theory and Experiment

Fig. 20.7 Conductance measurements in 1 G0 gold contacts under laser irradiation. A

comparison between the conductance-enhancement as a function of laser power for two
contacts (marked by dark and open circles) under irradiation with two wavelengths. The
solid lines correspond to a fit with Eq. (20.6) using α as an adjustable parameter (see
text). Adapted with permission from [974]. Copyright 2009 American Chemical Society.

in which the gold wires are fully anchored onto Si/SiO2 substrates [975]. As
a result the atomic junctions are mechanically highly stable even at room
temperature, and under irradiation their heat dissipation characteristics
are far more efficient than those of suspended MCBJs, resulting in only
residual heating. Thanks to this method the authors were able to carry out
a detailed study of the influence of laser light in the conductance of gold
atomic contacts with conductances equal to 1 G0 .
In this work, the junctions were irradiated with three different lasers
with wavelengths of 532 nm (2.33 eV), 658 nm (1.88 eV), 781 nm (1.58 eV).
The maximum used power of the lasers was ∼ 20 mW, all measurements
were performed under ambient conditions at room temperature and the
junctions were placed with their long axis parallel to the laser polarization.
Fig. 20.7 shows representative results of the conductance as a function
of laser intensity for two different contacts and two different wavelengths.
Notice that in the absence of light the conductance (measured at 30 mV)
is ∼ 1 G0 . In all cases the conductance is enhanced by laser irradiation and
the relative changes, which increase with decreasing wavelength, are below
10%. In order to establish a comparison with the results of PAT theory,
Eq. (20.6) was used with the transmission curve of Fig. 20.3(a). The results
for different values of α = eVac /~ω are shown in Fig. 20.3(b). Using α
as an adjustable parameter the authors were able to fit the experimental
results with a reasonable accuracy, see solid lines in Fig. 20.7. Notice in
particular the nonlinear behavior, which is a remnant of the Bessel functions
of Eq. (20.6).
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Optical properties of current-carrying molecular junctions 599

These results suggest that the irradiated transport at the wavelengths

in Fig. 20.7 is dominated by the PAT mechanism. However, for the 532
nm (2.33 eV) laser, deviations from the PAT theory were found. This
is understandable since at this wavelength the absorption of Au is not
negligible anymore (reflectance ∼ 0.64 to be compared with 0.97 and 0.95
for Au at 781 and 658 nm, respectively). In this case, the enhancement of
the conductance could additionally be due to the generation of hot electrons
(see discussion in section 20.2). Such mechanism was termed photoinduced
transport (PIT) in Ref. [974], where the authors presented a theoretical
estimate of the contribution of photo-excited electrons (hot electrons) to
the conductance and they showed that it could account for the discrepancy
with PAT theory. Further support for the important role of PIT under the
532 nm laser was the finding of a linear dependence of the conductance on
the laser power, which is expected from this mechanism.
In addition to the SERS experiments described in section 20.1, Natel-
son’s group reported transport experiments in which significant dc currents
in electromigrated molecular junctions under illumination were observed
using different molecules such as para-mercaptoaniline (pMA) and fluori-
nated oligomers (FOPE) [607]. According to the authors, their observations
are consistent with rectification at optical frequencies. Let us briefly repeat
the arguments of Ref. [607]. In the presence of an oscillating potential
V (t) = V + Vac cos(ωt) the current at small ac amplitudes can be written
via a Taylor expansion as
1 ∂2I
µ ¶ µ ¶
∂I
I(t) = I(V0 ) + Vac cos(ωt) + V 2 cos2 (ωt) + · · · .
∂V V 2 ∂V 2 V ac
(20.11)
2
Applying the trigonometric identity 2 cos (ωt) = 1 + cos(2ωt), we see that
the current nonlinearities lead to a second-harmonic ac signal as well as
an additional dc current, both linearly proportional to ∂ 2 I/∂V 2 . Notice
that the expression of the additional dc current was already obtained in
Eq. (20.8). This latter relation suggests that a comparison between mea-
surements of the ac current at 2ω and the correction of the dc current could
be used to test the occurrence of the rectification mechanism. In the case
of ideal rectification, Eq. (20.11) tell us that the ratio of those two currents
should be equal to one. Obviously, the current at 2ω can only be measured
at frequencies much lower than the optical laser frequencies (e.g. 200 Hz),
which weakens the usefulness of this test.
The measurements showed that the dc current under illumination
changes proportionally to the low frequency ac current at 2ω, which is
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600 Molecular Electronics: An Introduction to Theory and Experiment

an indication of the occurrence of rectification, although it does not ex-

actly follows the dependence expressed in Eq. (20.11). On the other hand,
the induced dc currents at optical frequencies were found to depend lin-
early on the incident intensity, which is again consistent with the optical
rectification mechanism. However, that this linear dependence (at small ac
amplitudes) is also expected from other mechanisms like the PIT mentioned
above. Therefore, the interpretation of these experimental results in terms
of rectification is not unambiguous.
The same group repeated similar experiments but this time at lower
temperature (80 K) and with radio frequencies (10 MHz). In this case it
was possible to measure directly the second derivative of the current in the
absence of radiation and it was shown that the rectified current followed
exactly Eq. (20.8) or Eq. (20.11). This is again a demonstration that at
low frequencies the rectification mechanism dominates the transport.
The last experiment that we shall mention in this section has been per-
formed by Ho’s group [976]. In this case the laser-assisted transport was
studied in a single-molecule double-barrier junction that was defined by
positioning a STM tip over an individual molecule adsorbed on a thin (∼
0.5 nm) insulating alumina film grown on a NiAl(110) surface. The two
tunnel barriers in the junction are the vacuum gap between the STM tip
and the molecule, and the oxide film between the molecule and NiAl. In
this case the target molecule was a magnesium porphine (MgP), a simple
metalloporphyrin molecule that is involved in photosynthesis, and the ex-
periments were conducted at low temperatures (∼ 10 K). In the absence of
laser illumination, the differential tunneling conductance (dI/dV ) spectra
were shown to exhibit stepwise changes (with well-defined threshold volt-
ages) and hysteresis. Upon illumination with three different lasers (532, 633
and 800 nm) the threshold voltage was shown to decrease linearly with the
photon energy, suggesting a resonant mechanism. The authors argued that
transport mechanism responsible for this behavior is a two-step process in-
volving excited states of the tip (i.e. photo-induced resonant tunneling), as
opposed to photon-assisted tunneling resonant tunneling from the tip di-
rectly to the molecule. In our opinion, without a quantitative analysis it is
not easy to discriminate between these two different mechanism since both
of them may lead to similar features in the current-voltage characteristics.
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Optical properties of current-carrying molecular junctions 601

20.5 Resonant current amplification and other transport

phenomena in ac driven molecular junctions

As it is clear from the previous section, photon-assisted tunneling is a crucial

transport mechanism in atomic-scale junctions even at optical frequencies.
However, it must be emphasized again that the PAT theory in the spirit of
Tien-Gordon original work has clear limitations and it is only valid when
the spatial dependence of the field-matter interaction along the junction
can be ignored. If such spatial dependence becomes important, which must
be always the case for large ac field amplitudes, the problem has to be
addressed with other formalisms such as the so-called Floquet theory (see
Ref. [211] for a review) or the NEGF approach detailed in section 8.3.
In most of the theoretical studies of the transport properties of irra-
diated molecular junctions the field-matter interaction is assumed to be
restricted to the molecular bridge. Moreover, the molecular wire is usually
described with a simple tight-binding Hamiltonian. This is represented
schematically in Fig. 20.8. In this type of models, the time-dependent
Hamiltonian that describes the molecular wire adopts a form like the fol-
lowing one [211]
N −1 ³ ´ X
c†n+1σ cnσ + h.c. +
X
Hwire (t) = ∆ [ǫn + xn a(t)]c†nσ cnσ , (20.12)
n=1,σ n,σ

where ∆ is the hopping matrix elements that describes the coupling of each
orbital to its nearest neighbors and ǫn stands for the one-site energies. The
time-dependent part in the second term of this Hamiltonian describes the
coupling to an oscillating dipole field that causes time-dependent level shifts
xn a(t), where xn = (N + 1 − 2n)/2 denotes the scaled position of site n.
The energy a(t), which is periodic in time, is determined by the electrical
field strength multiplied by the electron charge and the distance between
two neighboring sites.
At a first glance, one might have the impression that models based on
the Hamiltonian of Eq. (20.12) describe very different physics from the
Tien-Gordon PAT theory detailed above. However, one can show that
if the spatial dependence of the field-matter interaction in Eq. (20.12) is
neglected, i.e. if the driving shifts all the wire levels simultaneously, it is
possible to map the driving field by a gauge transformation to oscillating
chemical potentials. In other words, models based on Hamiltonians like
the one in Eq. (20.12) reproduce the simple Tien-Gordon-like results in the
limiting case of spatially homogeneous field-matter interaction.
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602 Molecular Electronics: An Introduction to Theory and Experiment

1
0
0
1 hω 0110
0
1 10
0
1
0
1 1010
0
1 ∆ 1010 ΓR
ΓL 1010 ∆
∆ ∆ 10
0
1
0
1 ε2 ε3 ε5
1010
0
1
0
1 ε1 ε4 1010
0
1
0
1 1010
L 101010 1010
1010
R
0
1
0
1
0
1 −2 −1 0 1 2
1010
0
1 x 10

Fig. 20.8 Schematic representation of the level structure of a molecular bridge with
N = 5 sites coupled to metallic electrodes.

