Solid State Physics

A primer
Solid State Physics
A primer

Luciano Colombo
University of Cagliari, Italy

IOP Publishing, Bristol, UK

ª IOP Publishing Ltd 2021

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Luciano Colombo has asserted his right to be identified as the author of this work in accordance
with sections 77 and 78 of the Copyright, Designs and Patents Act 1988.

ISBN 978-0-7503-2265-2 (ebook)

ISBN 978-0-7503-2262-1 (print)
ISBN 978-0-7503-2263-8 (myPrint)
ISBN 978-0-7503-2264-5 (mobi)

DOI 10.1088/978-0-7503-2265-2

Version: 20210301

IOP ebooks

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To my students.
 Contents

Foreword xii
Presentation of the ‘primer series’ xvi
Introduction to: ‘Solid state physics: a primer’ xviii
Author biography xx
Symbols xxi

Part I Preliminary concepts

1 The overall picture 1-1

1.1 Basic definitions 1-1
1.2 Synopsis of atomic physics 1-6
1.2.1 Atomic structure 1-6
1.2.2 Angular and magnetic momenta 1-7
1.2.3 Electronic configuration 1-8
1.3 Setting up the atomistic model for a solid state system 1-9
1.3.1 Semi-classical approximation 1-10
1.3.2 Frozen-core approximation 1-10
1.3.3 Non-magnetic and non-relativistic approximations 1-11
1.3.4 Adiabatic approximation 1-12
1.4 Mastering many-body features 1-15
1.4.1 Managing the electron problem: single-particle approximation 1-16
1.4.2 Managing the ion problem: classical approximation 1-18
References 1-19

2 The crystalline atomic architecture 2-1

2.1 Translational invariance, symmetry, and defects 2-1
2.2 The direct lattice 2-2
2.2.1 Basic definitions 2-2
2.2.2 Direct lattice vectors 2-4
2.2.3 Bravais lattices 2-6
2.2.4 Lattice planes and directions 2-9
2.3 Crystal structures 2-10
2.3.1 The basis 2-10
2.3.2 Classification of the crystal structures 2-12
2.3.3 Packing 2-15

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 Solid State Physics

2.4 The reciprocal lattice 2-16

2.4.1 Fundamentals of x-ray diffraction by a lattice 2-16
2.4.2 Von Laue scattering conditions 2-18
2.4.3 Reciprocal lattice vectors 2-20
2.4.4 The Brillouin zone 2-22
2.5 Lattice defects 2-23
2.5.1 Point defects 2-26
2.5.2 Extended defects 2-28
2.6 Classification of solids 2-30
2.7 Cohesive energy 2-32
References 2-35

Part II Vibrational, thermal, and elastic properties

3 Lattice dynamics 3-1

3.1 Conceptual layout 3-1
3.2 Dynamics of one-dimensional crystals 3-4
3.2.1 Monoatomic linear chain 3-4
3.2.2 Diatomic linear chain 3-7
3.3 Dynamics of three-dimensional crystals 3-12
3.4 The physical origin of the LO–TO splitting 3-15
3.5 Quantum theory of harmonic crystals 3-19
3.6 Experimental measurement of phonon dispersion relations 3-21
3.7 The vibrational density of states 3-24
References 3-27

4 Thermal properties 4-1

4.1 The lattice heat capacity 4-1
4.1.1 Historical background 4-1
4.1.2 The Debye model for the heat capacity 4-3
4.1.3 The general quantum theory for the heat capacity 4-7
4.2 Anharmonic effects 4-8
4.2.1 Thermal expansion 4-8
4.2.2 Phonon–phonon interactions 4-12
4.3 Thermal transport 4-15
References 4-21

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 Solid State Physics

5 Elastic properties 5-1

5.1 Basic definitions 5-1
5.1.1 The continuum picture 5-1
5.1.2 The strain tensor 5-3
5.1.3 The stress tensor 5-4
5.2 Linear elasticity 5-7
5.2.1 The constitutive equation 5-7
5.2.2 The elastic tensor 5-8
5.2.3 Elasticity of homogeneous and isotropic media 5-10
5.3 Elastic moduli 5-12
5.4 Thermoelasticity 5-14
References 5-18

Part III Electronic structure

6 Electrons in crystals: general features 6-1

6.1 The conceptual framework 6-1
6.2 The Fermi–Dirac distribution function 6-4
6.3 The Bloch theorem 6-8
6.4 Electrons in a periodic potential 6-10
References 6-14

7 Free electron theory 7-1

7.1 General features of the metallic state 7-1
7.2 The classical (Drude) theory of the conduction gas 7-3
7.2.1 Electrical conductivity 7-3
7.2.2 Optical properties 7-4
7.2.3 Thermal transport 7-7
7.2.4 Failures of the Drude theory 7-8
7.3 The quantum (Sommerfeld) theory of the conduction gas 7-10
7.3.1 The ground-state 7-10
7.3.2 Finite temperature properties 7-14
7.3.3 More on relaxation times 7-19
7.3.4 Failures of the Sommerfeld theory 7-23
References 7-23

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 Solid State Physics

8 The band theory 8-1

8.1 The general picture 8-1
8.1.1 Bands and gaps 8-1
8.1.2 The weak potential approximation 8-3
8.1.3 Band filling: metals, insulators, semiconductors 8-6
8.2 The tight-binding method 8-9
8.2.1 Bands in a one-dimensional crystal 8-9
8.2.2 Bands in real solids 8-13
8.3 General features of the band structure 8-18
8.3.1 Parabolic bands approximation 8-18
8.3.2 Electron dynamics 8-18
8.3.3 Electric field effects 8-20
8.3.4 Electrons and holes 8-23
8.3.5 Effective mass 8-24
8.4 Experimental determination of the band structure 8-28
8.5 Other methods to calculate the band structure 8-30
References 8-31

9 Semiconductors 9-1
9.1 Some preliminary concepts 9-1
9.1.1 Doping 9-2
9.1.2 Density of states for the conduction and valence bands 9-5
9.2 Microscopic theory of charge transport 9-7
9.2.1 Drift current in a weak field regime 9-7
9.2.2 Scattering 9-8
9.2.3 Carriers concentration 9-12
9.2.4 Conductivity 9-14
9.2.5 Drift current in a strong field regime 9-15
9.2.6 Diffusion current 9-16
9.2.7 Total current 9-18
9.3 Charge carriers statistics 9-18
9.3.1 Semiconductors in equilibrium 9-19
9.3.2 Chemical potential in intrinsic semiconductors 9-20
9.3.3 Chemical potential in doped semiconductors 9-20
9.3.4 Law of mass action 9-22
9.3.5 Semiconductors out of equilibrium 9-23

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 Solid State Physics

9.4 Optical absorption 9-24

9.4.1 Conceptual framework 9-24
9.4.2 Phenomenology of optical absorption 9-25
9.4.3 Inter-band absorption 9-27
9.4.4 Excitons 9-33
References 9-36

10 Density functional theory 10-1

10.1 Setting the problem and cleaning up the formalism 10-1
10.2 The Hohenberg–Kohn theorem 10-3
10.3 The Kohn–Sham equations 10-6
10.4 The exchange-correlation functional 10-9
10.5 The practical implementation and applications 10-10
References 10-11

Part IV Concluding remarks

11 What is missing in this ‘Primer’ 11-1

Part V Appendices

Appendix A A-1

Appendix B B-1

Appendix C C-1

Appendix D D-1

Appendix E E-1

Appendix F F-1

Appendix G G-1

Appendix H H-1

Appendix I I-1

xi
 Foreword

When Professor Colombo approached me to write the preface of this book, I was
both surprised and honoured. Surprised because there are many eminent scholars
out there that could ‘do the job’, and honoured because I have the greatest respect
for Professor Colombo and his work. I have known Professor Colombo for many
years—we interacted on several occasions (often at the MRS Fall Meeting in
Boston), he visited me in Montréal, I visited him in Cagliari (invited to give a short
course on models of laser ablation). I have always been impressed by the clarity and
rigour of his mind, his knowledge of various topics in physics, his desire to not only
do the ‘best physics’ but also to communicate it with clarity and passion. He has
been a model for me and I tried my best, over the years, to imitate his style.
Professor Colombo told me once that writing a book was some sort of ‘romantic’
adventure. I was not too sure what he meant (and I did not dare to ask!), but the idea
remained in my head. I suspect that he sees this as some sort of an idealised
accomplishment, because as a professor, you teach and you need (and love!) the
students to understand. This requires clarity and pedagogical aptitudes, and this is
precisely what one finds in this excellent book: a text written for students. Having
myself taught solid-state physics, in particular recently as an introductory course to
undergraduate students (one semester, three credits), this book would have been
perfect for my class, both with the material covered and with the level of knowledge
in quantum physics, statistical physics and molecular physics expected from
students. The amount of material covered is perhaps a bit large for a one-semester
course, but some more advanced topics may be left for personal reading. This book
fills a gap that is largely unoccupied. There are not many texts that address this topic
in such a comprehensive way, and at the beginner level.
I admit that the title of the book, Solid State Physics, surprised me a bit at first, as
it seems to me that ‘condensed-matter physics’ is perhaps more modern and
inclusive. But I thought ‘What’s in a name? That which we call a rose, by any
other name would smell as sweet’ (William Shakespeare, Romeo and Juliet). We
exchanged on this issue with Professor Colombo and I fully adhere to his views. The
solid state is somehow a subset of the condensed state, one that is idealised and can
be described using the fabulous mathematical toolbox that applies to periodic
systems, including periodic boundary conditions. And this is relatively easy for
students to understand and visualise. While the ideal solid state really does not exist
in nature (just as a circle does not exist), it provides a reference for understanding
more complex systems found in the real world. From a pedagogical viewpoint,
further, understanding the basics of the discipline, i.e. solid-state physics, is a
prerequisite to dealing with more complex situations, i.e. condensed-matter physics,
e.g. with such problems as amorphous systems (lack of periodic order), nano-
materials (importance of boundary effects), etc. To cite Professor Colombo, this
primer and the others in this series ‘are my personal battle to promote the need to
train students, especially at undergraduate level, by passing a strong, robust, general
and universally applicable set of fundamentals’. This statement, I would say, is the

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clear act of faith of a remarkable professor, fully dedicated to his art: kudos to
Professor Colombo.
The book is organised in three parts, each divided in several chapters. The first part,
‘Preliminary concepts’, presents the basic basics. Chapter 1, ‘The overall picture’, is
clever: it answers first the necessary question ‘what are we talking about?’ (crystalline
solids, which are approached from the point of view of diffusion—very nice!), then
moves on to discuss, upfront, concepts that percolate throughout the various topics
which will be addressed in subsequent chapters. The late George K Horton (Rutgers),
who was an authority in the area of dynamical properties of solids, told me once that
writing a paper (or, for that matter, a book chapter or a book) is not like writing a
novel: one should not keep the ‘punch’ for the end but say early where one is going,
and this will serve to light the route that one follows. This is precisely what this first
chapter does (and I wish I had followed George Horton’s advice more often myself!).
The connection between the atomic structure and the complicated many-body
electronic structure is introduced right at the outset (semi-classical approximation,
frozen-core approximation, Born–Oppenheimer approximation, etc). Chapter 2 goes
into the ‘heart of the matter’ and discusses in detail the crystalline structure, both in
direct and in reciprocal space (including an introduction to the Bloch theorem and
one-electron wavefunctions), and inevitable deviations from the ideal picture (defects).
The stage is set.
The second part, ‘Vibrational, thermal, and elastic properties’, emerges naturally
from chapter 2. Once again, Professor Colombo excels in communicating his
message effectively. As it should, the first chapter in this section, chapter 3, deals
with lattice dynamics: modes of vibration, dispersion relations (acoustic and optic
modes), dynamical matrix and, evidently, phonons. The presentation of phonons as
a gas of pseudo-particles within the framework of the quantum theory of the
harmonic crystal, is simple, direct and effective. From a communications viewpoint,
and this diffuses throughout the book, the author is clearly ‘speaking’ to the reader
—the student—just as if he was standing in front of the class. I would have loved to
have been one of his students (I have been on a few occasions)! Chapter 3 continues
with a clever discussion of the experimental determination of the dispersion relations
—an exercise that is not a priori simple and one that is, in my view, essential (and
sometimes overlooked)—and of the vibrational density of states, which saves one
from having to count the modes one by one! Chapter 4 is concerned with the thermal
properties of solids—heat capacity (including the Debye model and the quantum
framework that underlies the theory) and the very important anharmonic effects,
including a very nice section on phonon–phonon interactions for the most avid
readers (with an excellent discussion of the crystal momentum and normal/Umklapp
processes), and thermal transport (Boltzmann equation). The second part closes with
chapter 5 on a discussion of elastic properties (strain, stress, elastic constants),
moving from a discrete, atomic description of the crystal to a continuous description
of it. This topic allows one to connect to the real world: you know that solids are
made of atoms, but you see how they react to pressure or temperature, for example. I
would not normally cover this subject in my undergraduate course—but perhaps I
should!

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 Solid State Physics

The third part, ‘Electronic structure’, is where ‘the beef is’. It consists of five
chapters and begins with a general discussion of electrons in crystals (chapter 6),
more precisely methods for constructing the electronic structure, namely free-
electron, tight-binding and density-functional theory, all of which are developed
in detail in subsequent chapters. This chapter also reviews Fermi–Dirac statistics
and revisits in detail the Bloch theorem and Bloch wavefunctions. It ends with a
thorough discussion of the Kronig–Penney model which provides a simple, clear
introduction to the concept of band structure with its allowed regions and forbidden
regions (gaps). Chapter 7 is devoted to the free-electron theory of metals, and
includes a very nice discussion of the Drude model (and the reasons of its failure)
and of the Sommerfeld theory of conduction. This is one occasion to introduce the
electronic density of states, both at zero (ground-state) and at finite temperature,
showing very nicely the contributions of electrons and phonons to the specific heat,
and how the latter dominates at high temperatures. The chapter closes on a
discussion of relaxation times and scattering, and limitations of the Sommerfeld
theory. The latter topic provides a natural transition to chapter 8 (‘The band theory’)
that conveys many important concepts, notably the nearly-free electron gas (weak
potential) which permits the author to introduce such important ideas as energy
bands and energy gaps, hence explaining the difference between insulators, semi-
conductors and metals. Further to this, Professor Colombo discusses in detail the
tight-binding approach (and he is famous for his work in this area, among others)
and its success in describing the electronic structure. A thorough description of the
band structure follows—with such topics as electron dynamics, field effects,
electrons and holes, etc—and the chapter closes with a short overview of other
approaches to the band structure. Chapter 9, ‘Semiconductors’, is dedicated to the
theoretical framework underlying the physics of semiconductors—important for
applications—and, in particular, to electron transport. It introduces such important
concepts as doping, charge transport, carrier statistics and populations, optical
(photon) absorption and related carrier transitions and excitons. Finally, chapter 10
introduces the now ubiquitous density-functional theory—Hohenberg–Kohn and
Kohn–Sham theorems and practical implementation of the method; this is a great
idea, not common in solid-state physics books, as it allows students to connect to
topical research and shows them that quantum mechanics is a ‘practical tool’!
A few more notes; in line with the logic of ‘tell them what you are going to tell
them’, each chapter starts by a summarising ‘syllabus’. This is extremely useful and
helps give a direction to the reader. Further, the book includes several appendices
where specific and useful topics are discussed (like, for instance, ‘Essential thermo-
dynamics’, ‘The tight-binding theory’). Finally, the text contains many illustrations
that are very well done and very informative; graphical representations of complex
(or abstract) concepts that describe real-world objects is extremely useful for
students to develop their intuition.
This book is a ‘primer’ and, in that respect, cannot cover the subject matter in an
exhaustive manner. Yet, I find it to be remarkably complete, in particular for a
book intended to undergraduate students. And all of this in 250 pages is a genuine

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tour de force. True, there are some topics that could have been covered
(e.g. superconductivity), but there is already a lot of material for a one-semester
course. I would have loved to have this book for my class. Thanks to Professor
Colombo for this excellent addition to the literature. I predict that it will become a
‘classic’.
Laurent J Lewis
Département de physique
Université de Montréal
September 2020

xv
Presentation of the ‘primer series’

This is the second volume of a series of three books that, as a whole, account for an
introduction to the huge field usually referred to as ‘condensed matter physics’: they
are respectively addressed to atomic and molecular physics, to solid state physics,
and to statistical methods for the description of classical or quantum ensembles of
particles. They are based on my 20 year experience of teaching undergraduate
courses on these topics for bachelor-level programs in physical and engineering
sciences at the University of Cagliari (Italy).
The volumes are called ‘Primers’ to underline that the pedagogical aspects have
been privileged over those of completeness. In particular, I selected the contents of
each volume so as to keep limited its number of pages and so that the topics actually
covered correspond to the syllabus of a typical one-semester course.
More important, however, was the choice of the style of presentation: I wanted to
avoid an excessively formal treatment, preferring instead the exploration of the
underlying physical features and always placing phenomenology at the centre of the
discussion. More specifically, the main characteristics of this book series are:
• emphasis is always given to the physical content, rather than to formal
proofs, i.e. mathematics is kept at the minimum level possible, without
affecting rigour or clear thinking;
• an in-depth analysis is presented about the merits and faults of any
approximation used, incorporating also a thorough discussion of the con-
ceptual framework supporting any adopted physical model;
• prominence is always on the underlying physical basis or principle, rather
than to applications;
• when discussing the proposed experiments, the focus is given to their
conceptual background, rather than to the details of the instrumental setup.