The transport properties in models based on Eq. (20.12) are usually

computed within the Floquet theory. It is not our intention to describe
here this theory or other similar theoretical tools. Instead, the rest of the
section is devoted to a brief description of the main physical effects that
have been predicted with the help of models like the one in Eq. (20.12).
Most of these effects have not yet been confirmed experimentally, but there
are good reasons to believe that they will be observed in the near feature.
Resonant current amplification.– The application of laser fields in molec-
ular junctions can lead to resonant excitation of the molecular bridge states,
which in turn can be manifested as an enhancement of the dc current when
the driving field is in resonance. This phenomenon, sometimes referred to
as resonant current amplification, was already discussed in the context of
PAT in section 20.3.3. It was first predicted by several authors using mod-
els similar to that of Eq. (20.12). Thus for instance, treating the driving as
a perturbation, Keller et al. [978, 979] demonstrated that resonant electron
excitations result in peaks of the current as a function of the driving fre-
quency. Kohler et al. [980] studied the same problem including the driving
exactly within a Floquet master equation approach and later derived an an-
alytical expression [981]. In related work, Tikhonov et al. [968, 969] have
studied this problem with more realistic models for the molecular bridge
based on the extended Hückel approach. The central result of these studies
is that such resonant excitations enhance the current significantly. In par-
ticular, Kohler et al. [981, 211] have shown that at the resonant frequencies
the dc current decays linearly with the length of the molecule, in contrast
to exponential decay of the current in the absence of ac driving.
Ratchet effect and light-induced currents.– A widely studied phe-
nomenon in driven transport is the so-termed ratchet effect: the conver-
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Optical properties of current-carrying molecular junctions 603

sion of ac forces without any net bias into directed motion. In the context
of molecular junctions the question is whether it is possible to induce a
dc current with an ac field in the absence of a dc bias voltage. As we
commented in section 20.3.1, in the context of STM it is well-known that
this is indeed possible if there are left-right asymmetries in the junctions
[964, 965, 215, 216]. There, the ratchet effect is referred to as rectification
at zero dc bias. In the context of molecular junctions, this issue has been
extensively studied by Lehmann et al. [982–984] and the main results have
been reviewed in Ref. [211]. These authors have shown that it possible
to generate a dc current with a pure ac driving (i.e. a photocurrent) by
introducing certain asymmetries in the problem.15 For instance, using a
conductor with an asymmetric level structure, one can generate a dc cur-
rent even with a purely harmonic dipole driving. Another possibility is to
use a driving field in which several frequencies are mixed. In this case, a
dc current is generated even in spatially symmetric molecules bridges.
Related to the ratchet effect, Galperin and Nitzan [977] have predicted
that light-induced current in unbiased junctions (i.e. photocurrents) can
flow when the bridging molecule is characterized by a strong charge-transfer
transition. Such a current reaches its maximum when the light frequency
matches the internal transition frequencies of the molecule. Using realistic
estimates of molecule-lead coupling and molecule-radiation field interaction,
these authors showed that such an effect should be observable.
Coherent destruction of tunneling.– As we saw in our discussion of PAT
in section 20.3, the current as a function of the amplitude of the ac driv-
ing is modulated according to the behavior of the Bessel functions, see
Eqs. (20.4) and (20.5). If the parameter α is such that J0 vanishes, there
is a pronounced reduction of the current [see Exercise 20.2(iv)]. This phe-
nomenon appears in many different ac driven systems and it is known as
coherent destruction of tunneling [985]. For a detailed discussion of this
phenomenon in the context of molecular wires, see section 7 in Ref. [211]
and references therein.
Role of electron excitation in the leads.– As we discussed in section 20.2,
apart from modulating the electronic levels in the leads, the electromagnetic
field can produce hot electrons in the leads by direct photon absorption.
These electronic excitations can in turn contribute to the transport. Sim-
ple estimates of the contribution of these inelastic processes to the total
current have been put forward long ago in the context of the STM [961].
15 Tobe precise, the generation of a photocurrent requires the breaking of the so-called
generalized parity [211].
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604 Molecular Electronics: An Introduction to Theory and Experiment

More recently, Galperin and coworkers have studied in detail the influ-
ence of electronic excitations in the leads on the current through molecular
transport junctions [962]. These authors have concluded that in certain
situations such excitations can give a significant contribution to the cur-
rent and moreover, this contribution can be distinguished from the direct
current because it scales differently with the distance between the molecule
and the leads.
Let us conclude this section by saying that the different physical effects
discussed in the previous paragraphs can also have a big impact in other
transport properties like shot noise. The theoretical activities along these
lines have been reviewed by Kohler et al. in Ref. [211].

20.6 Fluorescence from current-carrying molecular junc-

tions

In this section we shall face the question number 5 of our list in the in-
troduction of this chapter. When a sufficiently high bias voltage is applied
to a molecular junction, two molecular orbitals can be partially populated
and then optical transitions between them become, in principle, possible
with the subsequent light emission (or fluorescence). Is the light emission
from a single molecule measurable? If so, what can we learn about the
junctions from this local optical spectroscopy? The goal of this section is
to briefly describe the recent experimental and theoretical efforts devoted
to answer these and other basic questions related to the current-induced
light emission from single molecules in transport junctions.
The fact that electron tunneling can lead to emission of light was first
discovered by Lambe and McCarthy in 1976 in the context of metal-oxide-
metal tunnel junctions [986]. In the context of atomic-scale junctions, light
emission has been frequently observed in STM experiments. Thus for in-
stance, it has been reported in clean metal [987, 988] and semiconductor
surfaces [989], as well as for atomic and molecular adsorbates on metal sub-
strates [990–993]. However, often the reported photon emission spectra do
not show identifiable molecule-related features [992]. On a metal surface,
the electronic levels of a molecule are considerably broadened whereas light
emission is strongly quenched, making it difficult to detect and identify any
molecule-specific emission.
In recent years, it has been demonstrated by means of STM experi-
ments that electric-current flow through a molecule may indeed cause the
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Optical properties of current-carrying molecular junctions 605

molecule to luminesce due to electronic transitions [994, 995]. Photon emis-

sion from single-molecule contacts had already been discussed by Buker
and Kirczenow in 2002 [996], and the appearance of those experiments has
motivated additional theoretical work [977, 952, 997]. However, a theoret-
ical understanding of single-molecule electroluminescence is in the earliest
stages and contact between theory and specific experiments is just begin-
ning to be made [998].
The basic idea of molecular electroluminescence as observed in STM ex-
periments is as follows: By positioning a STM tip above a single molecule on
a substrate and applying a bias voltage between the tip and the substrate,
electron transmission through the molecule may occur, mediated by the
molecule’s electronic orbitals, and the molecule may be found to luminesce.
In a simplified picture, when a bias voltage is applied, the molecule moves
out of equilibrium with a flux of electrons passing through it. If two molec-
ular orbitals are located in the energy window between the electrochemical
potentials of the STM tip and substrate, they will both be partially occu-
pied and if optical transitions between them are not forbidden, transitions
from the higher-energy orbital to the lower-energy orbital will occur re-
sulting in photon emission [996]. Such optical transitions will most likely
involve vibrational levels of both electronic states. This is schematically
represented in Fig. 20.9 as process B.
Molecular fluorescence always competes with the light emission channel
known as inelastic electron tunneling that takes place even in the absence of
molecules [961, 999, 1000]. This latter mechanism involves inelastic tun-
neling from the tip electronic states into the lower-lying states of the sample
with a simultaneous release of the excess energy in the form of a plasmon.16
The excited plasmon then decays into a far-field photon. The spectrum of
this emission is typically quite broad and has a characteristic energy cut-off
determined by the sample bias. This process is described schematically in
Fig. 20.9 (process A).17
In order to avoid the quenching of molecular fluorescence, the metal-
molecule coupling strength has to be reduced [996]. This was achieved by
Ho’s group [994] by adsorbing porphyrin molecules on an ultrathin alumina
film grown on a NiAl(110) surface and using the STM as a second weakly
16 By plasmon we mean here an electromagnetic mode of the tip-substrate system.
17 Other light-emitting processes involving the injection of hot electrons or the creation of
electron-hole pairs are also possible [961]. For photon energies ~ω < eV , where V is the
bias voltage, the single-electron process described above dominates the light emission.
However, some of the additional processes have no threshold voltage and therefore, they
can be responsible for the light emission with photon energies ~ω > eV [386].
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606 Molecular Electronics: An Introduction to Theory and Experiment

EF B
hω
A
eV
hω

EF

Tip Vacuum Molecule Substrate

Fig. 20.9 Diagram showing the two major processes contributing to STM-excited light
emission from a molecule adsorbed on a surface. In process A, the inelastic electron
tunneling channel, an electron tunnels from the Fermi level of the STM tip into an
unoccupied molecular orbital with simultaneous excitation of a plasmon. In process B,
the fluorescence channel, an electron tunnels into the higher unoccupied orbital of the
molecule. The charged molecule typically relaxes to a lower vibrational level of the same
electronic level, with subsequent radiative (excitation of a plasmon) transition to the
lower electronic level. The final step involves tunneling of this extra electron into the
substrate.

coupled electrode. Some characteristic photon emission spectra obtained

in this work are reproduced in Fig. 20.10. The results of panels A and B
were obtained with the STM positioned directly above a molecule. As one
can see, the spectra exhibit sharp features as compared with those related
to light emission from NiAl and oxide film (also shown in panels A and B).
Furthermore, the light-emission on top of the molecules was found to be
very sensitive to the tip position inside the molecule, which indicates that
the emission has submolecular resolution.
The series of emission spectra for different sample biases shown in panels
C and D in Fig. 20.10 further clarifies the nature of the observed spectral
features. In panel C, the spectral peaks do not shift when the bias voltage
is varied. This result indicates that these peaks did not originate from tran-
sitions between the electronic states of the tip and those of the substrate.
They were attributed to transitions inside the molecule. The existence of a
cut-off voltage (approximately 2 V in panel C) for excitation of the sharp
features is expected if an excited electronic level of the molecule participates
in the emission. The difference between this cut-off voltage and the photon
energy of the shortest-wavelength feature in the spectra (∼ 1.57 eV), lies
in the range of the low-energy dI/dV peak for this molecule, which further
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Optical properties of current-carrying molecular junctions 607

Photon energy (eV)

1.78 1.55 1.38 1.24 2.07 1.78 1.55 1.38
A 1000 B
2000

750 1
1500

3
500
1000 NiAl
1
2 2
500 3 250
Photon counts

NiAl 4
4
oxide oxide
0 0
C D
Counts

1500 2.4V 800

2.3V 0.0 0.4 0.8

Current (nA)
1000
400
2.2V
500
2.3V 2.1V
2.1V 2.2V
2.0V
0 0
700 800 900 1000 600 700 800 900
Wavelength (nm)

Fig. 20.10 (A-B) Light-emission spectra acquired on porphyrin molecules on an ultra-

thin alumina film grown on a NiAl(110) surface using the STM. The spectra acquired
on bare NiAl and Al2 O3 /NiAl(110) surfaces are also shown for comparison. The spectra
are offset vertically for clarity. Series A and B were taken with two different Ag tips. In
(A), the spectra were acquired at a voltage bias Vbias = 2.35 V and a current I = 0.5
nA, with an exposure time of 100 s; the NiAl and oxide spectra have been multiplied by
factors of 4 and 15, respectively. In (B), Vbias = 2.2 V, I = 0.5 nA, and exposure time
= 300 s; the oxide spectrum has been multiplied by a factor of 3. (C) Variation of curve
1 in (A) as a function of Vbias [the same tip was used as in (A)]. The inset shows the
dependence of the 800-nm peak intensity on current (Vbias = 2.35 V). Linear dependence
was found for all wavelengths in the measured spectral region. (D) Variation of curve 1
in (B) as a function of Vbias [the same tip was used as in (B)]. From [994]. Reprinted
with permission from AAAS.

supports this interpretation.

The authors concluded that the total light emission was a result of the
contribution of the two main processes discussed above (see Fig. 20.9). A
careful analysis of the emission spectra revealed that in the optical transi-
tions between two electronic states of the molecule, most likely the molecule
relaxed to the vibrational ground state of the excited electronic state be-
fore exciting a plasmon. Moreover, the vibrational features observed in
the light-emission spectra were found to depend sensitively on the different
molecular conformations and the corresponding electronic states.
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608 Molecular Electronics: An Introduction to Theory and Experiment

In another remarkable experiment Dong et al. [995] observed intrinsic

molecular fluorescence from porphyrin molecules on Au(100) by using a
nanoscale multimonolayer decoupling approach with nanoprobe excitation
in the tunneling regime. They observed well-defined vibrationally resolved
fluorescence excited by STM that matched nearly perfectly with the stan-
dard photoluminescence data of the molecule. The linewidths of spectral
peaks were found to narrow down with increased thickness, i.e. by making
the junctions more symmetric. On the other hand, a quantum efficiency of
∼ 10−5 photons per tunneling electron was obtained for the molecular fluo-
rescence at both polarities. Interestingly, emission of photons with energies
exceeding the energy of tunneling electrons was reported. The authors at-
tributed tentatively this phenomenon to an excitation mechanism via hot
electron injection from either tip or substrate.
From the theory side, simple bridge models have been used to elucidate
the basic facts related to the fluorescence of current-carrying molecular
junctions. Buker and Kirczenow [996] predicted that the photon emission
rate is more sensitive than the electric current to coupling asymmetries
between the molecule and contacts. They also showed that electrolumines-
cence may be used to measure the HOMO-LUMO gap and the location of
the Fermi level of the contacts relative to the HOMO and LUMO. This has
already become clear from our description of the experiments above.
On the other hand, Galperin and Nitzan [977, 952] have used the NEGF
formalism in combination with a simple two-level model to compute the de-
pendence of the emission rate on essential parameters such as bias voltage,
metal-molecule coupling strength and level separation. In particular, they
have derived a very transparent and intuitive relation in which the emission
rate is expressed in terms of the level occupations.
On the way to more quantitative descriptions of current-induced single-
molecule light emission, Harbola et al. [997] have developed a nonequilib-
rium superoperator Green’s function theory which can be combined with
DFT. More recently, Buker and Kirczenow [998] have presented a detailed
analysis of the experiments of Ref. [994] where the electronic structure cal-
culations were done using the extended Hückel approach.

20.7 Molecular optoelectronic devices

Time has come to address the last question posed in the introduction.
One of the dreams in molecular electronics is to use the amazing optical
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Optical properties of current-carrying molecular junctions 609

properties of molecules to develop novel optoelectronic devices. One idea is

to use compounds in which the excited states have very different character
as compared with the ground state. Thus, upon optical excitation one
could in principle populate those states and in turn change the conduction
properties of the junctions in which they are embedded. The problem with
this idea and similar ones is the need to continuously inject light into the
molecules to induce the transitions. As commented above, at present this
is challenging and the fundamental limitations, if any, are not known.
A class of molecules known as photochromic molecules offers a way
out for this problem. Photochromism is defined as a reversible photo-
transformation of a chemical species between two forms having different
absorption spectra. During the photoisomerization, not only the absorp-
tion spectra but also various physicochemical properties change, such as
the refractive index, dielectric constant, oxidation/reduction potential, and
geometrical structure. These molecular property changes can be exploited
in various photonic devices, such as erasable optical memory media and
photo-optical switch components, and they could also lead to applications
in the context of molecular electronics. A key idea is that photoisomer-
ization does not require to continuously photo-excite a molecule, and the
absorption of a single photon is enough to trigger its transformation.
There are many photochromic compounds, sometimes referred to as
photochromic molecular switches, but two classes of them have attracted
special attention in the field of molecular electronics. The first one is formed
by azobenzene and its derivatives. Azobenzene is composed of two phenyl
rings linked by a N=N double bond. One of the most intriguing properties
of azobenzene is the photoisomerization of trans and cis isomers. The two
isomers can be switched with particular wavelengths of light: ultraviolet
light for trans-to-cis conversion and blue light for cis-to-trans isomerization.
This is schematically represented in Fig. 20.11(a). The cis isomer is less
stable than the trans one and thus, cis-azobenzene thermally relaxes back
to the trans via cis-to-trans isomerization.
A second promising group of switches is formed by diarylethene
molecules, which were pioneered by Irie [1001, 1002]. These molecules
can be converted from a conjugated (“on” or closed state) to a cross-
conjugated (“off” or open) state upon illumination in the visible region,
see Fig. 20.11(b). The reverse process is possible with ultraviolet (UV)
light. Diarylethenes have additional attractive properties. First and fore-
most, they are fatigue resistant. Furthermore, their length change upon
isomerization is negligible. This allows for minimal mechanical stress when
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610 Molecular Electronics: An Introduction to Theory and Experiment

(a) (b)

Fig. 20.11 Photochromic molecular switches. (a) Azobenzene photoisomerization.

Dithienylethene (a member of the diarylethene class) photochemistry.

a molecule between two electrodes changes conformation.

Light-induced switching of conductance of a molecular junction based
on a photochromic molecule was first reported by Dulić et al. [587]. In
this work the MCBJ technique was used to investigate the transport of
single thiophene-substituted diarylethenes. The switching of the molecules
was observed from the conducting (closed) state to the insulating (open)
state upon illumination with visible light (546 nm). This switching results
in a significant resistance increase of over two orders of magnitude. How-
ever, the reverse process, which should occur upon illumination with UV
light (313 nm), was not observed. This one-way switching was further con-
firmed by means of UV/Vis spectroscopy to measure absorption of these
molecules self-assembled on gold. The authors attributed this observation
to quenching of the excited state of the molecule in the open form by the
presence of gold. Additional support for these conclusions was obtained by
the same group in STM measurements on monothiol thiophene-substituted
diarylethene switches in a dodecanethiol matrix [1003].
Theoretical studies have pointed out that the possibility to switch re-
versibly depends critically on the linker used [1004–1006]. In this sense,
it is believed that the one-way switching of the experiments above might
be due to the strong electronic hybridization between molecule and metal.
Indeed, He et al. [1007] reported photoisomerization in both directions for
diarylethenes with a methyl spacer and a phenyl linker in the para posi-
tion. In this case the transport data were obtained with a break junction
method and the authors reported single-molecule resistances of 526 ± 90
MΩ in the open form and 4 ± 1 MΩ in the closed form. It is important
to emphasize that the resistances of the two isomers were measured inde-
pendently, i.e. no conductance switching was observed in situ in the same
junction. In this experiment, the photoisomerization was demonstrated
with optical spectroscopy of self-assembled monolayers of these molecules
on gold surfaces. The crucial role of the linker was further illustrated by
Katsonis et al. [1008]. These authors demonstrated in STM experiments
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Optical properties of current-carrying molecular junctions 611

light-controlled reversible conductance switching for meta-phenyl-linked di-

arylethenes on gold. This is drastically different from the behavior of the
switch mentioned above [587], which had a thiophene linker. Interestingly,
the meta-phenyl spacer forms a cross-conjugated system, whereas the thio-
phene is fully conjugated with the switching unit. This suggests that the
reversibility of switching is directly related to the conjugation between the
switching unit and the substrate.
In a remarkable experiment, Whalley et al. [1009] used single-walled
carbon nanotubes (SWNTs) to contact single (or a few) diarylethene
molecules. These authors showed that the thiophene-based devices can be
switched from the insulating open form to the conductive closed form but
not back again, as in the experiments of Dulić et al. [587]. However, pyrrole-
based devices were shown to cycle between the open and closed states. In
particular, in a device with semiconducting nanotubes and pyrrole-based
diarylethenes initially in the open state, it was found that with UV irra-
diation the bridge transforms to the closed state and the current increases
by more than 5 orders of magnitude. Irradiation with visible light did not
restore the initial, low-conductance state; however, the low-conductance
state reappeared when the device aged at room temperature overnight.
The on/off cycle can be toggled many times.
One of the most impressive examples of reversible conductance switch-
ing in molecular devices has been reported by Kronemeijer et al. [1010].
In this work junctions with self-assembled monolayers of photochromic
diarylethene-based switches were studied. Large-area molecular junctions
were processed in vertical interconnects in an insulating photoresist matrix,
see left panel of Fig. 20.12. The diarylethene monolayer is self-assembled
in the individual interconnects and topped off with a highly conductive
organic top electrode. This organic top electrode is used to prevent the
formation of short-circuits from top to bottom electrode. Then, upon irra-
diation with a specific wavelength range, the conductance of these devices
can be optically switched. The major advantage of this approach is that
the two distinct isomers of the diarylethene can be individually synthesized
and, therefore, separately assembled in a device. Consequently, the ON
and OFF state can be independently measured in the devices, without any
involvement of a switching event. Optically induced switching of the con-
ductance of the devices in between these two states then provides a direct
proof of the molecular origin of the switching events.
In Fig. 20.12 (right panel) we reproduce the results of this experiment
for the current density versus voltage (J-V ) for devices with molecules ex-
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612 Molecular Electronics: An Introduction to Theory and Experiment

Fig. 20.12 Left panel: Schematic cross section of the device layout of a large-area
molecular junction in which the diarylethene is sandwiched between Au and poly(3,4-
ethylenedioxythiophene): poly(4-styrenesulphonic acid) (PEDOT:PSS)/Au. Using UV
(312 nm) illumination the open, nonconjugated isomer can be converted to the closed,
conjugated isomer. Visible irradiation of 532nm reverses the photoisomerization process.
Right panel: Current density (J) versus voltage (V ) of the closed (circles) and open
(squares) isomers as self-assembled in the molecular junctions, and J-V characteristics
of the junctions with the open isomer self-assembled and subsequently photoisomerized
to the closed isomer with UV irradiation (triangles). Averaged data (at least 35 de-
vices) from devices with diameters of 10-100 mm. Error bars by standard deviation.
Reproduced with permission from [1010]. Copyright Wiley-VCH Verlag GmbH & Co.
KGaA.