Despite the tutorial approach, I nevertheless wanted to follow the Italian academic
tradition, which provides even the elementary introduction to condensed matter
physics at a quantum level. I hope that my efforts have optimally combined ease-of-
access and rigour, especially conceptually.
The intentionally non-encyclopaedic content and the tutorial character of these
‘Primers’ should facilitate their use even for students not specifically enrolled in a
university curriculum in physics. I hope, in particular, that my textbooks could be
accessible to students in chemistry, materials science and also of many engineering
branches. In view of this, I have included a brief outline of non-relativistic quantum
mechanics in the first ‘Primer’, a subject that does not appear in the typical
engineering curricula. For the rest, classical mechanics, elementary thermodynamics
and Maxwell theory of electromagnetism are used, to which all students of natural
and engineering sciences are normally exposed.
Each ‘Primer’ is organised in parts, divided into chapters. This structure is
tailored to facilitate the planning of a one-semester course: these volumes aim at

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being their main teaching tool. More specifically, each part identifies an independent
teaching module, while each chapter corresponds to about two weeks of lecturing.
I cannot conclude this general introduction without thanking the many students
who, over the years, have attended my courses in condensed matter physics at the
University of Cagliari. Through the continuous exchange of ideas with them I have
gradually understood how best to organise my teaching and the corresponding study
material. As a matter of fact, the contents that I have collected in these volumes were
born from this very fruitful dialogue.
Luciano Colombo
Cagliari, June 2019

Acknowledgements
I am really indebted to Ms C Mitchell—Commissioning Editor of IOP Publishing—
and Mr Daniel Heatley—ebooks Editorial Assistant of IOP Publishing—for their
great enthusiasm in accepting my initial proposal and for shaping it into a
suitable editorial product. I also warmly acknowledge Ms Emily Tapp—
Commissioning Editor of IOP Publishing—and Mr R Trevelyan—Editorial
Assistant of IOP Publishing—for supporting my writing efforts of the second
Primer with great professionalism, always promptly replying to my queries and
clarifying my doubts. Overall, their assistance has been really precious.

xvii
 Introduction to: ‘Solid state
physics: a primer’

In my view the ‘solid state’ is an idealised state of condensed matter where

translational invariance rules most physical phenomena. By ‘idealised’ I simply
mean that real solid materials are in fact not at all perfectly periodic: they contain
defects, either point-like and extended, as imposed by thermodynamics; they are
contaminated or doped, as result of natural or intentional processes randomly
altering their pristine chemistry; they have a finite size, that is they have surfaces.
While all these phenomena are indeed fascinating topics of condensed matter
physics, a first pedagogical approach to solids is in my opinion better framed within
the idealisation of a surface-free and periodically invariant system, namely a crystal.
This allows one to develop a simple formalism—in several important cases analyti-
cally workable—which leads to quite a few important achievements like, for
instance, understanding the difference between metals and insulators or describing
the thermal transport in terms of the corpuscular phonon language.
A modern approach to the fundamental physics of crystalline solids does require
the use of quantum mechanics, which is the language adopted in any chapter,
although classical physics is often used in order to set up the first approach to a
problem or, through the manifestation of its inadequacy, to remark the real need of
a full quantum treatment. Furthermore, despite the tutorial approach, some robust
basic notion on atomic physics is nevertheless needed in order to fully appreciate the
physics discussed in this book.
A possible subtitle for this Primer could have been: ‘a first introduction to the
physics of phonons and electrons in crystalline systems’. As a matter of fact, these
are the only two topics—treated separately in view of the fundamental adiabatic
approximation—here discussed, among many other important ones which are
simply omitted, since this textbook is primarily intended as the study material for
a first undergraduate course in condensed matter physics. Despite this somewhat
narrow perspective, I believe that the focus on the physics of phonons and electrons
in periodic crystals is enough to define the minimal-complexity framework needed to
understand the fundamentals of condensed matter physics. It is understood that, in
order to complete the student’s background in this field, a more advanced course on
dielectric, optical, magnetic ordering, and many-body properties is mostly needed,
including topics like the physics of defects and surfaces, the mechanical properties of
solids, and superconductivity.
This Primer is divided in three parts: the first one provides, at first, a careful
account of the hierarchy of approximations adopted to treat the solid state and,
next, the formal tools to describe the crystalline atomic structures; the second part
deals with the vibrational, thermal, and elastic properties of crystal solids; the third
part is addressed to their electronic structure. A short presentation of the density
functional theory—which is nowadays considered the ‘standard model’ to treat

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 Solid State Physics

condensed matter systems—is offered in the last chapter. Nine appendices are added
to the text, each focussed on some technical development which, at first reading, can
be skipped without compromising the general understanding of the arguments
developed in the main text. A bibliography is added to each chapter as a guideline
for further reading.
The volume contains many figures, most of which are ‘conceptual’, that is: they
are basically intended to provide a graphical representation of the main ideas and
concepts developed in the written part. Tables with numerical values of important
physical properties are included as well, in the attempt to provide the reader with
information useful to ‘quantify’ the physical results presented. Finally, a list of all
the mathematical symbols used in the volume is given at the beginning as an
orientation guide while reading.
Luciano Colombo
Cagliari, February 2021

Acknowledgements
I am really indebted with many friends who helped me by critically reading (in part or
totally) the pre-editorial version of the book (for this special thanks go to Dr G Malloci
and Professor C Melis, both at the Department of Physics, University of Cagliari, Italy,
and to Dr S Giordano, CNRS and University of Lille, France) or by providing me with
the output of their calculations used to plot some figures (Dr G Fugallo, CNRS and
University of Nantes, France and Dr A Antidormi, ICN2, Barcelona, Spain) or for
suggesting updated bibliographic entries. Finally, Professor C Melis is once again
warmly thanked for assisting me in generating several figures reporting experimental
data taken from literature. The care these friends have taken in checking my original
manuscript has corrected several unclear passages and many misprints, thus greatly
improving my presentation. If there are still errors or omissions they should be
attributed solely to me.

xix
 Author biography

Luciano Colombo
Luciano Colombo received his doctoral degree in physics from the
University of Pavia (I) in 1989 and then he was a post-doc at
the École polytechnique fédérale de Lausanne (CH) and at the
International School for Advanced Studies (I). He became assistant
professor (tenured) at the University of Milano (I) in 1990, next
moving to the University of Milano-Bicocca (I) in 1996 for an
equivalent position. Since 2002 he has been full professor of
theoretical condensed matter physics at the University of Cagliari (I) and since
2015 fellow of the ‘Istituto Lombardo—Accademia di Scienze e Lettere’ (Milano, I).
He has been the principal investigator of several research projects addressed to solid-
state and materials physics problems, the supervisor of more than 80 students
(at bachelor, master, and PhD level), and the mentor of about 20 post-docs. He is the
author, or coauthor, of more than 276 scientific articles and 9 books (this included).
More about him can be found at: http://people.unica.it/lucianocolombo.

xx
 Symbols

α optical absorption coefficient

αT isothermal compressibility
β thermal expansion coefficient
γ weighted Grüneisen parameter
γsq Grüneisen parameter for (sq) phonon mode
Δ2R(t ) atomic mean square displacement
ϵ0 vacuum permittivity
ϵij ijth component of the second-rank strain tensor
ϵr(ω ) frequency-dependent relative permittivity of a material
κe thermal conductivity of electrons (generic)
κ eDrude thermal conductivity of electrons, calculated according to the Drude
theory
κ eSommerfeld thermal conductivity of electrons, calculated according to the Sommerfeld
theory
κl lattice thermal conductivity
κtot total thermal conductivity
λ first Lame’ coefficient
λe electron mean free path
λsq mean free path of the (sq) phonon mode
μ second Lame’ coefficient
μc chemical potential
μe electron mobility
μh hole mobility
μB electron Bohr magneton
μN nuclear magneton
μx exciton effective mass
ρ(r) electron density
ρGS (r) ground-state electron density
ρHEG electron density in a homogeneous electron gas
σe direct-current electron conductivity
σh direct-current hole conductivity
σtot direct-current total conductivity
τe electron relaxation time
τh hole relaxation time
τsq relaxation time (lifetime) of the (sq) phonon mode
φKS(r) Kohn–Sham orbital
Φ(r, R) total crystalline wavefunction
Φe flux of electrons
Φh flux of holes
Φph flux of photons
Ψ (eR)(r) total electron wavefunction (when ions are clamped in the R configuration)
Ψn(r) total ion wavefunction
ζ atomic valence
ωp plasma frequency
ωE Einstein frequency
ωD Debey frequency

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 Solid State Physics

a CB deformation potential for the conduction band

aH Bohr radius
aVB deformation potential for the valence band
a0 lattice constant of cubic crystals
{a1, a2, a3} translation vectors of the direct lattice
A atomic mass number
{b1, b2 , b3} translation vectors of the reciprocal lattice
B bulk modulus
B externally applied magnetic field
Bd Burgers vector of a dislocation
c speed of light
cVe constant-volume specific heat (electron contribution)
cVl constant-volume specific heat (lattice contribution)
cVtot total constant-volume specific heat
Cij ijth component of the elastic stiffness tensor in the compact Voigt
notation
Cijkh ijkhth component of the fourth rank elastic stiffness tensor
CVDrude(T ) constant-volume heat capacity (Drude interpolation scheme)
CVEinstein(T ) constant-volume heat capacity (Einstein model)
CVquantum(T ) constant-volume heat capacity (full quantum expression)
Voigt matrix representation of the elastic stiffness tensor in the compact Voigt
notation
De electron diffusivity
Dh hole diffusivity
Dij ijth component of the elastic compliance tensor in the compact Voigt
notation
Dijkh ijkhth component of the fourth rank elastic compliance tensor
e electron charge (absolute value)
ecohesive cohesive energy per atom of a crystal
E externally applied electric field
Ec energy at the bottom of the conduction band
EcHEG correlation energy of a homogeneous electron gas
EcLDA correlation energy in the local density approximation
Ecohesive cohesive energy of a crystal
Ee(R) total electron energy (when ions are clamped in the configuration R )
Ev energy at the top of the valence band
ExHEG exchange energy of a homogeneous electron gas
ExLDA exchange energy in the local density approximation
EF Fermi energy
ECB(k) dispersion relation for the conduction band
EVB(k) dispersion relation for the valence band
ET total energy of the crystal
Ex exciton binding energy
E x,n exciton discrete energy spectrum
E xc[ ρ(r)] exchange-correlation energy functional
ExcLDA exchange-correlation energy in the local density approximation
F [ ρ(r)] electron total energy functional

xxii
 Solid State Physics

g (r ) pair correlation function

gn(E ) electronic density of states per unit volume for the band n of a
semiconductor
ḡCB (E ) gn(E ) for the conduction band in parabolic bands approximation
ḡVB (E ) gn(E ) for the valence band in parabolic bands approximation
G lattice vector (reciprocal lattice)
G Gibbs free energy
G carrier generation rate
G (E ) electronic density of states (eDOS) of a metal
G(ω ) vibrational density of states (vDOS) or phonon density of states
Gn(E ) electronic density of states for the band n of a semiconductor
G D (ω ) vibrational density of states (Drude interpolation scheme)
h Planck constant
ℏ ℏ = h/2π
H enthalpy
î unit vector of the x-axis
ĵ unit vector of the y-axis
JCB electron charge density current (conduction band)
JVB hole charge density current (valence band)
Jq,tot total charge density current
Jh heat density current
k̂ unit vector of the z-axis
k electron wavevector
kB Boltzmann constant
kF Fermi wavevector
Ld line vector of a dislocation
me electron rest mass
m e* electron effective mass
m h* hole effective mass
mn neutron rest mass
mp proton rest mass
nˆ = (n1, n2 , n3) unit vector of a generic direction in space
NA Avogadro number
nBE(s q, T ) Bose–Einstein distribution for (population of) the (sq) phonon mode
nFD(E , T ) Fermi–Dirac distribution for (population of) the electron level with
energy E
n̄(s q, T ) non-equilibrium population of the (sq) phonon mode
Nc effective density of states in conduction band
Nv effective density of states in valence band
P pressure
q phonon wavevector (or wavevector of a lattice vibrational wave)
r generic electron position within a crystal
rx electron–hole orbital radius (exciton size)
R generic ion position within a crystal
R carrier recombination rate
Rb ion position within the basis
RH Rydberg constant for hydrogen
Rl lattice vector (direct lattice)
S entropy

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 Solid State Physics

T temperature
TD Debye temperature
TF Fermi temperature
Tij ijth component of the second rank stress tensor
T̂Rl translation operator such that TˆRl f (r) = f (r + Rl)
u elastic energy density
u ionic displacement vector
U internal energy
U (R) crystalline ion energy (for ions in configuration R )
vd electron drift velocity in a weak field regime
v th
e electron thermal velocity
vF electron Fermi velocity
vsat electron saturation velocity in a strong field regime
vCB(k) velocity of an electron in the conduction band
vVB(k) velocity of an electron in the valence band
V volume (generic)
Vc volume of the primitive unit cell (direct lattice)
Vcfp crystal field potential
Vee electron–electron Coulomb interaction potential
VHartree Hartree potential
Vne nucleus–electron Coulomb interaction potential
Vnn nucleus–nucleus Coulomb interaction potential
Vxc exchange-correlation potential
w atomic weight
Z atomic number

xxiv
 Part I
Preliminary concepts
 IOP Publishing

Solid State Physics

A primer
Luciano Colombo

Chapter 1
The overall picture

Syllabus—After defining what condensed matter (and, in particular, the crystalline solid
state) is, we build the theoretical framework of minimum complexity able to deal with
the corresponding atomic-scale physics. Proceeding from a limited set of basic
definitions, a hierarchy of approximations is introduced, like, e.g. the separation between
the nuclear and the electronic degrees of freedom and the single-particle picture.

1.1 Basic definitions

The first task we must accomplish is defining the physical system we are actually
interested in. This semantic exercise is quite important, since it will properly define
the topic treated in this textbook: the physics of crystalline solids.
A fruitful constitutive hypothesis to start from is embodied by the atomistic
picture [1, 2] which, relying on robust experimental evidence, states that ordinary
matter is made by elementary constituents known to be atoms1. We agree that
condensed matter forms whenever a very large number of such atoms (belonging to
just one or more chemical species) tightly bind together by electrostatic interactions.
Both features are indeed necessary in order to sharply define the state of aggregation
we are interested in: (i) the fact that the number of atoms is very large allows us to
exclude single molecules2 from the horizon of our interest, while (ii) the strong
character of their mutual interactions allows us to neglect the case of gaseous
systems.
The definition just given is actually very generic and it does not allow us to
distinguish between two paradigmatically different situations. In order to clarify and
resolve this ambiguity, let us consider a sample of condensed matter and let us label

1
By ‘ordinary’ we mean matter as typically organised in materials found on Earth.
2
A single molecule may have as many as thousands of atoms, like e.g. RNA or DNA nucleic acids: this is still
a comparatively smaller number than found in a material specimen with any dimension from the micro- to the
macro-scale.

doi:10.1088/978-0-7503-2265-2ch1 1-1 ª IOP Publishing Ltd 2021

 Solid State Physics

by R α(t ) the position of its αth atom at time t. We define the mean square atomic
displacement Δ2R(t ) as
N
1
Δ2R(t ) = ∑ R α(t ) − R α(0) 2 , (1.1)
N α= 1

where R α(0) represents the initial position of the αth atom and N is the total number
of particles in the system. It is understood that the system is in equilibrium at
temperature T. The calculation of Δ2R(t ) for a silicon sample is reported in figure 1.1
at two different temperatures, respectively, above and below its melting temperature
TmSi = 1685 K. If, according to atomic diffusion theory [3], we now link such a
quantity to the corresponding diffusion coefficient D(T )
1 Δ2R(t )
D(T ) = lim , (1.2)
t →+∞ 6 t
we immediately realise that below TmSi the sample does not show any self-diffusion
characteristics3 while above TmSi it flows. In other words, the definition of condensed
matter given above allows both solid and liquid systems to be called condensates,
despite their physics being largely different. Therefore, we make it clear that from
now on we will focus our attention only on the solid state, i.e. only on condensed
matter systems that do not show any diffusive behaviour (an introduction to the
fascinating physics of liquids can be found elsewhere [4]).
The description just given of solid state is still inconveniently too general, since it
is compatible with atomic arrangements showing quite unalike structural order

Figure 1.1. The mean square displacement of silicon at T = 300 K (black) and at T = 1800 K (red). Data are
calculated by molecular dynamics simulations. Arbitrary units (a.u.) are used in both axes.

3
The phenomenological Fick law of diffusion J(T ) = −D(T )∇C links the amount of matter J flowing through
a unit area in a unit time interval to the concentration gradient ∇C across that unit area. Accordingly, zero-
diffusivity systems do not flow.

1-2
 Solid State Physics

features: we must duly refine this notion. To this aim let us consider a monoatomic
solid state system and let us introduce its pair correlation function4 g (r ) which at a
generic position r is defined as
N
1
g (r ) = ∑ δ(r − R αβ) , (1.3)
Nρ α≠β=1

where ρ = N /V is the average number density in the system and R αβ = Rβ − R α is

the distance vector between atoms α and β. The 〈⋯〉 brackets indicate that the sum is
time- and direction-averaged by keeping constant the number of particles in the
system as well as its volume and temperature, while the δ-symbol is defined usually
as: it is worth one if its argument is null, otherwise it is worth zero.
Equation (1.3) describes how the number density varies within the system: more
specifically, it provides the probability of finding a pair of (α , β ) particles at a
distance r = Rβ − R α apart, relative to the corresponding probability calculated for
a uniform (random) distribution of similar atoms at the same density. In figure 1.2
we report the pair correlation function for two solid state silicon systems, both at
room temperature. The peaks of the g (r ) function indicate local densifications of
matter, while its minima indicate local rarefactions, in both cases referred to the

Figure 1.2. Pair correlation function for a crystalline (c-Si, red) and an amorphous (a-Si, blue) silicon sample
with same density and both kept at room temperature. The actual position of the first-, second-, and third-
coordination peak is reported in the case of c-Si. The dashed line represents the case of a uniformly distributed
mass with the very same density. Data are calculated by molecular dynamics simulations.

4
A thorough discussion of the pair correlation function is found elsewhere [3, 4].

1-3
 Solid State Physics

normalised value g (r ) = 1 corresponding, as anticipated above, to a uniformly

distributed mass. These peaks are precise markings of the discrete distribution of the
atoms forming the solid. For instance, the first peak reveals the position of the first
coordination shell, i.e. the average distance at which the first nearest neighbours are
found. Furthermore, because of the very definition of pair correlation function, we
argue that the integral of each peak represents the average number of neighbours
found at that distance.
It is apparent from figure 1.2 that there is a striking difference between the two
samples, namely: while a persistent pattern is found in one case, no structure at all is
observed beyond ∼4.5 Å in the other one. We conclude that figure 1.2 allows us to
distinguish between a discrete distribution of atoms characterised by a regular
repeated pattern and a discrete distribution of atoms showing no regularity above the
shell of second-next-neighbours. Solid state systems corresponding to the first
situation are characterised by long-range structural order and are named crystals;
on the other hand, solids showing just a short-range structural order are referred to as
amorphous systems. We state that from now on we will focus our attention only on
solid state crystalline materials.
In order to conclude the task of unambiguously defining the material systems to
be studied, we still have to address a last important issue. It is trivial to observe
that any object has a size, possibly even very large (especially when compared with
typical interatomic spacings or even molecular sizes), but in any case finite. In
short: a material system has boundary surfaces. This implies that the set of atoms
forming any condensed matter system can be divided into two categories: bulk
atoms and surface atoms. They feel a vastly different physics, as intuitively
understood by considering that surface atoms (i) are under-coordinated with
respect to bulk ones and that (ii) they are exposed to external perturbations (or,
equivalently, they interact with the environment surrounding the system). A
primary consequence is that surfaces usually show a lower degree of crystallinity
than the bulk regions: this is an annoying issue, in view of the above
comfortable definition of crystalline state. Furthermore, a large number of
phenomena only occur at surfaces, like growth, catalysis, or chemical bond
reconstruction to name just a few. Surface physics is, therefore, an industry with
many specifics and it represents a very lively and rapidly expanding sub-discipline
of modern condensed matter physics [5–9]; nevertheless, we will not deal with it
because it largely falls beyond the scope of this primer. Rather, we are primarily
interested in bulk properties and we will focus on them by invoking the following
justification: by assuming a nearly constant number density ρ, the number of bulk
and surface atoms, respectively, scales as ∼ρV and ∼ρV 2/3 for a system of volume
V. This implies that for a ‘large enough’ specimen, the number of surface atoms
becomes negligibly small as compared to bulk ones.
Now, the point is: what does ‘large enough’ mean? There is no rigorous answer to
such a seemingly innocent question. For instance, we could be tempted to provide a
simple pragmatic reply: such a condition is reached whenever bulk-like properties
become unaffected by surface phenomena. Unfortunately, this apparently reassuring
definition is not accurate enough to deal with a number of important issues like,

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 Solid State Physics

e.g. the proper definition of the work required to extract an electron from a solid5 or
the accurate description of polarisation effects in ionic crystals6.
The conclusion is that the present introduction to solid state physics, basically
focussed on the bulk properties of crystalline materials, is made conceptually clean
only by introducing an abstraction, namely by introducing suitable periodic boundary
conditions that allow for dealing with ideal surface-free systems7. The procedure is
illustrated in figure 1.3 in the case of a one-dimensional crystalline solid8: a number
N of atoms is considered so that they are arranged with the proper crystalline
regularity (that is: atoms are equally spaced by an amount a hereafter referred to as
the lattice spacing); next, we join the two terminal ends of the crystalline chain so
that the first and last atom actually overlap, their positions coincide, and they are
further treated as one particle. The formalisation of this procedure is

α
a
α

uα
α
Figure 1.3. Graphical representation of the Born–von Karman periodic boundary condition for a one-
dimensional crystalline solid with lattice spacing a.

5
Such an energy does depend on some surface-specific features like the distortion of the electronic charge
distribution with respect to the inner bulk.
6
Any arbitrary truncation of a solid of such a kind generates a different charge distribution at the surface
which, because of the long-range character of Coulomb interactions, differently affects the physics in the bulk
region.
7
We remark that selecting a solid state condensate with crystal order, among many other possible atomic
gatherings, was just a matter of choice. On the other hand, the use of conditions that free us from the existence
of something as real as a surface corresponds to adopting a model. Accordingly, the physics we will elaborate
within its framework will be paradigmatically representative, conceptually relevant, and predictive under many
concerns, but unavoidably incomplete.
8
Of course, the choice of a one-dimensional system is made just for convenience of graphical representation: as
a matter of fact, boundary conditions can be applied in one, two, or three dimensions, as needed.

1-5
 Solid State Physics

straightforward: let us indicate by u(R α ) the displacement of the αth atom along the
chain9. Imposing the above periodic boundary conditions is equivalent to placing
u(R α ) = u(R α + (N − 1)a ) and ψ (ri ) = ψ (ri + (N − 1)a ), (1.4)
where we have completed the picture by specifying as well how such conditions act
on the wavefunction ψ (ri ) of the ith crystalline electron10. We take this opportunity
to specify that throughout this book the positions of the atoms and electrons will be
indicated by upper and lower case letters, respectively, and labelled by Greek and
Latin indices, respectively. Equation (1.4), known as the Born–von Karman
condition, completes the reasoning aimed at removing surfaces from our problem.
We finally remark that following this construction we effectively reduce the number
of atoms of the system from N to (N − 1). However, provided that N is ‘large
enough’ we can set (N − 1) ∼ N . It is easy to understand that the expression ‘large
enough’ in this case is used in a less dense way of physical consequences than
previously discussed: it is just an allowable mathematical convenience.
In conclusion, we have eventually set all terms unequivocally defining what ‘a
bulk property of a crystalline system’ indeed means. After this semantic introduction
we are ready to start with the true solid state physics.

1.2 Synopsis of atomic physics

In order to fully exploit the atomistic picture that we are going to use as the
constitutive hypothesis to describe solid state materials, we need to recall some basic
notion of atomic physics [1, 2, 10], define the notation adopted throughout the
present volume, and set the value of some fundamental physical constants.

1.2.1 Atomic structure

We know that an atom is a bound system consisting of a nucleus with a positive
charge +Npe , where Np is the atomic number, that is the number of protons, and a set
of Z electrons, each carrying a charge −e . We recall that e = 1.602 19 × 10−19 C is
the elementary electric charge. If Z = Np then the atom is in a neutral configuration,
while if Z ≠ Np then we say that the atom has been ionised (either positively or
negatively provided that Z is smaller or larger than Np, respectively). The nucleus
also contains a number Nn of neutrons, carrying no electric charge. While all
electrons have the same mass me = 9.109 × 10−31 kg, the nucleus of each chemical
species has instead a specific mass M determined as: M = (Np + Nn )m p, where
m p = 1.672 × 10−27 kg is the proton mass11. We remark that A = Np + Nn is referred

9
The notion that atoms oscillate around their equilibrium position is here assumed on a purely intuitive basis,
but it will be extensively discussed in the next chapters.
10
We have anticipated an important twofold concept, namely that under some suitable approximations we can
treat electrons as single particles and we can separate atomic motions from electron dynamics. The next section
is devoted to this issue.
11
For our purposes it is sufficient to set the same mass to both protons and neutrons.

1-6
 Solid State Physics

to as the atomic mass number. Atoms with the same number of protons, but a
different number of neutrons are referred to as isotopes.
As for nuclei, we will further neglect their inner structure by treating them as
point-like, massive, and charged objects12. This is indeed a very good approximation
for any situation described in this volume and, therefore, protons and neutrons will
no longer enter as single objects in our theory. On the other hand, electrons will be
individually addressed. Nuclei and electrons are inherently non-classical objects and,
therefore, they must be duly described in quantum mechanical terms.

1.2.2 Angular and magnetic momenta

In addition to their charge and mass, electrons are further characterised by their spin
[2, 10]: an intrinsic angular momentum S, whose square modulus S2 and z-component
obey the following quantisation rules
S 2 = s(s + 1)ℏ2
(1.5)
Sz = ms ℏ,
with s = 1/2 and ms = ±1/2 (spin ‘up’ or ‘down’) known as the spin quantum
numbers and ℏ = h/2π = 1.054 46 × 10−34 J s is the reduced Planck constant. An
intrinsic spin magnetic moment MS , similarly quantised, is attributed to each electron
according to
μ
M S = −gS B S, (1.6)
ℏ
where μB = eℏ/2me = 9.273 2 × 10−24 J T−1 is the Bohr magneton, and gS ∼ 2 is the
spin g-factor.
Similarly, each nucleus, in addition to being charged, also carries a magnetic
moment MN [11] which for our purposes is conveniently defined as
μN
MN = gN N, (1.7)
ℏ
where gN is the nuclear g-factor (a dimensionless constant), μN is the nuclear
magneton
m
μN = e μB = 5.050 82 × 10−27 J T−1, (1.8)
mp

and N is the nuclear spin or, equivalently, the total nuclear angular momentum.
Electrons are also characterised by an orbital magnetic moment, since their orbital
motion around the nucleus corresponds to a current or, equivalently, to a magnetic
moment ML defined as [1, 2, 10]

12
Actually, nuclei are neither really point-like nor is their charge spherically distributed. Because of this,
nuclear electric quadrupole moment effects can be observed in high-resolution atomic spectroscopy measure-
ments [10].

1-7
 Solid State Physics

μB
ML = −gL L, (1.9)
ℏ
where gL is the orbital g-factor and L is the electron orbital angular momentum
obeying the quantisation rules
L2 = l (l + 1)ℏ2
(1.10)
Lz = ml ℏ,
cast in terms of the orbital quantum number l = 0, 1, 2, … and of the magnetic
quantum number ml = 0, ±1, ±2, … , ±l . The spectroscopic notation is widely
adopted to label quantum states differing by l: we will set l = 0→ s-states, l = 1→
p-states, l = 2→ d-states, and so on [2].

1.2.3 Electronic configuration

The central problem of the physics of atoms is to determine their ground-state
configuration, that is: the distribution of their electrons, among all available
quantum states, corresponding to the minimum total energy. For a multi-electron
atom this task is accomplished by following a rather complex procedure, qualita-
tively summarised below. A full account can be found elsewhere [1, 2, 10].
The first step consists in solving the complete Schrödinger equation13 for the
atom: a formidable indeed many-body quantum problem. The full scenario contains
electrostatic interactions (among electrons and between the nucleus and each
electron) as well as magnetic interactions (among all existing magnetic dipoles).
Coulomb interactions are by far the strongest ones and they determine the main
features of the atomic energy spectrum which can be calculated, for instance, within
the central field approximation (CFA)14. Here each electron is treated as a single-
particle undergoing an average central field due to the nucleus and the remaining
electrons. In this way the many-body problem is reduced to Z single-particle ones,
each separately solved by ordinary methods of atomic physics. The resulting CFA
electron wavefunctions ψnlmCFA
l
(r) = R̄nl (r ) Ylml (θ , ϕ ) are written in polar coordinates
(the central field has by construction a spherical symmetry!) as the product between
a radial function R̄ and a spherical harmonic function Y. Accordingly, each
quantum state is labelled by three quantum numbers, namely: the principal quantum
number n = 1, 2, 3, … and the l and ml ones already introduced in section 1.2.2
where their values have been assigned15. A twofold picture emerges that (i) the
energy spectrum is discrete and (ii) allowed atomic quantum states are organised in

13
For brevity we will hereafter adopt a widely used notation abuse: what in the text has been called the
‘Schrödinger equation’ should be more rigorously indicated as the eigenvalue equation for the Hamiltonian
operator describing the system of our interest. In short: time dependence has been already eliminated from our
problem.
14
Although some important many-body features are overlooked in this theory, it nevertheless contains all the
essential features needed for the present discussion.
15
We remark that the emerging organisation of quantum states is justified straightforwardly within the central
field approximation, but it is also fruitfully adopted in more refined quantum approaches like the Hartree, the
Hartree–Fock or the configuration-interaction theory.

1-8
 Solid State Physics

shells and sub-shells, respectively, corresponding to a given value of the n and of

l = 0, 1, 2, … , (n − 1).
Once the energy spectrum has been determined by only considering Coulomb
couplings, we can add magnetic interactions as twofold perturbations of electronic
and nuclear nature. The semi-classical vector model of the atom provides a very
direct and pedagogical treatment of magnetic interactions among electrons: each
electron pair i , j undergoes the following interactions
• orbit–orbit coupling, corresponding to an energy Eoo = ξLL Li · Lj
• spin–spin coupling, corresponding to an energy Ess = ξSS Si · Sj
• spin–orbit coupling, corresponding to an energy Eso = ξLS Li · Sj

where ξLL, ξSS and ξLS are the magnetic coupling constants. A major task of atomic
physics is providing their accurate determination. While electrostatic interactions
determine the main features of the atomic energy spectrum, magnetic interactions
provide its fine structure. Next, we eventually consider the nucleus by observing that
its dipole moment given in equation (1.7) feels both the magnetic field Belectrons
generated by the orbital motion of the electrons and the dipole–dipole interactions
between electron and nuclear spins. Accordingly, a new energy term Ehp comes into
play
E hp = −MN · Belectrons , (1.11)
providing the ultimate hyperfine structure of the atomic energy spectrum.
The next step eventually provides the electronic configuration we are interested in.
It basically consists in placing electrons on each allowed state, starting from the
lowest-energy one, by fulfilling both the Pauli principle and the Hund rules.
According to the Pauli principle, the maximum number of electrons that can be
accommodated in a sub-shell with orbital quantum number l is 2(2l + 1) and,
therefore, we can place up to 2n 2 electrons in a shell with principal quantum number
n. A sub-shell is said to be incomplete if the number of electrons there accom-
modated is smaller than 2(2l + 1). Following a standard picture, fully exploited in
the periodic table of elements, we will refer to electrons belonging to the highest-
energy and incomplete sub-shell as valence electrons, while electrons belonging to
low-energy and complete sub-shells will be named core electrons.

1.3 Setting up the atomistic model for a solid state system

Trying to plug the full atomistic picture into condensed matter physics is a hopeless
enterprise and an unreasonable choice as well: the resulting mathematical problem
would be too complicated to be solved by any analytical or numerical tool and,
furthermore, several details specific to the single atomic system are actually marginal
when matter is organised in condensates. In order to proceed, we need approxima-
tions. Far from being a fallback choice, this way of proceeding will allow us to bring
out the most salient physical aspects of the solid state, avoiding an excess of detail
that, in reality, would not translate into new meaningful knowledge. We are
therefore going to develop a hierarchy of approximations that will actually constitute

1-9
 Solid State Physics

the backbone of our working model for crystalline solids. In the following chapters
these approximations will be critically readdressed whenever some phenomenology
questions their validity.

1.3.1 Semi-classical approximation

In general, we will treat electric, charge current, and magnetic effects according to
the classical Maxwell electromagnetism; on the other hand, ion and electron physics
will be described according to quantum mechanics. However, there will be some
exceptions to this general choice.
First of all, we remark that the process of emission or absorption of electro-
magnetic energy by any material system will be described through the concept of
photon. This approach represents the most basic way to include the quantum nature
of electromagnetic radiation into our elementary theory. We will not go any further
because any improvement of this picture, admittedly simplified, would fall beyond
the scope of this tutorial introduction to solid state physics.
Finally, the dynamics of crystal lattices will be firstly treated by classical
mechanics in order to easily catch the phenomenology of ionic vibrations. Next, a
fully quantum picture will be developed through the concept of phonon.

1.3.2 Frozen-core approximation

To a large extent, the chemical properties of an atom are dictated by its valence
electrons [2]. In particular, valence electrons rule over the formation mechanism of
interatomic bonds, so ultimately affecting most of the physical properties in a
condensed matter system. This suggests that core electrons are expected to play a
minor role in determining most of solid state properties. We can exploit this
observation by introducing the frozen-core approximation which will greatly simplify
the picture. This approximation can be cast in a very simple form according to the
scheme
atom = 
nucleus
 + core
  + valence electrons
electrons
ion
= ion + valence electrons,
which suggests the following: we will implement the atomistic description of a
crystalline solid assuming that it consists of a collection of ions and valence electrons.
The former will be described as point-like objects with a nuclear mass specific to
their chemical species16 and carrying a positive charge. If there are in total
Z = Zc + Zv electrons (where Zc and Zv are the number of core and valence
electrons, respectively), then the ionic charge Q will be assigned the value Q = +Zv,
in units of the elementary charge e.
The main advantage of the frozen-core approximation is a dramatic reduction of
the number of electronic degrees of freedom to deal with: for instance, the main
features of the electronic structure in a silicon crystal will be studied by considering

16
The total mass of the core electrons is definitely negligible with respect to that of the bare nucleus.

1-10
 Solid State Physics

just four valence electrons for each ion, instead of the full 14 electron set found in a
silicon atom. In conclusion, hereafter when referring to ‘electrons’ we will actually
mean ‘valence electrons’, while core ones will never be addressed since they are
attached to nuclei forming ions.