clusively in the open or closed states. The conductance through the closed,
more-conducting state of the switch is shown to be 16 times higher at 0.75
V bias. On the other hand, devices with the open isomer were illuminated
for 15 min with 312 nm UV irradiation to convert the molecular switches in
the devices to the closed isomer. The J-V characteristics of the converted
open-state isomer after UV irradiation show an increase of the conductance
through the monolayer, as expected from the devices with closed isomers
present, see Fig. 20.12 (right panel). Following UV irradiation and consecu-
tive measurements, these devices were illuminated with 532 nm irradiation
(visible light) to achieve ring opening of the switches in the SAM. The
observed J-V characteristics show a significant decrease (by a factor 3)
of the conductance upon visible light irradiation, but the conductance of
the devices with the open isomer is not fully recovered. The origin of this
behavior was not fully understood (see Ref. [1010] for more details).
Let us mention that light-controlled conductance switching of molec-
ular devices based on photochromic diarylethene molecules has also been
demonstrated by van der Molen et al. [135]. In this case, the devices con-
sisted of ordered, two-dimensional lattices of gold nanoparticles, in which
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Optical properties of current-carrying molecular junctions 613

neighboring particles are bridged by the switchable molecules. In this work,

it was independently confirmed by means of optical spectroscopy that re-
versible isomerization of the diarylethenes employed is at the heart of the
room-temperature conductance switching.
As mentioned above, azobenzene and its derivatives are also possible
candidates for molecular switches [1011]. These molecules perform a cis-
trans isomerization upon illumination, accompanied by a significant change
in molecular length and dipole moment. Precisely, this significant change in
the length of the molecule can be a problem to fabricate photo-switchable
metal-molecule-metal junctions based on these compounds. Indeed, the ex-
periments reporting isomerization and reversible photo-switching of azoben-
zene molecules have been performed with these molecules bound to metallic
surfaces [1012–1015], i.e. without a second electrode. In these experiments a
STM (or a conductive AFM) tip was just used to probe the conformational
changes of the molecules after isomerization.

20.8 Final remarks

As we have seen in this chapter, the interplay between electromagnetic fields

and molecular transport junctions gives rise to fascinating possibilities and
new physical phenomena. We have shown that thanks to the phenomenon
of surface-enhanced Raman scattering the spectroscopy of single molecules
in transport junctions is indeed possible. The challenge is now to com-
bine SERS and transport measurements at low temperatures, which could
provide an unprecedented characterization of molecular junctions.
On the other hand, we have seen that different transport mechanisms
can play a role in the transport properties of irradiated junctions. The
existent experiments in atomic and molecular contacts suggest that the
photon-assisted tunneling mechanism (or rectification) is at work even at
optical frequencies, although more experiments are certainly needed to con-
firm it.
From the theory side, many novel transport phenomena have been pre-
dicted and await for experimental confirmation. The problems to observe
them are related both to the difficulty of injecting light in such tiny objects
surrounded by metallic electrodes and to the fact that other effects can
mask their appearance.
Current-induced light emission is an extraordinary physical phe-
nomenon that can provide the most direct spectroscopy of the electronic
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614 Molecular Electronics: An Introduction to Theory and Experiment

states in a molecular junction. So far, there have been very few experi-
ments where molecular signatures have been unambiguously identified in
the fluorescence spectra and more experiments are needed. For the theory
the challenge is now to provide quantitative predictions that can be directly
compared with the experiments.
The last topic that we have discussed in this chapter is very important
from the technological point of view. Molecular electronics is often seen as a
field that aims at reproducing the standard microelectronic components and
devices, but at a smaller scale. However, the future of molecular electronics
depends crucially on our ability to provide devices with new functionalities
out of the scope of more traditional technologies. The optical properties
of many molecules may offer a down-to-earth possibility for the future. In
principle, they can be used in transport junctions to control the current at
will and many researchers believe by now that molecular optoelectronics
will soon grow as a field of its own. However, so far it has been difficult
to take advantage of those optical properties and the only successful imple-
mentations have made use of photochromic molecular switches. At present,
it is not clear whether the difficulties encountered so far are just of techni-
cal nature or there are true fundamental limitations. In any case, the next
years will be certainly exciting for the scientists working on this subject.

20.9 Exercises

20.1 Photon-assisted tunneling in atomic gold chains: In the Exercise 7.5

a tight-binding model was used to show that the conductance of atomic chains
can exhibit parity oscillations, i.e. that it depends on whether the number of
atoms N in the chain is even or odd. Use that model to show that the sign of
correction to the linear conductance due to irradiation, ∆Gdc , can also exhibit
an even-odd effect. In particular, show that for low frequencies ∆Gdc < 0, if N is
odd; while ∆Gdc > 0, if N is even. Hint: For the last task use the low-frequency
formula of Eq. (20.9).
20.2 Photon-assisted tunneling in molecular junctions: The goal of this
exercise is to gain some insight into PAT in molecular junctions. For this purpose,
let us assume that the transmission function of a molecular junction is given by
the following double Lorentzian18

2
X 4ΓL ΓR
τ (E) = ,
i=1
(E − ǫi )2 + (ΓL + ΓR )2
18 This model was extensively used in section 19.3 to describe the thermopower in molec-
ular junctions.
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Optical properties of current-carrying molecular junctions 615

were ǫ1 and ǫ2 is the energy of HOMO and LUMO, respectively, and ΓL and
ΓR the broadenings by contacts L and R. For the sake of concreteness, we
shall assume throughout this exercise that ǫ1 = −7.0 eV, ǫ2 = −3.0 eV and
ΓL = ΓR = 30 meV.
(i) Use the second equality in Eq. (20.9), i.e. the classical rectification formula,
to analyze the low-frequency radiation-induced correction to the conductance
(∆Gdc ) as a function of the Fermi level position (EF ). In particular, show that
∆Gdc is negative only when EF is very close to one of the frontier orbitals.
(ii) Let us now assume that the Fermi level lies in the middle of the HOMO-
LUMO gap, i.e. EF = −5.0 eV and consider the limit of small ac amplitudes
(α ≪ 1). Compute ∆Gdc as a function of photon energy and show that for
~ω & 0.5 eV, the classical rectification formula fails to describe the correction to
the conductance given by the first equality in Eq. (20.9).
(iii) Now assume that EF = −6.0 eV and use Eq. (20.6) to compute ∆Gdc
as a function of the photon energy in the interval ~ω ∈ [0 eV, 4 eV] for different
values of α. For α ≪ 1 you will find the appearance of two peaks at 1 and 3 eV.
What is the origin of these peaks? Finally, compare the results obtained with the
exact formula of Eq. (20.6) and with the approximation of Eq. (20.9) to establish
in which range of α this approximation is valid.
(iv) One of the key signatures of PAT is the appearance of additional steps
in the I-V characteristics. Assume that EF = −6.0 eV and ~ω = 0.5 eV and
use Eq. (20.5) or Eq. (8.74) to compute the I-V curves and the corresponding
differential conductance for α = 0, 1, 2, 4. Discuss the origin of current steps (or
the corresponding peaks in the differential conductance) induced by the radia-
tion. Finally, analyze the phenomenon of coherent destruction of tunneling by
computing the I-V curves and differential conductance for values of α for which
J0 in Eq. (20.5) vanish, i.e. α/2 = 2.405, 5.520, . . . .
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616 Molecular Electronics: An Introduction to Theory and Experiment

January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Chapter 21

What is missing in this book?

At this stage in the development of molecular electronics it is already im-

possible to cover all the aspects of this multidisciplinary field in a single
monograph. Our selection of topics has been biased, as it could not be oth-
erwise, by our own backgrounds and research interests and we are aware of
the fact that important issues have been left out. Therefore, we would like
to close this manuscript by pointing out some of those topics and suggesting
some references where the reader can find information about them.
Among the topics not covered in this monograph, we believe that the
following ones are of special relevance:

• Molecules for molecular electronics: In section 3.2 we presented

a brief discussion about the typical molecules considered in molec-
ular electronics. We also mentioned briefly the electronic functions
for which they are well suited. A detailed discussion of this is-
sue can be found in several monographs and review articles, see e.g.
Refs. [583, 1002, 1016, 33, 47, 1017].
• Electron transfer: Electron transfer, the process by which an elec-
tron moves from one atom or molecule to another atom or molecule,
is one the simplest and most important reactions both in chemistry
and biology. In particular, electron transfer in donor-bridge-acceptor
complexes is in many respects very similar to the conduction of an
electron through a molecular transport junction.1 Thus, it is obvious
that molecular electronics can profit a lot from the much more ma-
ture field of electron transfer in chemistry and biology. In the sense,
researchers working in our field should at least know the basics of the
1 Although the driving forces are different in these two types of experiments, the mech-

anisms by which the electron is transferred through the bridge/molecular wire are essen-
tially the same.

617
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618 Molecular Electronics: An Introduction to Theory and Experiment

standard theory of electron transfer, known as Marcus theory. This

theory is nicely explained in the monograph of Kuznetsov and Ulstrup
[582], but we specially recommend chapters 16 and 17 of Nitzan’s book
on chemical dynamics in condensed phases [30]. For an introduction to
the techniques used to measure electron transfer rates, see Ref. [583].
This reference also contains countless experimental results for many
different chemical compounds. Finally, for a discussion of the connec-
tion between electron transfer rates and electrical conductance, see e.g.
Refs. [38, 830, 1018, 831].
• SAM-based molecular junctions: Since we are interested in the
basic conduction mechanisms in molecular systems, we have focused our
attention on the study of transport through single-molecule junctions.
However, the technological applications of molecular electronics will
surely come from devices containing of a large number of molecules,
like in the junctions based on self-assembled monolayers (SAM). In this
sense, there are many basic questions to be addressed like for instance
whether or not we can straightforwardly extrapolate the results for
single-molecule junctions to those devices.2 On the other hand, such
SAM-based devices require fabrication techniques that differ from the
ones described here. Some of these issues are discussed in the review
of Ref. [41].
• Scaling and integration of molecular devices: Important topics
for the future of molecular electronics, which are related to the pre-
vious issue, are the reliable mass production of molecular devices and
the integration of molecular junctions into macroscopic circuits. The
strategies and ideas explored so far in this respect have been reviewed
by Lu and Lieber in Ref. [1019].
• Carbon nanotubes: These molecules are considered to be something
in between a solid and a molecule and the study of their electrical and
thermal properties constitutes a field in its own. For this reason, we
have rarely talked about carbon nanotubes (CN) in this monograph.
However, it is obvious that many of the concepts, ideas and techniques
that we have discussed here are directly applicable to the problem of
transport through CN-based junctions. For a recent review on the
electronic and transport properties of carbon nanotubes, see Ref. [1020].

2 It is not obvious what is the role of inter-molecular interactions in these systems and in

some cases, the transport characteristics may differ significantly from the corresponding
ones of a single-molecule device.
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

What is missing in this book? 619

• Strongly correlated methods for molecular electronics: One of

the main challenges for the theory in our field is the development of
new methods that are able to describe properly the transport through
systems that exhibit strong electronic correlations, as it is often the case
in molecular junctions. Such methods should (and they will) replace
DFT in the near future as the main theoretical tool for the description
of transport in molecular junctions. We have not said much about
this topic in this monograph because those methods are still under
development and their performance has still to be established. For
recent advances on this subject we recommend [634, 1021–1023] and
references therein.
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620 Molecular Electronics: An Introduction to Theory and Experiment

January 12, 2010 11:27 World Scientific Book - 9in x 6in book

PART 5

Appendixes

621
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622
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Appendix A

Second Quantization

All the relevant systems in molecular electronics are composed of many

identical particles such as electrons, protons, phonons (or vibrations), etc.
As we all know, these particles obey their corresponding quantum statis-
tics depending on whether there are fermions or bosons. This statistics
is reflected in the symmetry of the many-particle wave functions. Thus
for instance, a fermionic wave function is expressed in the form of Slater
determinants to ensure its antisymmetry (Pauli’s exclusion principle).
The algebra with many-particle wave functions is quite cumbersome,
and it has been shown that the description of a many-body system can
be greatly simplified by using the so-called second quantization formalism
of quantum mechanics. This is just an alternative formalism (with no
new physics) in which the symmetry of the wave functions is transferred to
some convenient operators which fulfill simple commutation rules. Then, in
this formalism all the standard calculations can be done using the algebraic
properties of these operators, rather than using lengthy many-particle wave
functions.
The second quantization approach will be used throughout this book
and therefore, in order to make this manuscript more self-contained, we
have included a brief review of this formalism in this appendix. For a more
detailed discussion of the second quantization formalism, we recommend
the many-body textbooks of Refs. [173–176].

623
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624 Molecular Electronics: An Introduction to Theory and Experiment

A.1 Harmonic oscillator and phonons

A.1.1 Review of simple harmonic oscillator quantization

One convenient way to introduce the subject of creation and annihilation
operators, which is the essence of the second quantization formalism, is to
review the physics of a quantum-mechanical harmonic oscillator. Let us
consider a particle of mass m that is subjected to a one-dimensional (1D)
harmonic potential. The Hamiltonian describing this system can be written
as
p2 K
H= + x2 , (A.1)
2m 2
where x and p = −i~(∂/∂x) are the position and momentum operators,
respectively, which satisfy [x, p] = xp − px = i~.
To diagonalize this Hamiltonian, we introduce the frequency ω and the
dimensionless coordinate ξ:
K ³ mω ´1/2
ω2 = ; ξ=x . (A.2)
m ~
In terms of these new parameters, we can write the Hamiltonian simply as
∂2
µ ¶
~ω
H= − 2 + ξ2 . (A.3)
2 ∂ξ
The harmonic oscillator has a solution in terms of Hermite polynomials.
The corresponding eigenvalues are given by
µ ¶
1
Hψn = ~ω n + ψn , (A.4)
2
where n is an integer. From now on, we shall use the Dirac notation for
the eigenstate: |ni = ψn .
We now introduce two dimensionless operators as follows
µ ¶ ³ µ ¶
1 ∂ mω ´1/2 ip
a= √ ξ+ = x+ (A.5)
2 ∂ξ 2~ mω
µ ¶ ³ µ ¶
1 ∂ mω ´1/2 ip
a† = √ ξ− = x− .
2 ∂ξ 2~ mω
They are Hermitian conjugates of each other. They are sometimes called
raising and lowering operator (or ladder operators), but here we call them
creation (a† ) and annihilation (a) operators. In terms of these new opera-
tors, the Hamiltonian can now be written as
~ω £ †
aa + a† a .
¤
H= (A.6)
2
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Second Quantization 625

It is easy to show from their definitions that these operators satisfy the
following commutation relations
[a, a† ] = 1 ; [a, a] = 0 ; [a† , a† ] = 0. (A.7)
With these relations, the Hamiltonian adopts the following form
· ¸
~ω £ † †
¤ † 1
H= aa + a a = ~ω a a + . (A.8)
2 2
The three commutators above plus this Hamiltonian completely specify
the harmonic oscillator problem in terms of operators. With these four
relationships, one can show that the eigenvalue spectrum is indeed that of
Eq. (A.4). The eigenstates are
(a† )n
|ni = √ |0i, (A.9)
n!
where |0i is the ground state which obeys
a|0i = 0 (A.10)
and where the n! is for normalization. Operating on this state by a creation
operator gives
(a† )n+1
a† |ni = √ |0i = (n + 1)1/2 |n + 1i (A.11)
n!
the state with the next highest integer. In the same way, one can show that
a|ni = (n)1/2 |n − 1i, (A.12)
which shows that the annihilation operator a lowers the quantum number.
Then operating by the sequence
a† a|ni = a† (n)1/2 |n − 1i = n|ni (A.13)
gives an eigenvalue n, which verifies the eigenvalue relation A.4. Further-
more, using the original definition of Eq. (A.5) permits us to express x and
p in terms of these operators as
µ ¶1/2
~
x= (a + a† ) (A.14)
2mω
µ ¶1/2
m~ω
p=i (a† − a). (A.15)
2
The description of the harmonic oscillator in terms of operators is equiv-
alent to the conventional method of using wave functions ψn (ξ) of position.
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626 Molecular Electronics: An Introduction to Theory and Experiment

A.1.2 1D harmonic chain

With the material of the previous subsection, we are now in position to
discuss briefly our first many-body problem, namely the physics of phonons.
This will illustrate in an informal way the second quantization formalism
for bosons.
In a solid there are many atoms, which mutually interact. The vibration
modes are collective motions involving many atoms. A simple introduction
to this problem is obtained by studying the normal modes of an infinite
one-dimensional harmonic chain:
X p2 KX
i
H= + (xi − xi+1 )2 . (A.16)
i
2m 2 i
Here, we have assumed that an atom is only coupled to its nearest neighbors
and that all the atoms are identical.
The classical solution is obtained by solving the equation of motion:
−mẍj = mω 2 xj = K(2xj − xj+1 − xj−1 ). (A.17)
A solution is assumed of the form xj = x0 cos(kaj), where a is the inter-
atomic distance. Then, the normal modes have the solution
2K 4K
ωk2 = [1 − cos(ka)] = sin2 (ka/2). (A.18)
m m
To quantize the theory, let us impose canonical commutation relations
on the position and momentum of the lth and jth atoms: [xl , pj ] = i~δlj and
construct collective variables which describe the modes themselves (recall
k is wave vector, l is position):
1 X ikal 1 X −ikal
xl = √ e xk ; xk = √ e xl
N k N l
1 X −ikal 1 X ikal
pl = √ e pk ; pk = √ e pl , (A.19)
N k N l
which leads to canonical commutation relations in wave vector space:
1 X −ikal ik′ am
[xk , pk′ ] = e e [xl , pm ]
N
l,m
i~ X −ial(k−k′ )
= e = i~δk,k′ . (A.20)
N
l
Let us now express the Hamiltonian of Eq. (A.16) in terms of the new
variables. We have, with a little
X algebra,X
p2l = pk p−k (A.21)
l k
KX mX 2
(xl − xl+1 )2 = ωk xk x−k .
2 2
l k
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Second Quantization 627

Then the Hamiltonian may be written in wave vector space as

1 X mX 2
H= pk p−k + ωk xk x−k . (A.22)
2m 2
k k
Note that the energy is now expressed as the sum of kinetic plus po-
tential energy of each mode k, and there is no more explicit reference to
the motion of the atomic constituents. To second quantize the system, we
write down creation and annihilation operators for each mode k. We define
³ mω ´1/2 µ i
¶
k
ak = xk + p−k (A.23)
2~ mωk
³ mω ´1/2 µ ¶
k i
a†k = x−k − pk , (A.24)
2~ mωk
which can be shown, just as in the single harmonic oscillator case, to obey
commutation relations
h i h i
ak , a†k′ = δk,k′ , [ak , ak′ ] = 0, a†k , a†k′ = 0 (A.25)
and the Hamiltonian can be simply expressed as
µ ¶
1
~ωk a†k ak +
X
H= . (A.26)
2
k
These collective modes of vibrations are called phonons. They are the
quantized version of the classical vibrational modes in a solid. Each wave
vector state behaves independently, as a harmonic oscillator, with a possible
set of quantum numbers nk = 0, 1, 2, ... The state of the system at any time
is
Y Y (a† )nk
Ψ = |n1 , n2 , ..., nn i = |nk i = √k |0i (A.27)
k k
nk !
so that the expectation value of the Hamiltonian is
µ ¶
X 1
hHi = ~ωk nk + . (A.28)
2
k
In thermal equilibrium the states have an average value of nk which
is given in terms of the temperature β = 1/kB T by the Bose distribution
function:
1
hnk i ≡ Nk = β~ω ≡ nB (~ωk ). (A.29)
e k −1
So in summary, we have shown that the physics of these collective modes
can be described in terms of creation and annihilation operators that satisfy
simple commutation relations. In the next section we shall show in a more
formal manner that these basic ideas can be extended to any many-body
system, focusing on the case of fermionic particles.
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628 Molecular Electronics: An Introduction to Theory and Experiment

A.2 Second quantization for fermions

The systems that we will be dealing with are composed of many identi-
cal particles such as electrons or phonons. In quantum mechanics those
identical particles are indistinguishable. Thus for instance, no electron can
be distinguished from another electron, except by saying where it is, what
quantum state it is in, etc. Internal quantum-mechanical consistency re-
quires that when we write down a many-identical-particle state, we make
that state noncommittal as to which particle is in which single-particle
state. For example, we say that we have electron 1 and electron 2, and we
put them in states a and b respectively, but exchange symmetry requires
(since electrons are fermions) that a satisfactory wave-function has the form
Φ(r1 , r2 ) = A [φa (r1 )φb (r2 ) − φa (r2 )φb (r1 )] , (A.30)
i.e. the wave function is antisymmetric with respect to the exchange of the
two electrons: Φ(r1 , r2 ) = −Φ(r2 , r1 ). This is a consequence of Pauli’s
exclusion principle that states that there cannot be two fermions in the
same quantum sate.
If we have N particles, the wave functions must be either symmetric or
antisymmetric under exchange depending on the nature of the particles:
ΦB (r1 , ..., ri , ..., rj , ..., rN ) = ΦB (r1 , ..., rj , ..., ri , ..., rN ) (Bosons)
ΦF (r1 , ..., ri , ..., rj , ..., rN ) = −ΦF (r1 , ..., rj , ..., ri , ..., rN ) (Fermions).
In particular, in the fermionic case the antisymmetry of the wave func-
tion can be ensured by using Slater determinants. But, can we satisfy the
antisymmetry principle without using Slater determinants? Second quan-
tization is a formalism in which the antisymmetry property of the wave
function has been transferred onto the algebraic properties of certain oper-
ators. Second quantization introduces no new physics. It is just another,
although very elegant, way of treating many-electron systems, which shifts
the emphasis away from N -electron wave functions to the one- and two
electron matrix elements of the different operators. This has been illus-
trated already in the case of bosons with the analysis of the phonons in a
1D chain in the previous section, and we shall now concentrate on the case
of fermions.