1.3.3 Non-magnetic and non-relativistic approximations

A detailed calculation of the three coupling constants ξLL, ξSS and ξLS providing the
atomic fine structure leads to important information, namely: magnetic interactions
are comparatively much smaller than electrostatic ones [10]. This is confirmed
experimentally by observing that the splitting of the energy levels caused by the
above coupling terms is always some good order-of-magnitude smaller than their
separation predicted by only considering Coulomb interactions [2]. Therefore, our
choice will be to completely neglect magnetic interactions among electrons. This
choice is straightforwardly extended to the nucleus, following a simple argument.
The hyperfine energy terms given in equation (1.11) depend on MN , that is, on the
nuclear magneton. Now, it turns out that μN ≪ μB since the proton mass is much
larger than the electron one and, therefore, hyperfine interactions are also negligibly
small if compared to Coulomb ones. In summary, no magnetic couplings of whatsoever
origin will be in the first instance considered when developing our solid state theory17.
It is, however, important to remark the above non-magnetic approximation does
not imply that we are neglecting the true existence of the electron spin: as a matter of
fact, this cannot be done because we would lose the fundamental notion that
electrons are fermions and, therefore, they obey (i) the Fermi–Dirac statistics (see
section 6.2) and (ii) the Pauli principle. Rather, we reconcile the simplified non-
magnetic treatment with the need to take into account the spin by multiplying the
space part of the wavefunction by a spin function which is in charge of just
characterising the actual spin state of the electron. The resulting total wavefunction
will be labelled18 by a fourfold set of quantum numbers {n, l , ml , s} so that,
although no spin-related features are present in this simplified picture, it will not
be possible for two electrons to have the same set of four quantum numbers: if they
lie on the same energy level, then they must differ at least for their spin orientation.
Since spin is a relativistic feature [10, 12], neglecting any corresponding coupling
term is tantamount to developing a theory in the non-relativistic approximation. We
will take this in the widest meaning and, accordingly, we will overlook not only spin-
mediated interactions but also any other relativistic effect19. In particular, we will
use anywhere the rest mass me for electrons.
In conclusion, ions and electrons will interact only via Coulomb coupling: indeed
a major simplification for our developing theory.

17
Their effects are usually treated as a perturbation on the quantum energies predicted by only considering
Coulomb interactions. This is, for instance, the case of the spin-splitting of the energy bands.
18
This is strictly true only within the central field approximation.
19
They correct the electron kinetic energy and the nuclear Coulomb potential energy [10].

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 Solid State Physics

1.3.4 Adiabatic approximation

It is now time to formulate the full quantum mechanical problem describing the
physics of a solid; to this aim we are exploiting the set of approximations discussed
in the previous sections and using the labelling notation for ions and electrons
anticipated in section 1.1. In particular, the full set of electronic and ionic
coordinates will be indicated by the shortcuts r and R, respectively, while the total
crystalline wavefunction will be written as Φ(r, R).
The classical expressions for the Coulomb interactions at work within the crystal
(under the frozen-core, non-magnetic, and non-relativistic approximations) are

e2 Qα
Vne(r , R) = − ∑ ri − R α
4πϵ0 i, α

e 2 QαQβ
Vnn(r , R) = + ∑ (1.12)
4πϵ0 α>β
R α − Rβ
e2 1
Vee(r , R) = + ∑ ,
4πϵ0 i>j
ri − rj

describing, respectively, the ion–electron, the ion–ion, and the electron–electron

interactions. The corresponding Schrödinger equation is written as
⎡ ℏ2 1 2 ℏ2 ⎤
⎢−
⎢⎣ 2
∑ ∇α − ∑ ∇i2 + Vˆne(r, R) + Vˆnn(R) + Vˆee(r)⎥⎥Φ(r, R) = ET Φ(r, R), (1.13)
α
Mα 2m e i ⎦

where ET is the total energy of the crystal including any electronic or ionic
contribution, while the symbols ∇2α and ∇i2 imply derivatives with respect to the
αth ion and ith electron coordinates, respectively. We remark that quantum
operators will always be indicated with the same symbol used for their classical
counterpart, but topped with the ˆ hat symbol.
The quantum problem given in equation (1.13) is still unsolvable by any
analytical or numerical tool, in spite of the many drastic approximations we have
worked out to formulate it. This implies that in order to continue we need to further
simplify our theory. Good for us, we can take profit from a very basic fact: ions are
always much more massive than electrons and, therefore, their dynamics is expected
to be much slower. In other words, at any moment electrons feel ions as if they were
stationary in their instantaneous positions. The opposite view is equally valid: at any
moment electrons will be in their ground-state for the specific ionic configuration
occurring at that time. This conclusion is better understood by taking into account
that the maximum ionic oscillation frequency is about 1013 s−1, while the collective
excitations of the electron system20 have a typical 1016 s−1 frequency. Consequently,
electrons almost immediately respond to a perturbation due to ionic motion,

20
They are called plasma oscillations and will be studied in section 7.2.2.

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 Solid State Physics

re-adjusting in their new ground-state. In other words, electrons instantaneously

follow the atomic vibrations. The separation between vibrational and electronic
excitations is reflected in spectroscopic measurements: while emission/absorption of
photons caused by the oscillations of the ions around their equilibrium position
typically occurs in the infrared region, transitions occurring between electronic states
fall in the visible and ultraviolet regions of the electromagnetic spectrum. In short:
the physics of electrons and nuclei unfolds on quite different energy scales.
We exploit these phenomenological arguments by formally decoupling electronic
and vibrational degrees of freedom or, equivalently, by separating the r - and the
R-dependence of the total wavefunction Φ as
Φ(r , R) = Ψn(R) Ψ (eR)(r), (1.14)

where Ψn(R) and Ψ (eR)(r) are the total ionic wavefunction and the total electronic
wavefunction, respectively. More specifically, we state that Ψ (eR)(r) describes the
crystalline electronic structure corresponding to the clamped-ion configuration R
and, therefore, it must be understood that Ψe has an analytical dependence of the
r -coordinate set, while it has a parametric dependence on the R-coordinate set.
Equation (1.14) is usually referred to as the adiabatic approximation because it
implies that there is no exchange of energy between the system of electrons and the ion
lattice. Although we will also deal with non-adiabatic phenomena in the following
chapters, let us for the moment proceed below this approximation.
By inserting equation (1.14) into equation (1.13) we straightforwardly obtain the
following eigenvalue problem for the total electron wavefunction
⎡ ℏ2 ⎤
⎢−
⎢⎣ 2m e
∑ ∇i2 + Vˆne(r , R) + Vˆee(r)⎥⎥Ψ(eR)(r) = Ee(R)Ψ(eR)(r), (1.15)
i ⎦

where Ee(R) has the meaning of total electron energy when ions are clamped in the
configuration R. This is an important achievement: setting the problem as shown in
equation (1.15) implies that we need to separately calculate Ee(R) for each possible ion
displacement pattern. By combining equations (1.13) and (1.15), the corresponding
eigenvalue problem for the total ion wavefunction is immediately obtained as
⎡ ℏ2 1 2 ⎤
⎢− ∑ ∇α + Vˆnn(R) + Ee(R)⎥Ψn(R) = ET Ψn(R), (1.16)
⎢⎣ 2 α
Mα ⎥⎦

where in this case the difference ET − Ee(R) = U (R) must be understood as the
crystalline ion energy. In practice, the main conceptual consequence of the adiabatic
approximation is that solid state physics is traced back to the separate solution of
equations (1.15) and (1.16); any non-adiabatic effect will be treated as a perturba-
tion to their solutions.
Although we have based the adiabatic approximation on very robust phenom-
enological evidence, we still need to rigorously establish its accuracy. This task is
accomplished by a non trivial argument that we are developing in a simplified form,

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 Solid State Physics

just focussing on its main conceptual steps21. The Ψ (eR)(r) solutions of equation (1.15)
represent a complete set and, with no loss of generality, we can further assume that
they have been properly orthonormalised. Therefore, we can express the total
wavefunctions Φ as the following linear combination22
Φ(r , R) = ∑ Ξm(R) Ψ(e,R)m(r), (1.17)
m

where Ξm(R) are the expansion coefficients and in the present context the label m is a
shortcut for the full set of quantum numbers describing the states of the electron
system, found by solving equation (1.15). By inserting this representation into
equation (1.13) and using the orthonormality of the Ψ (eR)(r) wavefunctions, after
some algebra we obtain the eigenvalue equation for the Ξm(R)’s
⎡ ℏ2 1 ⎤
⎢−
⎢⎣ 2
∑ ∇2α + Vˆnn(R) + Ee(R) + ( ˆ
A + Bˆ
∑ mm ′ mm ′ ⎥⎥Ξm(R) = ET Ξm(R), (1.18)
)
α
M α m ⎦

where23
1 ⎡ ⎤
Aˆmm ′ = − ℏ2 ∑ ⎢
Mα ⎣
∫ *
d r Ψ (e,R)m(r)∇αΨ (e,R)m ′(r))⎥⎦ · ∇α
α
(1.19)
ℏ2 1
Bˆmm ′ = −
2
∑
Mα
∫ dr
*
Ψ (e,R)m(r)∇α2 Ψ (e,R)m ′(r).
α

We easily recognise that the adiabatic approximation is formally equivalent to

dropping off the terms ∑m (Aˆmm ′ + Bˆmm ′) Ξn(R) from equation (1.18): under this
condition, the Ξm(R) functions are nothing other than the total ion wavefunctions
Ψn(R) calculated when the electronic contribution to the total energy is provided by
the mth state. Now, the point is to assess how really accurate is the choice of
neglecting these terms in our theory.
To this aim we write the position of the αth atom in the form R α = R (0) α + κ uα,
where R (0)
α is its equilibrium position, u α its displacement, and κ a suitable expansion
parameter which allows us to write the total crystalline Hamiltonian appearing in
equation (1.13) in the form
ℏ2 1 2 ℏ2
Hˆ = − ∑ ∇α − ∑ ∇i2 + Vˆne + Vˆnn + Vˆee
2 α
Mα 2m e i
ℏ2 1 ℏ2 (0) (0) (1.20)
= −
2
∑ M ∇2α − 2m ∑ ∇i2 + Vˆee + Vˆne + Vˆnn
α α e i
(1) (2) (3) (1) (2) (3)
+ [κVˆne + κ 2Vˆne + κ 3Vˆne + ⋯] + [κVˆnn + κ 2Vˆnn + κ 3Vˆnn + ⋯],

21
The complete formal development is found in [13, 14].
22
In general, the eigenfunctions of an Hermitean operator represent a complete set and, therefore, they can be
used to expand any other wavefunction defined in the same volume of space and obeying the same boundary
conditions as a linear combination of them.
23
The asterisk * indicates the complex conjugate function.

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 Solid State Physics

(0) (0)
where V̂ne and V̂nn are the nucleus–electron and the nucleus–nucleus terms
calculated for ions at their equilibrium positions, respectively, while any term
written as Vˆ is λth order in the ionic displacements Uα (the derivatives appearing
(λ )

in these terms are evaluated with all ions in their equilibrium positions). Equation
(1.20) is known as the Born–Oppenheimer expansion. In the limiting case of
Mα → +∞ the total Hamiltonian operator reduces to

ℏ2 (0) (0)
lim Hˆ = − ∑ ∇i2 + Vˆee + Vˆne + Vˆnn , (1.21)
Mα → 0 2m e i

indicating that κ → 0 as well. Since such an expansion parameter is adimensional,

this conclusion suggests that we can set κ = (me /〈M 〉)ζ where 〈M 〉 is the average ion
mass and ζ > 0 a real number24.
By using the Born–Oppenheimer expansion for the V̂ne term, it is possible to get a
solution of equation (1.15) through a standard perturbative quantum mechanical
calculation: this provides the Ψ (eR)(r) functions. Next, by repeating a similar
expansion procedure to equation (1.18), the Aˆmm ′ and Bˆmm ′ operators are, after a
non trivial calculation, eventually found to be Aˆmm ′ ∼ O(κ 3) and Bˆmm ′ ∼ O(κ 4 ). In
conclusion, non-adiabatic terms appear at high order in the expansion parameter κ.
Since for any possible real crystal we have me /〈M 〉 ≪ 1, the conclusion is twofold: (i)
the Born–Oppenheimer expansion is quickly convergent and (ii) dropping off the
terms ∑m (Aˆmm ′ + Bˆmm ′) Ξm(R) from equation (1.18) is really a more than
acceptable approximation.
It is worth remarking that the formal argument we outlined above is centred on
the expansion parameter κ = (me /〈M 〉)ζ and, therefore, it is fully consistent with the
phenomenological argument used to set the discussion on the adiabatic approx-
imation: the very large difference in dynamical inertia between electrons and ions
supports a theoretical approach where their degrees of freedom are separately taken
into account.

1.4 Mastering many-body features

Equations (1.15) and (1.16) are the constitutive equations of solid state theory.
Although they have been derived under many (and sometimes strong) approxima-
tions, they both are a formidable many-body quantum problem complicated to the
extent that the present theoretical and computational knowledge does not allow for
their exact solution. In order to overcome this impasse two rather different strategies
will be adopted, as presented below.

24
Its is proved [13, 14] that by setting ζ = 1/4 , the total Hamiltonian expanded up to the second order
corresponds to the so-called harmonic approximation. While we will not provide the formal proof of this, we
will make an extensive use of this approximation when studying the lattice dynamics, although we will
formulate it in a much more phenomenological way.

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 Solid State Physics

1.4.1 Managing the electron problem: single-particle approximation

As far as the electronic problem given in equation (1.15) is concerned, we choose to
adopt another very drastic approximation, namely: we will handle the electron
problem within the single-particle approximation.
In principle, by exploiting a key result of atomic physics [2, 10], we could
represent the total electron wavefunction Ψ (eR)(r) as an anti-symmetrised product of
one-electron wavefunctions. In this context, each electron is described by a spin-
orbital function given by the product between a space- and a spin-wavefunction (this
latter is added to correct the limitations of the non-relativistic approximation and
thus enforcing the Pauli principle). Next, the best set of spin-orbitals is obtained by a
quantum mechanical variational procedure, where the total energy of the ground-
state of the crystalline system we are interested in is usually minimised. This leads to
rather complicated integro-differential equations, known as Hartree–Fock (HF)
equations [15], which are typically solved through a self-consistent procedure
implemented by a numerical calculation. We remark that the HF ground-state
energy is an excess approximation to the true value since it is obtained through a
variational principle. Furthermore, we remark that, while the HF method correctly
accounts for exchange effects (which are a truly many-body attribute), it is unable to
deal with quantum correlation features. The HF method [16] has been widely used
since it allows for qualitatively understanding many solid state physics problems.
Nevertheless, its deficiencies in fully catching all the many-body features and the fact
that a single anti-symmetrised product of spin-orbitals is in general found to be a
poor representation of the true Ψ (eR)(r) many-particle wavefunction have motivated
the development of more sophisticated many-body theories which are overviewed in
[15, 17] and fully developed in [18, 19].
A more accessible approach consists in adopting the single-particle approxima-
tion25: the full set of electron–ion and electron–electron interactions are represented
by an effective one-electron potential with the same periodicity of the underlying
crystal lattice, hereafter referred to as ‘crystal field’. In other words, we will assume
that each electron independently moves under the action of a local crystal field
potential describing its embedding into a crystalline environment made by all ions
and by all the remaining electrons. We will indicate such a crystal field potential
(cfp) as Vcfp(r).
The first feature of Vcfp(r) to account for is its periodicity. At this stage we have not
yet developed the mathematical background to rigorously treat the geometrical
characteristics of a crystal. We can nevertheless justify our assumption of periodic
potential by looking at the simple graphical rendering of figure 1.4: by embedding an
electron into a crystal, it will experience a net potential closely resembling that of an
isolated bare ion when closely approaching it; on the other hand, the net potential
will flatten off in the interstitial regions. Next, we remark that the crystal field
potential is local: its value only depends on the instantaneous position actually

25
This picture is equivalently referred to as independent electron approximation or one-electron approximation.

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 Solid State Physics

Vcfp (r)

Figure 1.4. Graphical representation of the potential acting on an electron in the case of a one-dimension solid.
Full line (blue): actual crystal field potential; dashed line (red): superposition of bare Coulomb potentials due
to isolated ions.

occupied by the single electron. This is another very convenient practical advantage
offered by the one-electron approximation with respect to many-body theories.
The resulting formalism is clean and conceptually simple: we are reduced to
solving the one-electron Schrödinger equation for each ith electron
⎡ ℏ2 ⎤
⎢− ∇2 + Vˆcfp(ri )⎥ψmi(ri ) = E m(Ri )ψmi(ri ), (1.22)
⎣ 2m e ⎦

where ψmi (ri) and Em(Ri ) are its wavefunction and energy, while mi stands for a
suitable set of quantum numbers describing the crystalline quantum state of the ith
electron. The adopted labelling once again makes it clear that the equation is solved
for ions clamped in the R configuration. The total electron energy approximating the
eigenvalue of equation (1.15) is calculated as
Ee(R) = ∑ E m(R).i (1.23)
i

It is manifest that equation (1.22) greatly simplifies the actual many-body problem
describing the electron system. However, we remark that we are not treating
electrons as non-interacting particles. Rather, we are assuming that a
suitable choice of Vcfp(r) will allow us to include in this simplified picture the most
relevant features of the electron–electron interactions.
Before even addressing the resulting description of the crystalline electronic
structure, we need to clarify, at least conceptually, how Vcfp(ri) can be defined. The
most effective procedure is implemented by the self-consistent-field method, which is
nowadays the most widely used approach in modern solid state physics since it
naturally translates into efficient computational algorithms allowing for its numer-
ical determination. Let us assume that a clever initial guess for the single-particle
wavefunctions ψm(0)i (ri) is possible. For instance, these functions could be atomic
orbitals, superpositions of plane waves, or any other orthogonalised set of
suitable wavefunctions. Then, the zero-order approximation of the electron–electron
Coulomb potential energy for the ith electron can be written as

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 Solid State Physics

e2 ⎡ψ (0)(r )⎤* 1
Vee(0)(ri ) = ∑∫ ⎣ mj j ⎦ r − r ψm(0)j (rj ) d rj , (1.24)
4πϵ0 j≠i j i

so that the classical total potential energy on such a particle is

(0)
Vcfp (ri ) = Vee(0)(ri ) + Vne(ri , R). (1.25)
(0)
By inserting the corresponding operator V̂cfp (ri) into equation (1.22) and solving it,
we get a new set of wavefunctions ψm(1)i (ri): they allow us to refine our initial guess on
electron–electron Coulomb potential energy, now calculated as
e2 ⎡ψ (1)(r )⎤* 1
Vee(1)(ri ) = ∑∫ ⎣ mj j ⎦ r − r ψm(1)j (rj ) d rj . (1.26)
4πϵ0 j≠i j i

By repeating the previous calculation, we solve again equation (1.22) for the new
estimate of the potential energy term. This will return a set of second-order
eigenfunctions ψm(2)i (ri) which we can further use to generate the second-order
approximation to the potential energy, and so on. The procedure is iterated until
the (n − 1) th order wavefunctions are found to differ from the nth order ones less
than an agreed degree of numerical accuracy. Full convergence is at this stage
proclaimed: we have both the ‘true’ effective crystal field potential and one-electron
wavefunctions.
In the next chapters the electronic structure of crystalline solids will be studied
under assumption that the one-electron local crystal field potential is known: this
will lead to the so-called electronic band theory. A rather different approach will be
outlined as well, named density functional theory (DFT). In this case, the key
quantities of the theory are the density (instead of wavefunctions) and the total
energy of the electron system. DFT is nowadays considered the ‘standard model’ for
the ab initio prediction of the physical properties of condensed matter systems.

1.4.2 Managing the ion problem: classical approximation

Let us now turn to considering equation (1.16). Even in this case the quantum
mechanical problem is inherently many-body and, therefore, quite difficult to
solve. However, we can adopt a simplification procedure based on the same
arguments put forward in support of the adiabatic approximation: ions are
comparatively very massive objects and, accordingly, we can in first approximation
treat them classically.
In practice, we will at first elaborate some suitable classical model for the force
field describing the interactions among ions. Next, we will investigate the ionic
displacements by solving their Newton equations of motion. Any relevant quantum
feature will be eventually added ex post, by a quantisation procedure of the classical
displacement field. As we will see, this will replace the classical description based on
lattice waves with the notion of quantum lattice vibrational field, whose quanta will
be referred to as phonons.

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 Solid State Physics

References
[1] Demtröder W 2010 Atoms, Molecules and Photons (Berlin: Springer)
[2] Colombo L 2019 Atomic and Molecular Physics: A Primer (Bristol, UK: IOP Publishing)
[3] Haile J M 1997 Molecular Dynamics Simulations: Elementary Methods (New York: Wiley)
[4] Hansen J-P and McDonald I R 2006 Theory of Simple Liquids 3rd edn (Oxford: Academic)
[5] Prutton M 1983 Surface Physics (Oxford: Oxford University Press)
[6] Zangwill A 1988 Physics at Surfaces (Cambridge: Cambridge University Press)
[7] Desjonqueres M-C and Spanjaard D 1996 Concepts in Surface Physics (Berlin: Springer)
[8] Bechstedt F 2003 Principles of Surface Physics (Berlin: Springer)
[9] Luth H 2014 Solid Surfaces, Interfaces and Thin Films 6th edn (Berlin: Springer)
[10] Bransden B H and Joachain C J 1983 Physics of Atoms and Molecules (Harlow: Addison-
Wesley)
[11] Povh B, Rith K, Scholz C and Zetsche F 2009 Particles and Nuclei 6th edn (Berlin: Springer)
[12] Greiner W 1990 Relativistic Quantum Mechanics (Berlin: Springer)
[13] Born M and Huang K 1954 Dynamical Theory of Crystal Lattices (Oxford: Oxford
University Press)
[14] Böttger H 1983 Principles of the Theory of Lattice Dynamics (Berlin: Akademie)
[15] Grosso G and Pastori Parravicini G 2014 Solid State Physics 2nd edn (Oxford: Academic)
[16] Pisani C, Dovesi R and Roetti C 1988 Hartree–Fock Ab Initio Treatment of Crystalline
Systems (New York: Springer)
[17] Ashcroft N W and Mermin N D 1976 Solid State Physics (London: Holt-Saunders)
[18] Inkson C J 1984 Many-body Theory of Solids–An Introduction (New York: Springer)
[19] Bruus H and Flensberg K 2004 Many-body Quantum Theory in Condensed Matter Physics:
An Introduction (Oxford: Oxford University Press)

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 IOP Publishing

Solid State Physics

A primer
Luciano Colombo

Chapter 2
The crystalline atomic architecture

Syllabus—The basic notions of crystallography are here outlined, setting the math-
ematical background to treat the physics of translationally invariant solid state
systems. The concept of direct lattice is developed, together with a number of formal
tools useful to describe the periodicity and symmetry of a crystalline array of atoms.
By discussing the main features of x-rays scattering, we further introduce the concept
of reciprocal lattice; then, we formally prove the Bloch theorem setting the general
mathematical form of any one-electron crystalline wavefunction. Thermodynamics is
then invoked to prove that the concept of perfect crystal is just an idealisation: solids do
contain both point- and extended-defects, whose classification and characteristics are
outlined. A qualitative classification of crystalline solids and a rudimental model for
their cohesive energy are eventually discussed.

2.1 Translational invariance, symmetry, and defects

In section 1.1 we have qualitatively defined a crystal as a periodic arrangement of
atoms, displaying long-range order. We have developed this notion in contrast to the
case of an amorphous solid, where just short-range coordination is observed, while
no regular repeated pattern is found beyond the shell of first- or second-nearest
neighbours. In addition, in disordered solids even the local environment surrounding
each atom is different from place to place. This definition of crystal is correct, but
definitely insufficient to provide us with all the mathematical tools needed to
characterise the translational invariance and the symmetry imposed by the perio-
dicity. The discipline in charge of elaborating these tools is called crystallography
[1–3]; it makes use of the mathematical methods of the group theory which can be
developed at a tutorial [4, 5] or advanced level [6, 7]. In this formal environment it is
possible to characterise any possible existing crystal structure, unambiguously
defining its periodicity and symmetry properties which, in turn, can be fruitfully
used to describe any physical property of the crystal. Only the most simple crystal
structures and their basic physics will be here addressed and, therefore, we will not

doi:10.1088/978-0-7503-2265-2ch2 2-1 ª IOP Publishing Ltd 2021

 Solid State Physics

need the full theoretical machinery of group theory. As a matter of fact, we will
make use of just the basic notions of crystallography, such as the concept of direct
and reciprocal lattice, together with their most important geometrical characteristics.
These are the topics of sections 2.2 and 2.4, respectively, while in section 2.3 a
detailed classification of the crystals structures is reported.
In defining the concept of lattice we will explicitly resort to the idealisation of
perfect crystal, namely: we will assume that the periodic atomic architecture forming
a crystalline solid is completely defect-free. While useful under many circumstances
and physically sound in a large number of prototypical cases, this picture does not
faithfully correspond to the real situation: thermodynamics dictates that at any finite
temperature a crystal lattice must contain defects. Therefore, in section 2.5 we will
describe the most common lattice defects, stressing the difference between point and
extended ones. The chapter is eventually concluded by a phenomenological
classification of solids (Section 2.6) and a tutorial introduction to cohesive theory
(section 2.7).

2.2 The direct lattice

2.2.1 Basic definitions
The existence of a discrete regular distribution of matter within a crystal suggests
that its organisation consists in a space periodic repetition of identical structural units,
each of which may contain one or more atoms1. In this latter case, the fundamental
structural unit could be formed by atoms belonging to the same or to different
chemical species: we will refer to elemental or compound systems in the two cases,
respectively.
By some formal rules discussed below we can distinguish between the geometry
and the structure of a crystal. Its geometry is described by introducing a
suitable discrete grid of points in the space fulfilling the fundamental requirement
of translational invariance. In doing that, we have in general many different options
and, in particular, we could define the grid such that all its points are equivalent in
any respect to the origin of the frame of reference set to define positions. This
requirement imposes that the arrangement and the orientation of the points on the grid
must appear the same from whichever site is selected to look at their distribution.
In general, not all atoms of a crystal occupy a position with these characteristics,
as illustrated in figure 2.1 for graphene2: a single atomic plane of carbon atoms
placed at the corners of regular hexagons. It is clear from figure 2.1(left) that atoms
labelled by capital letters A, B, C, … lie at positions fully equivalent (under both the
arrangement and orientation criteria) to the position of the origin O, while atoms
labelled by lower letters a, b, c, … do not have this property. The grid of points A, B,
C, … shown in figure 2.1(right) is named the lattice of graphene.

1
Or molecules in the case of molecular crystals [8, 9] which, however, we will never consider.
2
It will be sometimes convenient to illustrate new key concepts by discussing the case of two-dimensional
lattices, where pictures are more easy to understand. Conclusions will be straightforwardly generalised in three
dimensions.

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 Solid State Physics

Figure 2.1. The graphene crystal (left) and its two-dimensional lattice (right). The red (blue) atoms are
equivalent (non-equivalent) to that at the origin (grey).

Figure 2.2. Generating a two-dimensional crystal by combining a square lattice with a three-atom basis.

Once a lattice is defined, we can decorate it by associating each of its points with a
set of atoms, possibly differing in chemical nature. This set is referred to as the
crystal basis. The procedure is shown in figure 2.2 in the hypothetical case of a two-
dimensional square lattice with a basis formed by three unlike atoms, while in the
previously discussed case of graphene, the basis consists in just two atoms like the
pairs (O,c), (A,d), (E,o), and so on. This picture should make it clear that in general
the geometry and the structure of the crystal are not the same.
In summary, the lattice and the basis carry different, but equally important,
information: geometrical and chemical, respectively. Therefore, a crystal can be
defined as the sum of a lattice and a basis. This synthesis not only provides us with an
operational definition of ‘crystal’, but it also defines the sequence of the topics

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 Solid State Physics

addressed in the rest of this section. We remark that, for reasons that will be clear
later in this chapter, the lattice is better defined as direct lattice, meaning that it is
hosted in the space where distances are measured in units of meter.

2.2.2 Direct lattice vectors

The vector positions Rl of the lattice points are defined as
Rl = n1a1 + n2a2 + n3a3, (2.1)
where {a1, a2, a3} are named translation vectors and n1, n2 , n3 = 0, ±1, ±2, ±3, ….
Translation vectors must not all lie on the same plane. Through equation (2.1) an
infinite lattice is generated (for this reason Rl is also referred to as lattice vector), with
translational invariance: the geometrical situation is just the same if viewed from any
two positions r and r′ such that r′ = r + Rl as illustrated in figure 2.3 in the case of a
two-dimensional square lattice.
The choice of translation vectors is not unique, as shown in figure 2.4: the same
lattice can be equivalently spanned by different sets of translation vectors. We
accordingly distinguish between primitive translation vectors and conventional trans-
lation vectors following a very simple criterion: if lattice points are found only at the
corners of the parallelepiped whose edges are defined by {a1, a2, a3}, then the
translation vectors are primitive. This is the case of the red and blue sets of vectors
in figure 2.4; conversely, the magenta vectors represent a conventional set. Lattices
generated by primitive translation vectors are referred to as Bravais lattices. In this
case, lattice points closest to a given point are named its nearest neighbours. Their
number (necessarily equal for each lattice point because of the translational
invariance property) is a characteristic of the specific Bravais lattice: it is called
the coordination number.
The volume Vc of the parallelepiped defined by the translation vectors {a1, a2, a3} is
Vc = a1 · a2 × a3 , (2.2)

Rl = 3a1 + a2

r′
a2
r
a1

Figure 2.3. Pictorial representation of the translational invariance in the case of a two-dimensional square
lattice. Since Rl is assigned according to equation (2.1),the physics looks the same if observed in r or in
r′ = r + Rl .

2-4
 Solid State Physics

Figure 2.4. Some different choices of translation vectors in a three-dimensional lattice. The red, blue, and
magenta sets of vectors equally span the lattice.

as imposed by vector algebra, and the corresponding portion of space is referred to

as the unit cell of the crystal. In the case where it is defined by primitive translation
vectors, it is more precisely labelled as a primitive unit cell; otherwise it is named a
conventional unit cell. Such a volume has the remarkable property that it fills all
space without overlapping and without leaving voids when translated through
equation (2.1).
A conventional unit cell is typically larger than the primitive one, but it could
have the exact symmetry of the lattice it represents. This is illustrated in figure 2.5 in
the case of the three-dimensional face-centred cubic crystal3. In this case the
conventional cell has four times the volume of the unit one, but it fully reflects
the leading cubic symmetry of the lattice. The use of conventional cells explains
some phenomenological results, like e.g., the remarkable similarities between
different specimens of the same material or even the empirical law originally
formulated by N Steno in 1761, still used in mineralogy, according to which the
faces of crystals made by the same substance form identical angles.
In generating the primitive unit cell of a crystal we can add an additional request,
namely that it defines the region of space closer to a given lattice point than to any
other one. This special kind of primitive cell is called Wigner–Seitz cell. It is not
difficult to generate such a cell: just draw a line connecting the selected lattice point
with its first next neighbours; then, at the midpoint of each segment draw a plane

3
Once again, we are anticipating some information discussed later in this chapter; here it is sufficient to say
that in such a lattice the points are placed at the corners of a cube, as well as at the centre of its faces.

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 Solid State Physics

a3

a2 a3

a2
a1
a1
Figure 2.5. Primitive (red) and conventional (black) unit cell for the face-centred cubic lattice. The
corresponding translation vectors are indicated with the same colour code.

normal to the segment itself; the portion of space contained within these planes has
by construction the required property. The Wigner–Seitz cell is a particular case of
primitive unit cell and, like all the others, contains only one lattice point.

2.2.3 Bravais lattices

Translational invariance represents the dominant structural feature of any crystal,
largely dictating its physics. Nevertheless, it is not the only operation taking the
lattice in itself. For instance, let us consider the face-centred cubic lattice shown in
figure 2.5: it is easy to recognise that any rotation of a π /2 angle about a line normal
to a face and passing through its centre leaves the lattice unchanged. Similarly, a
reflection in any plane defined by the cube faces takes the lattice in itself. These are
just simple examples of non-translational symmetry operations: their full description
is the core business of crystallography [4–7]. Here we limit ourselves to defining some
general features allowing for the classification of the Bravais lattices.
First of all, we understand that all the operations we are dealing with are rigid,
that is, they do not change the distance between lattice points. In other words, we are
not considering deformations. Under this constraint, we can distinguish between
pure translations and other operations that leave just one lattice point fixed. For
example, imagine a two-dimensional square lattice and a rotation of a π /2 angle
about a line normal to the plane and passing through a lattice point. It is a key result
of crystallography that by combining a translation with an action leaving just one
lattice point fixed we get a symmetry operation for the selected lattice. We do not
formally prove this result, but the graphical example shown in figure 2.6 makes it
plausible. In summary, all operations taking a lattice in itself are either pure
translations or leave a particular lattice point fixed or are a combination of the two.

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 Solid State Physics

π/2

π/2

Figure 2.6. Graphical proof that a symmetry rotation about a line normal to the plane containing a two-
dimensional square lattice and passing through a non-lattice point (top) is equivalently represented by a
sequence of a lattice translation and a rotation leaving fixed a lattice point. The labelling of lattice points is just
a guide to figure out the action of any operation: in fact, they are all equivalent.

The full set of symmetry operations of the three kinds is known as the space group or
symmetry group of the Bravais lattice. On the other hand, the subset of symmetry
operations that leave a particular lattice point fixed is referred to as the point group
of the Bravais lattice; their elements are called point operations.
Crystallography classifies the existing Bravais lattices according to the space
group or to the point group they belong to. The seminal work by M L Frankheim
and A Bravais, developed in the middle of the XIXth century, has proved that if we
follow the space group criterion we get 14 Bravais lattices, while if we follow the
point group criterion we get seven crystal systems. These results hold for three-
dimensional lattices, while two-dimensional ones present a much simpler situation:
there exist just five two-dimensional Bravais lattices. More precisely, they are known
as the oblique, the square, the hexagonal, the rectangular, and the centred
rectangular lattice: their graphical representation is straightforward. Since Bravais
lattices are more numerous than crystal classes, we understand that each of the latter
can be attributed one or more lattice(s). The key concept is however that Bravais
lattices belonging to the same crystal class have the same point group or, equivalently,
they are characterised by the same set of point operations.
The seven three-dimensional crystal systems are classified by using the lengths of
the edges of their conventional cell and the angles they form. We will use the

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 Solid State Physics

γ γ
γ
c c
c
α β α β α β
a b a b a b

γ γ
c c
α β α β
a b a b

γ
γ
c c
α β α β
a b a
b
Figure 2.7. The conventional unit cell of the seven crystal systems, each one characterised by a different point
group symmetry. Black Latin symbols (a, b, c ) indicate the length of the cell sides while blue Greek symbols
(α, β , γ ) indicate the angles between cell edges.

standard notation to indicate with (a, b, c ) the edge lengths and with (α , β, γ ) the
angles. Also, (a, b, c ) will be referred to as the lattice constants. The crystal systems
are shown in figure 2.7 and named:
1. Cubic: it is characterised by a = b = c and α = β = γ = π /2. Its point group
contains all the symmetry operations of a cube. There are three Bravais
lattices for this system, namely the simple cubic (lattice points at the corners
of the cube), the body-centred cubic (lattice points at the corners of the cube
and at its centre), and the face-centred cubic (lattice points at the corners of
the cube and at the centre of its faces). We will refer to them by using the
acronyms sc, bcc, and fcc, respectively. The sc lattice is the only one for
which the conventional cell shown in figure 2.8 is a primitive cell.
2. Tetragonal: it is characterised by a = b ≠ c and α = β = γ = π /2. Its point
group contains all the symmetry operations of an orthogonal prism with a
square basis and an unequal height. There are two Bravais lattices for this
system, namely the simple tetragonal (lattice points at the corners of the
orthogonal prism) and the centred tetragonal (lattice points are the corners of
the orthogonal prism and at its centre).
3. Orthorhombic: it is characterised by a ≠ b ≠ c and α = β = γ = π /2. Its
point group contains all the symmetry operations of an orthogonal prism
with three unequal lengths. There are four Bravais lattices for this system,
namely the simple orthorhombic (lattice points at the corners of the prism),
the base-centred orthorhombic (lattice points at the corners of the prism and
at the centre of the two bases), the body-centred orthorhombic (lattice points

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 Solid State Physics

at the corners of the prism and at its centre), and the face-centred
orthorhombic (lattice points at the corners of the prism and at the centre
of its faces).
4. Monoclinic: it is characterised by a ≠ b ≠ c and α = β = π /2 ≠ γ . Its point
group contains all the symmetry operations of a prism with three unequal
lengths and a non-rectangular basis. There are two Bravais lattices for this
system, namely the simple monoclinic (lattice points at the corners of the
prism) and the centred monoclinic (lattice points at the corners of the prism
and at its centre).
5. Triclinic: it is characterised by a ≠ b ≠ c and α ≠ β ≠ γ . Its point group
contains all the symmetry operations of a three-dimensional object with
parallel opposite faces, but no restrictions both on the side lengths and on the
angles between edges. There is just one Bravais lattice for this system which is
simply known as triclinic.
6. Trigonal: it is characterised by a = b = c and α = β = γ < 2π /3 (all angles
not equal to π /2). Its point group contains all the symmetry operations of a
rhombohedron (i.e. a cube which has been stretched along its diagonal).
There is just one Bravais lattice for this system which is equally known as
trigonal or rhombohedral.
7. Hexagonal: it is characterised by a = b ≠ c and α = β = π /2, γ = 2π /3. Its
point group contains all the symmetry operations of an orthogonal prism
with a regular hexagon as basis. There is just one Bravais lattice for this
system which is simply known as hexagonal.