A.2.1 Many-body wave function in second quantization

The total wave function for the ground state and excited states of non-
interacting particles is the product of single-particle wave functions. How-
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Second Quantization 629

ever, because we are considering identical fermions, this product must be

anti-symmetrized (Pauli’s exclusion principle) and the proper wave function
is the Slater determinant
¯ φk1 (r1 ) φk1 (r2 ) · · · φk1 (rN ) ¯
¯ ¯
¯ ¯
1 ¯¯ φk2 (r1 ) φk2 (r2 ) · · · φk2 (rN ) ¯¯
Φk1 ,...,kN (r1 , ..., rN ) = √ ¯ .. .. .. .. ¯, (A.31)
N ! ¯¯ . . . . ¯
¯
¯ φ (r ) φ (r ) · · · φ (r ) ¯
kN 1 kN 2 kN N

where N is the number of particles and the φk ’s are a set of single-particle

states with ki as quantum number (e.g. energy). If the particles are allowed
to interact with each other, or with an external potential, then the exact
wave functions of the system are no longer that of Eq. (A.31), but a linear
combination of Φ’s:
X
Ψ(r1 , ..., rN ) = Ak1 ,...,kN Φk1 ,...,kN (r1 , ..., rN ). (A.32)
k1 ,...,kN

That is, the Φk1 ,...,kN (r1 , ..., rN ) for the non-interacting system are the basis
states used to describe the interacting system.
Now these are rather clumsy expressions to carry around, so it would
be desirable to have a more compact way of writing them. This may be
achieved by noting that since all particles are indistinguishable, the essential
information in Eq. A.31 is just how many particles there are in each single-
particle state. Therefore, we could equally well specify the state of the
non-interacting system by writing Φ as
Φk1 ,...,kN (r1 , ..., rN ) = Φnp1 ,np2 ,...,npi ,... (r1 , ..., rN ). (A.33)
For short, we shall represent this as
Φnp1 ,np2 ,...,npi ,... (r1 , ..., rN ) ≡ |np1 , np2 , ..., npi , ...i (A.34)
meaning: np1 particles in state φp1 , np2 in φp2 , etc., where nk = 0 or 1
by the Pauli principle. This is called “occupation number notation”. For
brevity, from now on we shall drop the p’s and just use the numerical
subscripts. Then
Φ = |n1 , n2 , ..., ni , ...i. (A.35)
It is important to remember that the |n1 , n2 , ..., ni , ...i are orthonormal
because the Φk1 ,...,kN are, and we may write this in the following way
hn′1 , n′2 , ..., n′i , ...|n1 , n2 , ..., ni , ...i = δn′1 ,n1 δn′2 ,n2 ...δn′i ,ni ... (A.36)
Up to this point we have been dealing with systems containing a fixed
number of particles. Now we take an important step, and, even though the
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630 Molecular Electronics: An Introduction to Theory and Experiment

particle number in a real system is fixed, allow N to be variable, running

from 0 to ∞. This generates the following set of basis functions

0 Φ0 |000...i
1 Φ1 , Φ2 , Φ3 , ... |100...i, |010...i, |001...i, ...
2 Φ12 , Φ13 , Φ23 , ... |1100...i, |1010...i, |0110...i, ...
.. .. ..
. . .
N Φk1 ,k2 ,...,kN |n1 , n2 , ..., ni , ...i
.. .. ..
. . . (A.37)

The state Φ0 or |000...i with no particles at all in it is called the true

vacuum. The set of all |n1 , ..., ni , ...i in Eq. (A.37) is a complete orthogonal
set of basis functions in an extended Hilbert space in which the number
of particles is variable. This set is often called occupation number basis,
and the whole formalism is sometimes referred to as occupation number
representation.
States like the ones appearing in Eq. (A.37) can describe the state of
a non-interacting fermionic systems. In the presence of interactions the
correct eigenstates of the system can be obtained as linear combination of
the states |n1 , ..., ni , ...i, i.e.
X
Ψ= An1 ,...,ni ,... |n1 , ..., ni , ...i. (A.38)
n1 ,...,ni ,...

A.2.2 Creation and annihilation operators

We shall go on constructing the formalism of second quantization by show-
ing how the properties of determinants can be transferred onto the algebraic
properties of operators. For this purpose, we begin by associating a creation
operator c†i and an annihilation operator ci with each single-particle state
φi . We define c†i and ci by their action on an arbitrary Slater determinant
|n1 , ..., ni , ...i as follows

c†i |n1 , ..., ni , ...i = (−1)Σi (1 − ni )|n1 , ..., ni + 1, ...i (A.39)

Σi
ci |n1 , ..., ni , ...i = (−1) ni |n1 , ..., ni − 1, ...i, (A.40)

where

Σi = n1 + n2 + ... + ni−1 . (A.41)

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Second Quantization 631

That is, we get a factor of (−1) for each particle (i.e., each occupied state)
standing to the left of the state i in the wave function. For example
ci |..., 0i , ...i = 0 , c†i |..., 1i , ...i = 0
c3 |11111000...i = +|11011000...i
c†4 |1110100...i = −|11111000...i
c†2 c3 c†1 c2 c†3 c1 |1100...i = −|1100...i. (A.42)
One of the nice properties of the c†i operators is that by applying them
repeatedly to the true vacuum state (state with no particles in it), it is
possible to generate all other states, thus:
|n1 , n2 , ...i = (c†1 )n1 (c†2 )n2 ...|0000...i. (A.43)
For example
|011000...i = c†2 c†3 |0000...i. (A.44)
Another important property of the c†i , ci operators is that they are
hermitian adjoint of each other, i.e. c†i = (ci )† . The demonstration is left
to the reader. This property shows that c†i , ci are non-hermitian and are
therefore not observables. It is, however, easy to construct a hermitian
operator from c†i and ci as follows. The combination

n̂i = c†i ci (N̂ =

X †
ci ci ) (A.45)
i

is obviously hermitian and is an extremely important observable called num-

ber operator (N̂ = total number operator). To understand its properties,
let it operate on some typical state vectors:
c†i ci |n1 , n2 , ..., 1i , ...i = (−1)Σi c†i |n1 , n2 , ..., 0i , ...i
= (−1)Σi +Σi |n1 , n2 , ..., 1i , ...i
= (+1)|n1 , n2 , ..., 1i , ...i.
Similarly
c†i ci |n1 , n2 , ..., 0i , ...i = 0|n1 , n2 , ..., 0i , ...i,
so that in general
c†i ci |n1 , n2 , ..., ni , ...i = ni |n1 , n2 , ..., ni , ...i. (A.46)
Thus, the eigenvalue of the number operator for the state φi is just the
occupation number for that state. Hence, in the occupation number ba-
sis, all number operators are diagonal and the total system wave function
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632 Molecular Electronics: An Introduction to Theory and Experiment

|n1 , ..., ni , ...i are just simultaneous eigenfunctions of the number operators
n̂1 , ..., n̂i , ....
The c†i , ci operators obey the following important fermion commutation
rules:
{cl , c†k } = cl c†k + c†k cl = δlk ; {cl , ck } = {c†l , c†k } = 0. (A.47)
These can be easily proved from the definitions of Eqs. (A.39) and (A.40).
Thus for instance, the second relation can be shown as follows:
cl ck |n1 , ..., nl , ..., nk , ...i = (−1)Σk nk cl |n1 , ..., nl , ..., nk − 1, ...i (A.48)
Σk +Σl
= (−1) nk nl |n1 , ..., nl − 1, ..., nk − 1, ...i
Σl
ck cl |n1 , ..., nl , ..., nk , ...i = (−1) nl ck |n1 , ..., nl − 1, ..., nk , ...i
= (−1)(−1)Σk +Σl nk nl |n1 , ..., nl − 1, ..., nk − 1, ...i,
where the extra (−1) on line four comes from the fact that there is one less
particle to the left of state k. Adding the two equations yields the second
rule in Eq. (A.47). The other rules may be established in a similar fashion.
The importance of the above set of anti-commutation relations lies in
the fact that all the antisymmetry properties are built into them. Therefore,
by using them in the right places, we do not have to worry either about the
symmetry of the wave functions themselves, or even about the awkward
(−1)Σ factors.