In figure 2.8 a set of primitive translation vectors for the three cubic lattices are
shown for illustration purposes.
We conclude this section by introducing the concept of crystallographic axes:
they are imaginary lines drawn within the lattice, defining a useful frame of
reference in the crystal since their intersection point will be used as the origin. They
are not all coplanar; for cubic lattices, they are parallel to the edges of the
conventional unit cell. This implies that in most lattices three crystallographic axes
are found, corresponding to the (a, b, c ) edges of figure 2.7. On the other hand, the
hexagonal lattice needs four crystallographic axes: three of them fall in the same
basal plane of the hexagonal prism, intersecting with a 2π /3 angle at its centre; the
fourth axis passes through the intersection point at a right angle to the plane
formed by them.

2.2.4 Lattice planes and directions

Within a crystal lattice we can identify sets of planes with the threefold property of
(i) containing lattice points, and being (ii) parallel and (iii) equally spaced. Such
lattice planes play an important role in determining the diffractions of whatever
waves are travelling within the crystal.
Since any plane is determined by three non collinear points in the space, the
conventional procedure to identify a family of parallel lattice planes is based on

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 Solid State Physics

a1
a2
a3
a3 a1 a2
a2
a1 a3
Figure 2.8. Primitive translation vectors for the sc, bcc, and fcc lattices. Black lines represent the underlying
symmetry of the lattice; magenta lines represent just a guide to the eye helping to figure out the space relation
among lattice point positions.

the intercepts made on the crystallographic axes by the nearest plane to the origin
(not considering the plane that possibly contains the origin for the reason that
will be immediately clear). The first step consists in finding such intercepts in
terms of the lattice constants (a, b, c ): this defines a set of three integer numbers;
next, the reciprocal of these numbers is taken; finally, these reciprocals are
reduced to the smallest three integer numbers (h, k , l ) having the same ratio. For
instance, let us consider the case shown in figure 2.9: the plane has intercepts with
the crystallographic axes given by 3, 1, and 2 in units of the (a, b, c ) lattice
constants. Their reciprocal are 1/3, 1 ad 1/2. The smallest three integers with the
same ratio are (263): they label the plane. The three numbers (h, k , l ) are known
as the Miller indices of the plane (in fact, they identify the family of its parallel
lattice planes). In figure 2.10 some important planes in Bravais lattices of the
cubic crystal system are shown. We finally remark that whenever a plane
intercepts an axis on its negative side with respect to the origin, the corresponding
Miller index is negative: in order to keep this information, this index will be
labelled by placing a bar on it.
Similarly to planes, even directions can be identified within a lattice. In this case, a
set of three integers is used and put in square parenthesis as [u, v, w ]: they represent
the set of smallest integers with the same ratio as the components of a vector
pointing along the selected direction, referred to the crystallographic axes. In
figure 2.11 we show some important directions in Bravais lattices of the cubic
crystal system. We remark that only in this special case is the [u, v, w ] direction
always normal to the (u, v, w ) plane.

2.3 Crystal structures

2.3.1 The basis
Once the lattice has been determined, we can generate the actual crystal structure of
a solid material by simply assigning to each lattice point the very same basis, whose
definition was provided in section 2. In this case the actual position R of an ion is

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 Solid State Physics

Figure 2.9. The shaded lattice plane is identified by the Miller indices (263). The (a, b, c ) lattice constants are
shown by red segments. In their units, the plane intercepts are (3, 1, 2).

Figure 2.10. Some important planes and corresponding Miller indices for lattices of the cubic crystal system.

Figure 2.11. Some important directions for lattices of the cubic crystal system.

written by summing the lattice vector Rl to the vector Rb providing the relative
position of the atom within the basis
R = Rl + R b , (2.3)
where we have placed the origin of the frame of reference for each basis on the lattice
points Rl , without any rotation from place to place (in the case of a monoatomic

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 Solid State Physics

Bravais crystal, we obviously have Rb = 0). The resulting periodic crystal structure
eventually corresponds to a physical object, while the lattice was just an abstract
geometrical entity. We nevertheless remark that such a crystal structure is a still
idealised physical system since, at variance with real solids, (i) it is infinite and (ii) it
does not contain any imperfection.
In the most general case, a basis will contain two or more atoms (or molecules),
whose relative positions must be the same everywhere4 in order to ensure the
translational invariance. On the other hand, when the basis contains just one single
atom, the resulting structure is referred to as a monoatomic Bravais crystal. This is
indeed a special case of crystal structure, since it can be equivalently described by
using a lattice with a basis, provided that a non-primitive unit cell has been
chosen5.

2.3.2 Classification of the crystal structures

A large variety of solids (mostly metallic) crystallise as Bravais crystals, mainly in
the bcc and fcc form6. More specifically (standard chemical symbols hereafter
appearing are defined in the periodic table reported in appendix A) we have:
1. The bcc monoatomic Bravais crystal structure: it is typically assumed by
alkali metals (Li, Na, K, Rb, Cs), by some transition metals (V, Cr, Fe, Nb,
Mo, Ta, W), by some alkaline-earth metals (Ba) or by other metals (Tl).
2. The fcc monoatomic Bravais crystal structure: it is typically assumed by noble
elements (Ne, Ar, Kr, Xe), by some transition metals (Co, Ni, Cu, Rh, Pd,
Ag, Ir, Pt, Au), by some alkaline-earth metals (Ca, Sr), by other metals (Al,
Pb), by some lanthanides (La, Pr, Yb) or actinides (Th).
The set of non-Bravais crystals is of course much more rich, including
elemental as well as compound solids. Some important crystal structures with
a basis are:
3. The sodium chloride structure: it consists of an equal number of atoms of
two A and B chemical species, placed at alternate points of an sc lattice.
The resulting crystal structure is described as an fcc lattice with a basis of
two different atoms in position (0, 0, 0)a and (1/2, 1/2, 1/2)a , where a is
the cubic lattice constant. This crystal structure is typical of AB ionic
solids, where A is any alkali metal (Li, Na, K, Rb, Cs) and B any halogen
atom (F, Br, Cl, I). Similarly, A-B compounds crystallise in this form,
where A is a metal (Ag, Mg, Ca, Sr or Ba) and B is a non-metal (O, S, Se
or Te) element.
4. The cesium chloride structure: it consists in bcc lattice whose sites are
occupied by atoms of two A and B chemical species so that each A-atom
has eight nearest neighbouring B-atoms (and vice versa). The resulting

4
Meaning: the same in any basis assigned to each lattice point.
5
A bcc Bravais crystal with lattice constant a can be equivalently described as an sc lattice with a two-atom
basis by adopting a cubic conventional unit cell with side a and a basis of two atoms in positions (0, 0, 0)a and
(1/2, 1/2, 1/2)a . The crystallographic axes provide the frame of reference.
6
The sc form is instead really very rare in normal conditions of temperature and pressure.

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 Solid State Physics

crystal structure is described as an sc lattice with a basis of two different

atoms in position (0, 0, 0)a and (1/2, 1/2, 1/2)a , where a is the cubic
lattice constant. We find in this crystal structure Cs-based materials, like
CsCl, CsBr, and CsI.
5. The diamond structure: it consists of two inter-penetrating fcc monoatomic
Bravais lattices, displaced along the diagonal of the cubic conventional unit
cell by a/4, where a is the cubic lattice constant. The two sub-lattices are
occupied by atoms of the same chemical species. The resulting crystal
structure is described as an fcc lattice with a basis of two identical atoms
in position (0, 0, 0)a and (1/4, 1/4, 1/4)a . Elemental semiconductors (Si, Ge,
α-Sn) and diamond (C) crystallise in this structure.
6. The zincblende structure: it is similar to the diamond structure, but the two
sub-lattices are occupied by atoms of two different chemical species.
Compound semiconductors crystallise in this structure, in any IV–IV (SiC)
or III–V (where III = Al, Ga, In and V = P, As, Sb) or II–VI (where II = Zn,
Cd, Hg and VI = S, Se, Te) combination. Other crystals assuming this
structure are CuA (where A = F, Cl, Br, I), BeB (where B = S, Se, Te), MnC
(where C = S, Se).
7. The hexagonal structure: it consists of two inter-penetrating hexagonal
Bravais lattices, shifted from one another by a displacement vector
(a /3, b /3, c /2), where a = b and c are defined in figure 2.7. The ‘ideal’
hexagonal structure has a c /a = 8/3 ratio, while actual hexagonal
crystals deviate from this value. The ideal c/a ratio is calculated by
assuming that each lattice point is occupied by a hard sphere, a situation
which is referred to as hexagonal close packing (see next section for more
detail). However, atoms are not rigid spheres and, therefore, in real
materials this ratio can assume different values. We find in crystal
structure many elemental solids like those made by Cd, Mg, Nd, Os,
Sc, Ti, Zn, and Zr with a c/a ratio of 1.89, 1.62, 1.61, 1.58, 1.59, 1,59,
1.86, and 1.59, respectively.

For all the formal developments discussed in the next chapters it is important
to visualise the seven crystal structures, which are therefore shown in
figure 2.12.
We conclude this section with an important remark. A basis is a group of
atoms which, in principle, is not required to have full spherical symmetry like,
instead, the single atom (in fact a totally symmetric object) assigned to each
point of Bravais lattices. The consequence is that the number of symmetry
groups existing for general crystal structures is greatly increased with respect to
the case of monoatomic Bravais lattices. More specifically, crystallography has
classified 230 space groups and 32 crystallographic point groups for general
crystal structures (these numbers should be compared to 14 and 7, respectively,
found in monoatomic Bravais lattices). While more details are found in [7], here
we limit ourselves to mentioning the various kinds of symmetry operations

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 Solid State Physics

Figure 2.12. The seven crystal structures discussed in the text. Atoms are shown in blue or red colour. Black
lines represent the underlying symmetry of the lattice; magenta lines represent just a guide to the eye helping
one to figure out the space relation among atom positions.

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 Solid State Physics

found in crystallographic point groups. Basically, we can distinguish among: (i)

rotations through integer multiples of 2π /n angles about some given axis, with n
limited to the set {2, 3, 4, 6} of values; (ii) reflections in a mirror plane; (iii)
inversions through a fixed point; (iv) roto-reflections where a rotation is followed
by a reflection in a plane, thus providing a symmetry through a net rotation of
an angle not equal to 2π /n; (v) roto-inversions where a rotation through a 2π /n
angle is followed by an inversion in a point found on the rotation axis.
Rotational symmetry deserves a clarification. In listing all possible rotations
by 2π /n angles we restricted n to just the four values {2, 3, 4, 6}. More
specifically, we did not cover the case with n = 5 or n = 7. In order to
figure out the fundamental reason for this limitation, it is useful to start by
considering the full set of two-dimensional Bravais lattices (which have been
listed in section 2): by very intuitive arguments it is easy to prove that they can
have neither a five-fold nor a seven-fold rotation axis (we cannot exactly cover a
floor with pentagonal or heptagonal tiles!). Then, it is really easy to prove that
an n-fold rotation axis cannot exist in three-dimensional Bravais lattices if it
does not exist in some two-dimensional one. In conclusion, even in three
dimensions we cannot have five-fold or seven-fold rotational symmetry.
There is, however, a notable exception to this general rule which we discuss
in appendix B where the concept of quasi-crystal is outlined.
Two main classification schemes have been elaborated to label the crystallo-
graphic point groups [7, 10], among which the Schoenflies nomenclature system is the
most widely used; here symmetry groups are put in the following categories:
• Cn: groups containing only a n-fold rotation axis;
• Cnv: groups containing a n-fold rotation axis and mirror planes containing it;
• Cnh: groups containing a n-fold rotation axis and just one mirror plane
normal to it;
• Sn: groups containing only n-fold roto-reflection axis;
• Dn: groups containing a n-fold rotation axis and other two-fold rotation axes
normal to it;
• Dnh: groups containing the same operations of Dn and an additional mirror
plane normal to the n-fold axis;
• Dnd: groups containing the same operations of Dn and mirror planes
containing the n-fold axis and bisecting the angles between the two-fold axes.

The second classification scheme is named international notation. Its correspond-

ence with the Schoenflies nomenclature system is found elsewhere [7].

2.3.3 Packing
There is still a remaining criterion for classifying atomic architectures to be
discussed. It is based on the assumption to treat atoms as attracting hard spheres.
While this is clearly a very crude approximation, it is reasonably well satisfied by
metals and this represents the phenomenological foundation for its applicability.

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 Solid State Physics

A A A A A A
B C B C B C B CB C
A A A A A A
C B C B C B C B C B
A A A A A A
B C B C B CB CB CB
A A A A A A

Figure 2.13. A close-packed sequence of hard-sphere layers. The first layer is marked grey (A lattice), the
second in red (B lattice). The third layer can be placed on the A lattice or, alternatively, on the blue one
(C lattice).

Since atoms are looked at as ‘hard’ spheres, they cannot overlap; however,
because of their mutual attraction, they tend to assume an arrangement that
minimises the total energy of the system. This implies that they tend to pack as
closely as possible7.
Let us start by arranging identical hard spheres on a plane: in the closest packing
configuration the centres of the spheres lie on a two-dimensional triangular lattice, in
positions marked by A letters in figure 2.13. By looking at this configuration from
the top, we can identify for the second layer the new triangular lattice (lying on a
plane parallel to the first layer) marked by B letters. While this choice is unique,
when adding a third layer we can add spheres on the triangular lattice marked by C
letters or, alternatively, on the triangular lattice once again marked by A letters (in
both cases the third layer lies on a plane parallel to the two previous ones). The
corresponding stacking sequence is ABCABCABC⋯and ABABABAB⋯, respectively:
they are named fcc structure and hexagonal close-packed (hpc) structure. The atomic
layers so generated correspond to (111) planes of the fcc lattice or to the basal plane
of the hexagonal lattice. In both configurations each atom has 12 nearest neigh-
bours: any model according to which the total energy of a crystal only depends on
the number of nearest neighbours must necessarily predict the very same energy for
fcc and hpc structures.
The number of different ways to pack hard spheres in arrangement other than the
close-packed one is actually infinite. In table 2.1 we summarise some properties of
the packing in cubic lattices. The packing fraction is the volume fraction occupied by
the hard spheres: the labelling ‘close packing’ for fcc is justified by the fact that its
packing fraction is maximum.

2.4 The reciprocal lattice

2.4.1 Fundamentals of x-ray diffraction by a lattice
The construction of crystal structures according to the formal rules developed in the
previous section finds full experimental evidence by means of x-ray crystallography

7
In essence this corresponds to the practical exercise of stacking cannonballs on top of each other.

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 Solid State Physics

Table 2.1. Some features of cubic lattices (volumes and distances in units of the a lattice constant).

sc bcc fcc
3 3
Volume of the conventional unit cell a a a3
Volume of the primitive unit cell a3 a 3 /2 a 3 /4
Number of nearest neighbours 6 8 12
Distance of the nearest neighbours a 3 a /2 a/ 2
Packing fraction π /6 3 π /8 2 π /6

Figure 2.14. Pictorial representation of x-ray diffraction by a crystalline sample. Left: an incoming plane wave
is scattered by each single atom forming the crystalline sample. Right: constructive and destructive interference
phenomena between the diffused spherical waves provide light points and low brightness regions in the
diffractogram collected by the detector.

[10–12], where an electromagnetic radiation with a typical wavelength in the range

0.01–10nm is made incident on and then diffracted8 by a solid state crystalline
sample.
The experimental situation is qualitatively summarised in figure 2.14: an
incoming x-ray beam is collimated on a crystalline sample and the corresponding
diffracted beam is collected by a detector as a distribution of light points,
separated by regions of low brightness. They correspond, respectively, to the
positions of maximum and minimum intensity of the diffracted beam; this picture
is known as diffractogram. Assuming that the sample consists of a discrete
distribution of atomic scattering centres and analysing by the laws of optics [13]
all the angles formed between the direction of the incident and diffracted beams
(as well as their intensities), it is possible to reconstruct the space distribution of
the centres. In other words, from the distribution of the light points on the
diffractogram it is eventually possible to specify the atomic architecture of the
investigated sample.

8
Diffraction [13] is the phenomenon of elastic diffusion of a wave by an ordered array of scattering centres. In
the physics we are discussing, we will accordingly assume that the scattering centres are crystalline atoms.

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 Solid State Physics

2.4.2 Von Laue scattering conditions

We now perform a detailed analysis of the scattering events occurring in x-ray
diffraction. Let k in be the wavevector of the incoming monochromatic plane wave
with amplitude
Ain(r , t ) = A 0 exp [i (k in · r − ωt )], (2.4)
where ω is its angular frequency, while r and t indicate the position in space and
time, respectively. Our goal is to predict the amplitude A out of the scattered waves. To
this aim we adopt a model originally developed by M von Laue and based on two
simplifying assumptions: (i) the incoming beam is weak enough that its interaction
with the sample does not affect the underlying crystal structure and (ii) the scattering
events are elastic, that is, x-rays do not lose energy by diffusion (i.e. their intensity is
unaffected by scattering).
We now remember that, according to the Huygens–Fresnel principle of elemen-
tary optics [13], any point-like object invested by a plane wave becomes the source of
a scattered spherical wave. Accordingly, with reference to figure 2.15, we can write
the amplitude A out of the spherical wave emerging from the atom at position R and
revealed by a detector at a distance D from the origin of the adopted frame of
reference as
exp [ik in∣D − R∣]
A out = 
A 0
exp [i
(k · R − ωt
in  )] · fR · ,
∣ D − R∣ (2.5)
incoming plane wave
⏟
atomic form factor   
scattered spherical wave

where the last term on the right-hand side accounts for phase change and amplitude
decrease of the scattered wave. Such changes are due to the complex of quantum
phenomena occurring in the interaction between the electromagnetic wave and the
atom at position R: their overall effect is summarised by the atomic form factor fR
which depends on the atomic number Z of the atomic scatterer, as well as by its
electron charge density ρ(r) through the general expression

fR = ∫ ρ(r) exp[i K · (r − R)]d r, (2.6)

D−R

R D

Figure 2.15. Geometry of the M von Laue model for x-ray scattering.

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 Solid State Physics

where K = k out − k in is the scattering vector (we have named k out the wavevector of
the scattered wave). The atomic form factor is calculated by the atomic theory of
elastic scattering [14] and we will consider it as known.
Given a typical experimental setup for x-ray diffraction measurements, with no
loss of generality we can assume D ≫ R, since the detector is usually placed at a
comparatively very large distance from the sample which, instead, is near the origin
O of the laboratory frame of reference. Furthermore, at such a large distance the
diffused spherical wave can be approximated to a very good extent by a plane wave
with wavevector k out ; under the assumption of elastic scattering, we have k in = kout .
The geometry sketched in figure 2.15 suggests that K is nearly parallel both to D
(measuring the sample–detector distance) and to D − R (measuring the distance
from the atom at position R and the detector). By combining the geometrical
information collected so far, we can write
k in∣D − R∣ ∼ k out · (D − R) = koutD − k out · R = k inD − k out · R , (2.7)
which leads to the following expression for the amplitude of the scattered wave
A 0 exp [i (k inD − ωt )]
A out = · fR · exp[ −iK · R]. (2.8)
D
We note that the first term on the right-hand side is easily calculated once the
characteristics of the incoming wave (k in and ω) and the geometry of the experiment
(D) are known. We accordingly set Ω = A 0 exp [i (k inD − ωt )]/D = constant.
The total diffused wave has an amplitude Atot resulting from the superposition of
all waves scattered by single atoms. We therefore write
Atot = Ω ∑ fR exp[ −iK · R], (2.9)
R

where the lattice sum spans all the atoms in the crystal. Following the notation
introduced in section 2.3.1, their position is R = Rl + Rb. Equation (2.9) is then
written as
⎡ ⎤⎡ ⎤
Atot = Ω ∑ fRb exp( −i K · Rb) ∑ exp( −i K · Rl)⎥,
⎢ ⎥ ⎢ (2.10)
⎢⎣ ⎥⎦⎢⎣ ⎥⎦
Rb Rl

where we have exploited the fact that each basis atom with position Rb has exactly
the same form factor in any basis9.
The maxima of the diffracted beam (corresponding to the bright light points in
figure 2.14) correspond to the maxima of Atot , which we are now going to determine.
We already know that Ω is a constant; the same is also true for the second term on
the right-hand side of equation (2.10): it is calculated once for all, as soon as the
chemical nature and the geometrical structure of the basis has been specified (the fRb
form factors are provided by atomic physics). We conclude that the maxima of Atot

9
The sum in the second term on the right-hand side of this equation is known as the structure factor [15].

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 Solid State Physics

correspond to the maxima of the last term on the right-hand side of equation (2.10).
Since Rl vectors are given by equation (2.1), the maximum condition occurs
provided that
K · a1 = 2πl K · a2 = 2πm K · a3 = 2πn , (2.11)
where l , m, n are integer numbers. Equations (2.11) are known as Laue conditions:
they are remarkably important since they state that allowed scattering vectors K lie
on a discrete lattice.
This is indeed quite a different kind of lattice with respect to the direct one, since
here distances are measured in units of (meter)−1: for this reason it is referred as the
reciprocal lattice. While the direct lattice spans the space where ionic positions are
framed and their displacements take place, the reciprocal lattice is the natural
environment where the wavevectors of whatever wave-like phenomena within a crystal
are defined. This will affect the whole solid state physics.

2.4.3 Reciprocal lattice vectors

The reciprocal lattice is formally described by the same concepts developed in
section 2 for the direct one. More specifically, its points are given by
G = m1b1 + m2 b 2 + m3b 3, (2.12)
where {b1, b2 , b3} are named reciprocal translation vectors and m1, m2 , m3=
±1, ±2, ±3, …. The maximum scattering vectors K entering equation (2.10) lie
on this reciprocal lattice and, therefore, they must fulfil equation (2.12); accordingly,
by setting K = G , after some little algebra we obtain that the Laue conditions are
satisfied if
a2 × a3 a3 × a1 a1 × a2
b1 = 2π b 2 = 2π b 3 = 2π . (2.13)
a1 · a2 × a3 a1 · a2 × a3 a1 · a2 × a3
This result provides the formal definition of b-vectors and stresses the fact that the
reciprocal lattice is closely related to its direct counterpart. A number of formal
relations hold, including the following most important ones:
ai · bj = 2πδij with i , j = 1, 2, 3
G · ai = 2πmi with i = 1, 2, 3 (2.14)
exp(i G · Rl) = 1 for any pair of direct/reciprocal vectors,

where δij is Kroenecker delta-symbol. Furthermore, we observe that bi and bj with

i , j ≠ k are normal to ak , while bi is not in general parallel to ai : this holds only in
crystals with orthogonal axes.
The reciprocal lattice has a number of important formal properties which are very
easy to prove by applying the above definitions:
• the reciprocal lattice is a kind of Bravais lattice;
• through equation (2.12) an infinite lattice is generated, once again charac-
terised by translational invariance;

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 Solid State Physics

• it is possible to identify volumes such that, upon translations by a G vector,

they entirely fill the reciprocal space, without overlapping or leaving empty
spaces: such volumes, therefore, play the same role as unit cells in the direct
lattice;
• the reciprocal of the reciprocal lattice is nothing other than the original direct
lattice;
• interestingly enough, two-dimensional direct lattices have a remarkable
property, namely they are self-reciprocal: the direct and the reciprocal lattice
have the very same symmetry.

In order to complete the discussion of cubic and hexagonal direct lattices

developed in section 2, we discuss here the generation of their reciprocal lattice
vectors. Let us consider at first sc, bcc, and fcc direct lattices: for them it is natural to
adopt a Cartesian frame of reference whose three axes coincide with the edges of the
cubit conventional unit cell; this set of Cartesian axes will be identified by the set of
mutually normal unit vectors {iˆ , j ˆ , kˆ }. For the hexagonal lattice just the c axis is
collinear with a Cartesian one. The corresponding sets of direct and reciprocal
vectors are reported in table 2.2 for the four cases of interest.
Another interesting feature is that any reciprocal lattice vector G is normal to a
family of planes in the direct lattice. It is easy to prove it. Let exp(iG · r) be a plane
wave whose maximum (unity) amplitude values occur on parallel planes in real
space, separated by a distance d = 2π /G . We know that G is normal to such a family
of planes; we also know (see equation (2.14)) that exp(iG · Rl) = 1 for any Rl lattice
vector. Therefore, we conclude that the family of parallel planes in real space must
necessarily contain all those lattice planes which are spanned by suitable Rl vectors.
Since the reciprocal lattice vector G is normal to them, we can use its three integers

Table 2.2. Direct {a1, a2, a3} and reciprocal {b1, b2, b3} translation vectors of cubic (sc, bcc, fcc) and hexagonal
(hex) lattices. Just one lattice constant a is used in the cubic case, while two lattice constants (a, c ) are needed
for the hexagonal lattice (see figure 2.7).

sc bcc fcc hex

a ˆ
a1 aî a ˆ
(j + kˆ − iˆ ) 2
(j + kˆ ) aî
2

+ iˆ − j
ˆ) + iˆ )
a ˆ a ˆ
a2 aĵ 2
(k 2
(k
(
a − 2 iˆ +
1
2
3 ˆ
j )
a3 a ˆ ˆ − kˆ ) a ˆ ˆ)
+j c k̂
a k̂
2
(i +j 2
(i
2π ˆ 2π ˆ
+ kˆ ) + kˆ − iˆ )
2π
b1 a
î a
(j a
(j 4π
3a (
2
3
iˆ + 2 j
1ˆ
)
+ iˆ ) + iˆ − j
ˆ)
b2 2π 2π ˆ 2π ˆ 4π
a
ĵ a
(k a
(k ĵ
3a

b3 2π 2π ˆ ˆ) 2π ˆ ˆ − kˆ ) 2π
a
k̂ a
(i +j a
(i +j c
k̂

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 Solid State Physics

{m1, m2 , m3} to label this family of direct lattice planes. In conclusion, reciprocal
lattice vectors provide an alternative definition for Miller indices10.

2.4.4 The Brillouin zone

The conventional way to generate the reciprocal primitive unit cell is by following the
Wigner–Seitz construction: the resulting cell is referred to as the first Brillouin zone.
By construction, it contains all the wavevectors that are not linked by a reciprocal
translational vector G . Its volume is (2π )3/Vc , where Vc is the volume of the primitive
unit cell of the corresponding direct lattice defined in equation (2.2). We remark that
the use of the adjective ‘first’ will be clear when discussing the vibrational and
electronic properties of crystalline solids. We will hereafter make use of the acronym
1BZ to indicate the first Brillouin zone.
The boundaries of the 1BZ are given by planes which, as explained in the previous
section, are in turn defined by means of reciprocal lattice vectors. The general
principle is that the 1BZ is the smallest volume in the reciprocal space which is
enclosed by planes normally bisecting reciprocal lattice vectors drawn for the origin.
With reference to table 2.2, we can calculate that the boundary planes of the 1BZ for
the three cubic lattices are defined as follows (the 1BZ of the hexagonal lattice is
added for completeness):
• simple cubic lattice: take the six planes normal to the vectors ±2πiˆ /a , ±2π j ˆ /a,
ˆ
and ±2π k/a at their midpoints: they define a cubic volume;
• body-centred cubic lattice: take the 12 planes normal to the vectors
ˆ ± k)/a , 2π ( ±kˆ ± iˆ )/a , and 2π ( ±iˆ ± j
2π ( ± j ˆ )/a at their midpoints: they
define a rhombic dodecahedron volume;
• face-centred cubic lattice: take the eight planes normal to the vectors
2π ( ±iˆ ± j ˆ ± kˆ )/a at their midpoints and further cut them by another set
of six planes bisecting the reciprocal lattice vectors ±4πiˆ /a , ±4π j ˆ /a , and
ˆ
±4π k/a : the resulting volume is a truncated octahedron.

A number of high symmetry lines and points can be identified as shown in

figure 2.16 by red lines and black dots, respectively. High-symmetry points of the
1BZ typically lie at the centre of the zone, edges, and faces, as well as at the corner
points. They play an important role in solid state physics: whenever we need to
visualise a crystalline physical property depending upon a wavevector11, the
conventional choice is to follow the path marked in red colour in figure 2.16,
corresponding to the edges of the so called irreducible part of the 1BZ. For further
reference, we report a standard labelling used for fcc crystals to indicate some high-
symmetry directions: the three directions connecting the Γ zone-centre to the X, K,
and L point are indicated as Δ, Σ, and Λ, respectively.

10
In order to avoid any ambiguity, the widely adopted convention is that the shortest reciprocal vector parallel
to G is used to label the family of lattice planes normal to it.
11
Anticipating the topics of the next chapters, the physical property could be the frequency of a vibrational
normal mode of the lattice or the energy of a crystalline electron.

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 Solid State Physics

R P
Γ Γ H
X N
M

L A
U L H
Γ X
K W M K Γ

Figure 2.16. The first Brillouin zone of the sc, bcc, fcc, and hex lattices. Black dots and red lines represent the
high symmetry points and the edges of the irreducible part, respectively. Blue arrows indicate the direction and
the orientation (but not the modulus) of the reciprocal translation vectors {b1, b2, b3} defined in table 2.2.

2.5 Lattice defects

The property of translational invariance extensively discussed in the previous
sections generates ideally perfect crystalline structures. This is valid if we either
consider an infinite lattice or apply Born–von Karman periodic boundary con-
ditions. While useful in many circumstances to develop the constitutive ideas of solid
state physics, this idealised situation is surely a strong approximation to reality: in
fact, perfect crystals do not exist since at any finite temperature a solid state system
does contains defects, i.e. lattice imperfections, that locally break the translational

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 Solid State Physics

invariance. Their role is key in affecting many physical properties like, for instance,
the transport of electric charge or thermal energy.
We will prove the unavoidable presence of defects in crystals by applying a simple
thermodynamical argument to a monoatomic Bravais lattice containing N atoms
and kept at constant non-vanishing temperature T and pressure P. Its energy content
is provided by the Gibbs free energy G = U − TS + PV = H − TS , where U and
H = U + PV are the internal energy and enthalpy, respectively (in appendix C,
reference is made to the thermodynamic potentials used in this demonstration). The
generation of a lattice defect (we mean: the local alteration of the crystal structure)
requires a work
ΔHf = H − H 0 , (2.15)

known as the formation energy of the defect. In this equation H0 represents the
enthalpy of the pristine ideal crystal. In order to make physical concepts clear, we
consider the actual case of a lattice vacancy and a self-interstitial defect: in the first
case, a single atom is removed and taken far away from the crystal, while in the former
case an extra atom of the same chemical nature is added to the crystal in a position not
corresponding to any lattice point12. These defects are named native, since the
chemistry of the crystal is unaffected by their existence. While these are specific (but
realistic) situations, the reasoning developed below will lead to conclusions of general
validity. We will further assume that the crystal is always in thermodynamical
equilibrium, even after defects have been generated in it. A more thorough discussion
on the formation of crystal defects is found elsewhere [16, 17].
The generation of a native defect will of course affect the crystal entropy and,
therefore, we need to calculate its variation ΔS with respect to the pristine situation.
Let us suppose to generate n similar defects at random and far away13 positions. The
number W of unalike configurations where the same number of native defects is
differently distributed within the Bravais crystal is given by simple combinatorics
N!
W= , (2.16)
( N − n ) ! n!

so that the entropy variation ΔS due to defects is

⎡ N! ⎤ N!
ΔS = S − S0 = kB ln ⎢ ⎥ − kB ln[1] = kB ln , (2.17)
⎣ ( N − n ) ! n! ⎦ (N − n )!n!

where kB is the Boltzmann constant and S0 is the entropy of the perfect crystal for
which W = 1 since there exists just one ideal lattice configuration. In order to process

12
This means that there does not exist a lattice vector of the kind given in equation (2.1) giving the position of
the self-interstitial defect.
13
This assumption allows us to neglect the interactions among defects, by further considering that the vacancy
and the self-interstitial are not electrically charged.

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 Solid State Physics

equation (2.17) we make use of another combinatorics result, namely the Stirling
approximation, according to which for any sufficiently large number Ω it holds that

ln Ω! = Ω ln Ω − Ω . (2.18)

By using the Stirling approximation in equation (2.17) we get

ΔS = kB[ln N! − ln(N − n )! − ln n!] =
= kB[N ln N − N − (N − n ) ln (N − n ) + (N − n ) − n ln n + n ] = (2.19)
= kB[N ln N − (N − n ) ln (N − n ) − n ln n ].