A.2.3 Operators in second quantization

We have seen that we can represent determinants by using creation and
annihilation operators, which obey a set of anti-commutation relations,
and a vacuum state. To be able to develop the entire theory of many-
electron systems without using determinants, we must express the many-
body operators in terms of the creation and annihilation operators. This is
the goal of this subsection.
All the operators that we shall encounter, in particular for electronic
systems, can be written in first quantization as the sum of two types of
operators. The first type is a sum of one-electron operators
N
X
O1 = h(i), (A.49)
i=1

where h(i) is any operator involving only the ith electron. These operators
represent dynamic variables that depend only on the position or momentum
of the electron in question, independent of the position or momentum of
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Second Quantization 633

other electrons. Examples are operators for the kinetic energy, attraction
of an electron to a nucleus, dipole moment, and most of the other operators
that one encounters. The second type of operator is a sum of two-electron
operators
N
1X
O2 = V (i, j), (A.50)
2
i6=j

where V (i, j) is an operator that depends on the position (or momentum)

of both the ith and jth electron. The Coulomb interaction between two
electrons
e2
V (i, j) = (A.51)
|ri − rj |
is a two-electron operator.
Obviously, the expression for an operator O in second quantization must
be such that the value of the matrix element hK|O|Li, |Ki and |Li being
two arbitrary Slater determinants, is the same irrespective of whether we
obtain it using the properties of determinants or using the algebra of cre-
ation and annihilation operators. The appropriate expressions for O1 (our
sum of one-electron operators) and O2 (the two-electron operator) in second
quantization are
hij c†i cj
X
O1 = (A.52)
ij
1X
O2 = Vijkl c†i c†j cl ck , (A.53)
2
ijkl

where the sums run over the set {ψi }. Here, the different matrix elements
are defined as follows
Z
hij ≡ dr1 ψi∗ (r1 )h(r1 )ψj (r1 ) (A.54)
Z Z
Vijkl ≡ dr1 dr2 ψi∗ (r1 )ψj∗ (r2 )V (r1 , r2 )ψk (r1 )ψl (r2 ). (A.55)

Let us now sketch the demonstration of this result. Consider the

following N -electron Slater determinant |Ψi = |ψ1 , ..., ψa , ψb , ..., ψN i =
|11 , ..., 1a , 1b , ..., 1N i. From the first quantization formalism (this is a simple
exercise), we know that the expectation value of the one-electron operator
O1 is equal to
X
hΨ|O1 |Ψi = hii . (A.56)
i
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634 Molecular Electronics: An Introduction to Theory and Experiment

Let us now demonstrate that we can also recover this result using the second
quantization expression for the operator O1 . In this case,
hij hΨ|c†i cj |Ψi.
X
hΨ|O1 |Ψi = (A.57)
ij

Since both cj and c†i are trying to destroy an electron (cj to the right and
c†i to the left), the indices i and j must belong to the set {a, b, ...} and thus
X
hΨ|O1 |Ψi = hab hΨ|c†a cb |Ψi. (A.58)
ab

Using
c†a cb = δab − cb c†a (A.59)
to move c†a to the right, we have
hΨ|c†a cb |Ψi = δab hΨ|Ψi − hΨ|cb c†a |Ψi. (A.60)
The second term on the right is zero since c†a
is trying to create an electron
in ψa , which is already occupied in |Ψi. Since hΨ|Ψi = 1, we finally have
X X
hΨ|O1 |Ψi = hab δab = haa . (A.61)
ab a

in agreement with the first quantization result above. We can proceed

in a similar way to demonstrate the result for two-electron operators (see
Exercise A.4 at the end of this appendix).
Note that the form of the operators above is independent of the number
of electrons. One of the advantages of second quantization is that it treats
systems with different numbers of particles on an equal footing. This is
particularly convenient when one is dealing with infinite systems such as
solids or molecular junctions.

A.2.4 Some special Hamiltonians

Our description of the electronic structure of any electronic system will
start always by presenting the corresponding Hamiltonian. In this sense,
it is important to get familiar with the form that some basic Hamiltonians
adopt in second quantization.
The Hamiltonian of an electron system has the following generic form
in first quantization
X · p2 ¸
1X
i
H= + U (ri ) + V (ri − rj ) (A.62)
i
2m 2
i6=j
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Second Quantization 635

with the electrons interacting with a potential U (r), such as the lattice
potential in a solid, and with each other through particle-particle interac-
tions V (ri − rj ), typically the Coulomb interaction. As we have learned
above, this Hamiltonian can be written in terms of the fermionic creation
and annihilation operators as
1X
hij c†i cj + Vijkl c†i c†j cl ck ,
X
H= (A.63)
ij
2
ijkl

where
· 2 2 ¸
~ ∇
Z
hij = dr ψi∗ (r) − + U (r) ψj (r)
2m
Z Z
Vijkl = dr1 dr2 ψi∗ (r1 )ψj∗ (r2 )V (r1 − r2 )ψk (r1 )ψl (r2 ).

The precise form of this Hamiltonian depends primarily on the single-

particle basis {ψi } used. For the study of electron in solids, a popular basis
set is plane waves: ψi (r) = ψk,σ (r) = L−3/2 eik·r uσ , where k is the electron
momentum, σ is the spin index and uσ is a spinor. The Hamiltonian then
has the form
1 X
ǫk c†kσ ckσ + vq c†k+qσ c†k′ −qσ′ ck′ σ′ ckσ ,
X X
H= U (q)ρq +
q
2V
kσ ′ ′ kk qσσ
(A.64)
where V is the total volume of the system and ǫk = ~2 k 2 /2m. The second
term represents the interaction between the electrons and the atoms or ions
of the solid, where ρq is the electron density operator given by
X †
ρq = ck+qσ ckσ . (A.65)
kσ

Finally, vq is the Fourier transform of the Coulomb potential e2 /r and it is

given by vq = 4πe2 /q 2 .
The full electron gas Hamiltonian of Eq. (A.64) is too complicated and
it is often approximated by a model Hamiltonian which has a simpler form.
Some of these popular models are discussed next.
The homogeneous electron gas is a model which is studied frequently to
learn about correlation effects. It has the Hamiltonian
1
ǫk c†kσ ckσ + vq c†k+qσ c†k′ −qσ′ ck′ σ′ ckσ .
X X
H= (A.66)
2V ′
kσ kk ,q6=0,σσ
′

The basic premise is to get rid of the atoms and to replace them with
a uniform positive background charge of density n0 . The homogeneous
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636 Molecular Electronics: An Introduction to Theory and Experiment

electron gas is also called jellium model. One can think of taking the
positive charge of the ions and spreading it uniformly about the unit cell
of the crystal. Of course, the homogeneous electron gas has no crystal
structure. To preserve charge neutrality, the average particle density of the
electron gas must also be n0 . This model, although a bit academic, has
played a key role to understand basic issues about the Coulomb interaction
of a many-particle system. A detailed discussion of this model can be found
in Ref. [174].
The plane-wave model is often a poor approximation of electron be-
havior in solids where the electrons are localized on atomic sites and only
occasionally hop to neighboring sites. This behavior is described by the
tight-binding model, where the basis is formed by localized atomic-like or-
bitals.3 One simple form of this model is bilinear in the operators:
tij c†iσ cjσ .
X
H= (A.67)
ijσ

The index j denotes a site at point Rj , while i represents the nearest

neighbor atoms. The matrix elements tij are given by
~2 2
Z · ¸
∗
tij = dr φ (r − Ri ) − ∇ + U (r) φ(r − Rj ), (A.68)
2m
where the orbitals φ(r) are localized in the sites Ri and Rj . Thus, the
element tij (for j 6= i) represents processes where the electron jumps from
site j to i, while tii is the site energy. Simple versions of the model usually
have a single orbital state for each atomic site. More realistic versions of
the tight-binding model allow for multiple orbitals characteristic of p- or
d-electrons (see Chapter 9).
The tight-binding Hamiltonian may also contain the Coulomb interac-
tion between electrons. In its most general form, the interaction term is
1X
Vijkl c†i c†j cl ck (A.69)
2
ijkl

where
e2
Z Z
Vijkl = dr1 dr2 φ∗ (r1 − Ri )ψ ∗ (r2 − Rj ) ψ(r1 − Rk )ψ(r2 − Rl ).
|r1 − r2 |
The four orbitals could be centered on four different sites. These are called
four-center integrals. They are usually small and often neglected in many-
body calculations.
3 Tight-binding models and their used in molecular electronics are the subject of Chapter
9.
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Second Quantization 637

The Hubbard model [177] retains only the Coulomb integral which is the
largest, namely that in which all four orbitals φ(r) are centered on the same
site. This term describes the interaction between two electrons which are
on the same atom. Since two electrons cannot be in the same state, the two
on the same atom must be in different atomic states. In the simplest model,
which considers only a single orbital state on each atom, the two electrons
must have different spin configurations. One has spin up, while the other
has spin down. The Hubbard model considers the following Hamiltonian

c†iσ ciσ + t c†iσ cjσ + U

X X X
H = ǫ0 ni↑ ni↓ (A.70)
i ij i

e2
Z Z
U = Viiii = dr1 dr2 |φ∗ (r1 )|2 |φ∗ (r1 )|2 . (A.71)
|r1 − r2 |

The hopping term is usually limited to nearest neighbors. The Hamilto-

nian was also introduced by Gutzwiller [178], who studied the properties of
electrons in d-bands in ferromagnets. It is thought to be a good model for
electron conduction in narrow band materials, for example, in transition
metal oxides. The Hubbard model has been investigated thoroughly over
the past forty years, and its properties are starting to be understood [179].
A simplified version of the Hubbard model is the so-called Anderson
model [180]. In this model the Hubbard-like interaction is considered to
be only present in a single site. This model was introduced to study the
interaction of localized magnetic impurities with the conduction electrons
of a metal. In recent years, it has been widely used to study the electronic
and transport properties of quantum dots and molecular transistors (see
Chapters 15 and 17).

A.3 Second quantization for bosons

The second quantization formalism for bosons was already outlined when
we discussed the physics of phonons in section A.1.2. Anyway, for the sake
of completeness, we summarize here the main results of this formalism for
the case of bosons:

(1) The many-body wave functions for a bosonic system has to be sym-
metric with respect to the particle exchange. In this sense, the Slater
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

638 Molecular Electronics: An Introduction to Theory and Experiment

determinant of the fermionic case is replaced now by

µ ¶1/2 X
n1 !n2 !...
Φk1 ,k2 ,...,kN (r1 , ..., rN ) = (+1)P [φk1 (r1 )...φkN (rN )]
N!
P
= Φn1 ,...,ni ,... (r1 , ..., rN ) = |n1 , ..., ni , ...i, (A.72)
where P is the permutation operator which interchanges the ri ’s in all
possible ways. Moreover, the occupation number can take any integer
value: ni = 0, 1, 2, 3, ....
(2) The c†i , ci operators are now defined by
√
c†i |n1 , ..., ni , ...i = ni + 1|n1 , ..., ni + 1, ...i (A.73)
√
ci |n1 , ..., ni , ...i = ni |n1 , ..., ni − 1, ...i. (A.74)
(3) The commutation relations now read:
h i h i
cl , c†k = cl c†k − c†k cl = δlk ; [cl , ck ] = c†l , c†k = 0. (A.75)

(4) The one- and two-body operators are expressed in terms of the creation
and annihilation operators in the same way as in the fermion case.

A.4 Exercises

A.1 Find c1 c†5 c2 |111000...i.

A.2 Find hΨ|N̂ |Ψi, where |Ψi = A|100...i + B|111000...i, and N̂ is the total
number operator.
A.3 Demonstrate the first and third fermion commutation rules of Eq. (A.47).
A.4 Verify that for a two-particle system, the matrix elements of the two-body op-
erator O2 in Eq. A.53 between two-particle states h0...1p ...1q ...| and |0...1r ...1s ...i
are the same as the matrix elements of the first quantization version of O2 taken
between the corresponding two-particle Slater determinants.
A.5 Prove that the components of the total spin operator, S, in second quantized
form are:
1 X“ † ”
Sx = ck↑ ck↓ + c†k↓ ck↑
2
k
i X“ † ”
Sy = − ck↑ ck↓ − c†k↓ ck↑
2
k
1 X“ † ”
Sz = ck↑ ck↑ − c†k↓ ck↓ .
2
k
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

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January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Index

1/2 rule, 510 Ni, 336, 339, 346

Pb, 298, 315
absorption sidebands, 531 permalloy, 344
adiabatic hypothesis, 148 Zn, 303
alkanes, 47, 62, 361, 395, 480, 563, atomic force microscope (AFM), 21
564, 567, 577 attenuation factor, 377
alkenes, 48 azobenzene, 609
alkynes, 48
anchoring groups, 52, 408 back-gate, 34, 58
amine, 409, 410, 412 ballistic anisotropic
carboxylic acid, 410 magnetoresistance (BAMR), 348
cyano (or nitril), 374, 412 ballistic magnetoresistance, 344
methyl phosphine, 410 benzene, 49, 248, 405
methyl sulfide, 410 benzenedithiol, 392, 576
nitro, 374, 412 blinking, 581
thiol, 59, 62, 374, 395, 408, 410, Bloch sums, 239
412 Born-Oppenheimer approximation,
Anderson model, 137, 170, 449, 637 265
Friedel’ sum rule, 171 Breit-Wigner formula, 201, 212, 366
Anderson-Holstein model, 525
anisotropic magnetoresistance C60 , 51, 61, 411, 417, 457, 464, 493,
(AMR), 335, 347 498
atomic contacts, 19, 297 carbon monoxide (CO), 54, 62, 65
Ag, 300, 321 carbon nanotubes, 53, 618
Al, 298, 303, 307, 315, 321 cis conformation, 48
Au, 297, 299, 300, 315, 320 coherent destruction of tunneling, 603
Co, 336, 339 coherent transport, 359, 392
Cu, 321 coherent tunneling, 359, 392
Fe, 336, 339 conductance fluctuations, 319
Mg, 303 conductance quantization, 98, 298,
Na, 302, 321 336
Nb, 315, 321 conductance quantum, 80, 198, 296,

697
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698 Molecular Electronics: An Introduction to Theory and Experiment

297 local density approximation

conduction channels, 96, 210 (LDA), 273
conjugation, 47, 250 local spin-density approximation
contacting molecules, 45 (LSDA), 274
ensembles, 45, 66 pseudopotentials, 282
single-molecule contacts, 45, 55 Thomas-Fermi model, 268
Coulomb blockade, 57, 424 density of states, 115
addition energy, 435 diarylethenes, 609
constant interaction model, 432 direct tunneling, 360
Coulomb oscillations, 427 DNA, 51, 546
Coulomb peak, 442
Coulomb staircase, 429 elastic cotunneling, 451
orthodox theory, 432 electroluminescence, 605
stability diagram, 437 electromigration, 13, 31, 35, 36, 38, 56
coupling, 56, 61 electron transfer, 617
strong, 58, 61, 66 electron-phonon interaction, 217, 473
weak, 61, 66 electrospray ionization (ESI), 54
current rectification, 584 energy bands, 240
cyclic voltammetry, 36 equation-of-motion method, 136
extended Hückel method, 241
delocalization energy, 250
density functional theory (DFT), 263 Fano factor, 556
basis sets, 280 Fano resonances, 383
contracted Gaussian functions Feynman diagrams, 151
(CGF), 281 electron-electron interaction, 152
double-zeta, 281 energy space, 158
Gaussian-type-orbitals external potential, 157
(GTOs), 280 Hartree diagram, 169
minimal basis sets, 281 irreducible diagrams, 165
numerical basis functions, 281 reducible diagrams, 165
polarization functions, 282 skeleton diagrams, 167
Slater-type-orbitals (STO), vertex, 154
281 field operators, 128
split-valence, 282 fluorescence, 604
electron density, 268 Fowler-Nordheim tunneling, 360
exchange-correlation energy, 271 Franck-Condon blockade, 530, 560
exchange-correlation functional, Franck-Condon matrix elements, 527
273 fullerene, 47, 51, 54, 65
generalized gradient approximation functionalization, 62
(GGA), 275
gradient expansion approximation giant magnetoresistance (GMR), 335
(GEA), 275 gold nanoparticles (GNPs), 59, 62
Hohenberg-Kohn theorems, 269 graphene, 252
hybrid functionals, 277 Green’s functions, 114
Kohn-Sham equations, 272 advanced function, 114, 128
LCAO approach, 278 causal function, 129
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Bibliography 699

Dyson’s equation, 117, 165 Kondo effect, 170, 236, 453

retarded function, 114, 128 Kondo resonance, 173, 454
self-energy, 117, 164 Kondo temperature, 454
spectral representation, 116, 132 spin-flip cotunneling, 453
Grundlach oscillations, 88 unitary limit, 236, 456

Hammett constant, 416 Landauer approach, 78

Hartree-Fock approximation, 169, 266 Landauer formula, 80, 96, 197, 210,
Coulomb integrals, 267 296
exchange integrals, 267 Langmuir-Blodgett film (LB), 54
Hartree-Fock equations, 267 light-induced current, 603
Hartree-Fock potential, 267 liquid environment, 63, 64
Koopman’s theorem, 268 lowest unoccupied molecular orbital
heat generation, 560, 565 (LUMO), 250, 364
heat transport, 560
heating, 565 magnetoresistance, 343
Heisenberg picture, 113 Marcus theory of electron transfer,
heterojunction, 12, 21, 22 618
highest occupied molecular orbital mechanically controllable break
(HOMO), 250, 364 junctions (MCBJ), 11, 24, 35, 36,
Holstein model, 503 38, 64, 297, 301
homogeneous electron gas, 635 lithographic MCBJ, 25
hopping conduction, 361 notched-wire MCBJ, 25
hopping regime, 537 thin-film MCBJ, 25, 36
hot electrons, 584 Meir-Wingreen formula, 222
Hubbard model, 636 methane, 54, 65
hydrocarbons, 47 molecular switches, 609
hydrogen, 54, 62, 65, 401, 558 molecular tunnel junctions, 368
monolayer, 53, 54
I(s) method, 418 multiple Andreev reflection (MAR),
I(t) method, 418 306, 590
inelastic cotunneling, 452
inelastic electron tunneling nanopore, 12
spectroscopy (IETS), 476 nanotubes, 47, 53
interaction picture, 144 negative differential conductance
(NDC), 466, 530, 534
jump to contact, 21 negative differential resistance
junction temperature, 565, 583 (NDR), 12, 41, 385
nitrogen, 54
Keldysh formalism, 179, 180 noise power, 101
Dyson’s equation, 184 nonequilibrium Green’s function
Feynman diagrams, 184 formalism (NEGF), 179
Keldysh contour, 181 NRL tight-binding method, 254
Keldysh function, 187
Keldysh space, 183 oligo(phenylene-ethynylene) (OPE),
triangular representation, 188 480
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700 Molecular Electronics: An Introduction to Theory and Experiment

oligo(phenylenevinylene) (OPV), 480, creation operator, 624

496 fermion commutation rules, 632
oligophenyleneimine, 544 number operator, 631
Onsager relation, 104 Seebeck coefficient, see thermopower,
oxygen, 54, 65 569
self-assembled monolayer (SAM), 52,
pair tunneling, 530 53
PDMS, 64 shot noise, 99, 199, 554
Peltier coefficient, 104 side-groups, 413
phenylenes, 574, 577 Simmons model, 86, 360, 362
phonon sidebands, 522, 528 single-electron transistor (SET), 424
phonon-assisted resonant tunneling, single-molecule magnet (SMM), 465
522 single-molecule transistor (SMT), 424
photochromic molecules, 609 spin-coating, 28, 53
photochromism, 609 spintronics, 335
photoconductance, 230, 588 STM controlled experiments, 416
photocurrent, 589 subharmonic gap structure, 306, 590
photoinduced transport (PIT), 599
submonolayer, 54
photoisomerization, 609
substituents, 413
photon-assisted tunneling (PAT),
surface-enhanced Raman
224, 231, 584, 586
spectroscopy (SERS), 568, 580
piezo, 25, 26, 30
point contact spectroscopy (PCS),
thermal conductance, 104, 562
483
thermal noise, 102, 554
polaron, 524, 541
thermionic emission, 361
shift, 526, 530
thermoelectric coefficients, 104, 105
transformation, 521, 526
polyenes, 48 thermoelectric effects, 104
propensity rules, 517 thermopower, 104, 569
proximity effect, 315 thiol, 52
three-terminal device, 45
ratchet, 602 tight-binding approach, 237, 636
rectification, 371, 594 time-evolution operator, 146
resonant current amplification, 602 time-ordering operator, 129, 147
resonant tunneling, 92 trans conformation, 48
resonant tunneling diode, 385 transmission coefficients, 96, 296
resonant tunneling model, 231, 364 transmission electron microscope
(TEM), 23, 58, 316
scanning tunneling microscope tunnel effect, 84
(STM), 10, 19, 23, 30, 31, 60, 64, tunneling magnetoresistance (TMR),
297, 301 335
scattering approach, 78 tunneling traversal time, 423
Schrödinger picture, 112
screening approximation, 288 ultra high vacuum (UHV), 21, 54
second quantization, 623
annihilation operator, 624 variational principle, 265
January 12, 2010 11:27 World Scientific Book - 9in x 6in book

Bibliography 701

water, 65, 491

Wick’s theorem, 150
wide-band approximation, 126, 197,
201
Wiedemann-Franz relation, 106