By combining equations (2.15) and (2.19) we eventually obtain the Gibbs free energy
variation ΔG caused by the generation of n native defects in the crystal
ΔG = G − G0 = nΔHf − T (S − S0) =
(2.20)
= nΔHf − kBT [N ln N − (N − n ) ln (N − n ) − n ln n ].
Since the crystal is in equilibrium we have
d ΔG N−n ΔHf
= 0 → ΔHf − kBT [ln(N − n ) − ln n ] = 0 → ln = , (2.21)
dn n kBT
which leads to14
⎡ ΔHf ⎤
n(T ) = N exp⎢ − ⎥, (2.22)
⎣ kBT ⎦

which provides the number n(T ) of native defects found in the crystal in equilibrium at
temperature T. This result holds in general15, proving that at any finite temperature a
crystal must necessarily contain any kind of defects, the number of which varies
according to their formation energy ΔHf .
In order to catalogue the various kinds of defects, in the next section we will
follow a double criterion, according to which defects are distinguished both on the
basis of their dimension (point and extended defects) and by considering their
chemical nature (native and non-native character). Defects of any kind can combine
within the same crystal, generating a structure that can markedly deviate from the
ideal case of perfect lattice. Among the many possible configurations, two situations
play an important role in solid state and materials physics, namely alloys and
polycrystals; they are presented in appendix B. Here we also outline the intriguing
case of quasi-crystals: atomic architectures made by regular repetitions of structural
building blocks with no translational invariance.

14
We remark that ln[(N − n )/n ] = ln(N /n − 1) ≃ ln N /n since N ≫ n .
15
It is important to remark that we have implicitly assumed a static lattice approximation, that is: crystalline
atoms have been considered as clamped at their lattice positions. Actually, as will be extensively discussed in
the next chapters, atoms oscillate around their positions: therefore, we should also consider the vibrational
energy contribution to G . For the sake of simplicity, we have overlooked this feature, but this does not affect
the general validity of the final conclusion.

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 Solid State Physics

2.5.1 Point defects

Lattice imperfections involving just a single atom are named point defects: their
pictorial representation is reported in figure 2.17. They are more specifically referred
to as native or non-native: the former ones correspond to the vacancy and self-
interstitial case, previously introduced; the latter ones, instead, are generated whenever
an atom is added of a chemical species not found in the pristine crystal. Of course, in
lattices with a basis various kinds of vacancies and self-interstitials in fact exist.
Furthermore, in two-atom compound crystals we can also find the anti-site defect,
corresponding to a position exchange between two unalike nearest neighbouring
atoms. Sometimes point defects gather to form small defect aggregates. The most
common aggregates are defect clusters (where, for instance, a number of vacancies or
self-interstitials precipitate, thus forming, respectively, a small void or an inclusion in
the host crystal) and Frenkel pairs (where a bound pair of a vacancy and a self-
interstitial is formed). Finally, an atom can replace a regular crystalline atom,
belonging to a different chemical species: this configuration is known as a substitutional
impurity.
An important feature missing so far in our discussion is that the underlying lattice
is deformed by the presence of defects. We can say that this is per se another kind of
defect which, in this case, is described in terms of an induced lattice strain field:
interatomic distances are varied with respect to the ideal case and the bond network
is accordingly distorted. In figure 2.18 we provide a rendering of this concept in the
simple case of a two-dimensional square lattice which offers the possibility of a very
intuitive graphics.
Impurity defects are generated in the host crystal by natural contamination or by
some artificial process. In the first case, we refer to the very common contamination
from atmospheric environment which injects impurities and, by affecting its chemical
purity, alters many intrinsic physical properties of the crystal. Among the most

Figure 2.17. Point defects in a model two-dimensional compound crystal.

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 Solid State Physics

Figure 2.18. Pictorial representation of the lattice strain field (shaded regions) generated by an interstitial
impurity (left) and a vacancy (right). The black lines represent the distorted network of the interatomic bonds.

common impurities we find hydrogen, nitrogen, carbon and oxygen. Another

example of contamination is represented by the deposition on the surface of oxygen
atoms, always present in the air humidity: this process, called oxidation, causes the
formation of a surface oxide layer that often aggregates in a non-stoichiometric and
structurally disordered form.
In the case of intentional contamination, impurities are implanted into the host
lattice in arbitrary concentration. This process is normally called doping: it is widely
used in technology whenever it is necessary to alter the intrinsic number of charge
carriers of a material, so as to obtain the exact value useful for the application of
interest. The most common doping technique is ion implantation: an ion beam is
accelerated against a target sample; if the kinetic energy of the accelerated ions is
sufficiently high (in practice this energy varies in the range 103–106 eV), they can
penetrate the sample and, through a series of multiple inelastic collisions with the
atoms of the target crystal, they lose their initial kinetic energy, eventually stopping
inside the sample. This bombardment obviously generates quite a large lattice
damage because many atoms of the implanted region are knocked out from their
lattice position by collisions with the ions of the beam. Therefore, as a result of the
implantation process, the target sample ends up having a very messy and chemically
contaminated structure. By now applying suitable thermal annealing cycles, the
lattice damage can be recovered: during the temperature-activated recrystallisation
process the implanted impurities are incorporated into the lattice, mainly in the form
of substitutional defects. The implantation process is sketched in figure 2.19 (top).
In order to understand how doping affects the electrical property of a material, we
qualitatively discuss the prototypical case of a silicon sample which has been
implanted by As atoms. At this stage it is useful to recall that Si and As belong
to the group IV and group V, respectively (see appendix A), and therefore their
chemical valence is 4 and 5: for this reason silicon crystallises as an elemental solid in
the diamond structure with four-fold coordination. Whenever an As atom replaces a

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 Solid State Physics

Figure 2.19. Top: pictorial representation of the ion implantation process. The grey shaded area indicates the
region of maximum lattice damage. Bottom: graphical rendering of the doping process. For each As atom
implanted into a silicon sample, an electron is made free to join a conduction gas of charge carriers.

Si one upon implantation, its first four electrons provide the same bonding as in the
pristine sample; the fifth electron, instead, is available to join a conduction gas of
free similar charge carriers generated by the other implanted dopant impurities. The
number density of this gas is basically determined by the actual number of implanted
As atoms per unit volume: its overall conduction properties can be therefore
engineered as needed. The doping process is sketched in figure 2.19 (bottom).

2.5.2 Extended defects

Lattice defects involving multi-atomic configurations are called extended defects and
represent lattice errors. The two most significant cases we limit our attention to are
dislocations and grain boundaries.
Dislocations are line defects: a crystal lattice is ‘dislocated’ with respect to a line
defined by an appropriate vector Ld . The concept is illustrated in figure 2.20 (top) for
the two different cases of edge dislocation and screw dislocation. Dislocations are
described crystallographically by a set of two vectors: the first one is Ld , while the
second vector (indicated with the symbol Bd ) is called Burgers vector and it is
graphically represented in figure 2.20 (bottom) in the case of an edge dislocation.
Basically, Bd represents the difference in path when the dislocation core is short-
circuited in the defective lattice or when the same path is followed in the perfect
lattice16. By means of the pair {Ld , Bd} we can distinguish the two kinds of extended
line defects in that Ld and Bd are normal or parallel in edge or screw dislocations,
respectively.

16
It is understood that the individual steps of the path are represented by translation vectors.

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 Solid State Physics

Ld
Ld Bd Bd

Bd

Figure 2.20. Top: an edge dislocation (left) and a screw dislocation (right) are shown together with their line
vector Ld and Burgers vector Bd . Bottom: the graphical definition of the Burgers vector in the case of an edge
dislocation. The magenta arrows mark the path defining Bd . The symbol ⊥ indicate the position of the
dislocation core.

Dislocations play a fundamental role in plasticity17. Qualitatively, a plastic

deformation is due to the generation–migration–accumulation sequence of a
dislocation forest. Another situation where dislocations are found is at the interface
between two lattice-mismatched crystals. In this case, in an attempt to accommodate
the unalike interatomic spacings on the planes parallel to the interface, the two
crystals deform (by stretching or by compression provided that the lattice constant is
smaller or larger, respectively) and thus they store elastic energy. When the lattice
mismatch is sizeable, the accumulated elastic energy can be sufficient to be converted
into formation work of dislocations, whose generation allows recovery of the
pristine lattice spacings far away from the dislocation core.
Grain boundaries are planar defects: they form at the interface between two
differently oriented crystal lattices. There are basically two different ways to
generate a grain boundary (GB), both obtained by cutting a crystal along an
imaginary plane: (i) a twist GB is formed by rotating one of the two semi-crystals
around the normal direction of the imaginary plane; (ii) a tilt GB is instead
generated by inclining one of the two semi-crystals with respect to the other one,
through the imaginary plane. When the tilt or twist angle is small, a regular bond
network is reconstructed at the interface between the two semi-crystals, although
with a different topology with respect of the ideal crystal. On the other hand, when
such angles are large enough, the accumulated elastic energy of lattice distortion is

17
In solid mechanics plasticity is the tendency of a body under load to undergo permanent deformations.

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 Solid State Physics

Figure 2.21. Left: a triple junction among three tilt GBs in silicon with small tilt angles; the shaded area
represents the GB region. Right: a twist GB in silicon with a large twist angle; some structural disorder is
found, both in atomic coordination and topology of the bond network. Pictures are obtained by atomistic
simulations. Left: picture adapted from [18]. Right: picture adapted [19].

converted in formation work of an amorphous-like boundary layer; here atoms are

found with coordination other than the pristine one, and bonds are randomly
oriented. GBs of both kinds are shown in figure 2.21.
Grain boundaries represent accumulation regions for impurities or precipitation
sites for dopant species. Qualitatively, we can explain this evidence by considering
the possibly very disordered character of the bonding network in the boundary
region: as a matter of fact, here a very high density of unsaturated bonds with high
chemical reactivity is found (see for instance figure 2.21(right)): they work as
gatherers of chemical species different from those of the host lattice.

2.6 Classification of solids

Symmetry has been largely used in the previous sections to rigorously classify
crystalline solids. Other criteria can in fact be followed, based on physical properties
of any kind. For instance, by looking at the mechanical response under load we can
discriminate between elastic or plastic materials, as well as classify their fracture,
failure, and yield behaviours; similarly, by investigating the interaction between
electromagnetic waves and solids, we can separate them according to their optical
properties. However, the most fruitful classification scheme based on a physical
property is likely based on the configuration assumed by the crystalline valence
electrons. This approach relies on the frozen-core approximation presented in
section 1.3.2; since at this stage we have not yet developed any knowledge about
the electronic structure of a solid state system, we will proceed at a more qualitative
(but nevertheless useful in the next chapters18) level than allowed by symmetry
classification.

18
Also, it provides a justification for the previous reference to metallic or non-metallic solids we did in
sections 2.3.2 and in appendix B.

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 Solid State Physics

The first step consists in making a difference between metals and insulators. While
the most rigorous definition is only provided by the quantum band theory (see
chapter 8), for the present discussion it is sufficient to rely just on phenomenological
inputs: metals are good electrical conductors, while insulators are not. Whenever
subjected to the action of an external electric field, metals are crossed by a current of
charge carriers. This evidence is qualitatively attributed to the fact that valence
electrons form a conduction gas of nearly-free charged particles: under the action of
an external electric field, the conduction gas is accelerated, giving rise to current
phenomena. On the other hand, in an insulator the density of such a gas is so small,
that the resulting current density is comparatively negligible. From these consid-
erations we draw a qualitative picture: in a metal system, ions sitting at lattice
positions are embedded into a ‘glue’ of valence electrons which form an almost
uniform charge distribution.
The case of insulators is definitely more complex, since quite different situations
are found. Basically, they are broadly distinguished into four kinds:
• molecular crystals: solids made of noble elements Ne, Ar, Kr, Xe belonging to
the group VIII A (see appendix A); their electronic structure is characterised
by the fact that they completely lack valence electrons, since the configuration
of the isolated atoms is only slightly affected in the crystalline state; their
binding is due to weak van der Waals or electric dipole forces;
• ionic crystals: solids made of positive and negative ions, as consequence of
electron transfer from cations to anions; their electronic structure is charac-
terised by the fact the valence electron density is very highly localised nearby
the cation and anion cores; consequently, they can be roughly treated as a
periodic array of impenetrable positively or negatively charged spheres,
respectively. Their physical properties are therefore largely dominated by
electrostatic interactions; the most common ionic crystals are formed by
atoms belonging to the I–VII A (also known as alkali-halides), or II–VI A,
and III A–V A groups19;
• covalent crystals: solids where the valence charge is mainly localised along
chemical bonds; this configuration is explained by the so called octet rule20,
according to which elements tend to bond through electron sharing so that
each atom reaches the same electronic configuration as a noble gas with eight
electrons in the outer shell; covalent solids are made of atoms belonging to the
IV A, or II–VI A, and III A–V A groups;
• hydrogen-bonded crystals: solids formed by electrostatic attraction between a
hydrogen atom (working as the positive centre) and a negatively charged ion;
since hydrogen is comparatively smaller than any other crystalline ion, it can
easily fit into the crystal structure; typical hydrogen-bonded solids are organic
ones or ice (solid water).

19
In fact, II–VI A and III A–V A solids are better defined as partly ionic, partly covalent solids.
20
The octet rule is used in chemistry as a rule of thumb for predicting the properties of the elements [20].

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 Solid State Physics

Figure 2.22. A pictorial representation of the valence charge distribution (grey) in metals and ionic or covalent
insulators. Positive and negative ions are in red and blue colour, respectively.

The metallic and insulating bonding structure is schematically shown in

figure 2.22, where a pictorial view of the valence charge distribution is drawn.

2.7 Cohesive energy

A final question still remains to be answered: how much work Ecohesive is needed to
assemble a set of atoms into a crystalline solid? This quantity is defined as the
difference in energy between a configuration where atoms lie at infinite distance and
a configuration where they form a bound crystal. Let us name ETfree atoms the total
energy of N free atoms (possibly of different chemical species) and ETcrystal the energy
of their crystalline configuration, then we calculate the cohesive energy per atom as
1 free atoms
ecohesive = [ET − ETcrystal ], (2.23)
N
while of course Ecohesive = Necohesive .
The most fundamental approach for calculating the cohesive energy is quantum
mechanical [21], as formally shown in equation (1.13) (since Ecohesive does depend on
the ground-state energy of the solid). This is the only way to proceed for metals and
covalent crystals where valence electrons are totally or partially delocalised. On the
other hand, for molecular and ionic crystal it is possible to follow a much simpler,
although phenomenological, approach [10, 11, 15] where a number of simplifications
are assumed, in that: atoms are treated classically (we place them at rest exactly in
positions corresponding to crystalline lattice sites with no quantum zero-point
energy effects included; the valence electron charge distribution is treated by
classical electrostatics); the calculation is performed by imposing an arbitrary value
of the lattice constant(s); and the temperature is set to zero (no free energy
contributions are accounted for). By imposing that the derivative of the cohesive
energy with respect to the lattice constant(s) is zero, we obtain their equilibrium values:
those ones at which the real crystal is found in stable equilibrium at zero pressure.
In molecular crystals binding is due to the balance between interatomic attractive
dipolar interactions (long-ranged and weak) and Pauli repulsion between electron
clouds (occurring at short distance and very strong). Usually the resulting pair
interaction potential is written in the form known as the 6–12 Lennard-Jones
potential [22]

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 Solid State Physics

⎡⎛ σ ⎞12 ⎛ σ ⎞6⎤
VLJ(R ) = 4ϵ⎢⎜ ⎟ − ⎜ ⎟ ⎥ , (2.24)
⎣⎝ R ⎠ ⎝R⎠ ⎦

where ϵ represents the attraction strength and σ the radius of the repulsive core
(assumed spherical and centred on each lattice site). In equation (2.24) R is the
interatomic distance. The cohesive energy Ecohesive of a molecular crystal crystal
containing N atoms is
N
Ecohesive = ∑ VLJ(Rl), (2.25)
2 Rl ≠ 0

where the origin Rl = 0 has been placed on an arbitrary lattice site. If we indicate by
Rnn the first nearest-neighbour distance, then any translation vector is cast in the
form Rl = α l Rnn; in this way, we can more conveniently write the cohesive energy as
⎡ ⎛ σ ⎞12 ⎛ σ ⎞6⎤
Ecohesive(R nn ) = 2Nϵ⎢A1⎜ ⎟ − A2 ⎜ ⎟ ⎥, (2.26)
⎢⎣ ⎝ R nn ⎠ ⎝ R nn ⎠ ⎥⎦

where A1 = ∑R ≠0α l−12 and A2 = ∑R ≠0α l−6 , both only depend on the crystal struc-
l l
ture. Minimising Ecohesive(Rnn ) leads to the theoretical prediction of equilibrium
nearest-neighbour separation Rnn = (2A1/A2 )1/6σ . Finally, by a standard thermody-
namics argument (see appendix C) and using the volume per atom v = V /N (where
V is the total volume of the N-atom crystal) and the cohesive energy per atom
ecohesive = Ecohesive /N we easily get the bulk modulus B of molecular crystals as
∂ 2ecohesive 4ϵ
B=v = 3 A1−3/2 A25/2 (2.27)
∂v 2
σ
The prediction of the Lennard-Jones potential, despite its phenomenological
character, is remarkably in good agreement with experimental data, as shown in
table 2.3.
In ionic crystals binding is due to the balance between electrostatic interactions
(long-ranged and both attractive or repulsive, according to the signs of the charges
for any selected ion pair) and Pauli repulsion between electron clouds (occurring at

Table 2.3. Lennard-Jones parameters (ϵ, σ), cohesive energy per atom (ecohesive ), nearest-neighbour separation
Rnn , and bulk modulus B of some fcc molecular crystals made by noble elements. Values in parenthesis
represent experimental data. For any crystal here represented we have A1 = 12.132 and A2 = 14.454 .

ϵ [10−4 eV] σ [Å] ecohesive [eV/atom] Rnn [Å] B [atm]

Ne 32.11 2.74 0.02 2.99 (3.13) 1.79 (1.09)

Ar 107.25 3.40 0.08 3.71 (3.75) 3.15 (2.67)
Kr 144.49 3.65 0.12 3.98 (3.99) 3.42 (3.46)
Xe 205.50 3.98 0.17 4.34 (4.33) 3.77 (3.56)

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 Solid State Physics

short distance). The first term is by far the dominating one in these crystals and it is
hard to evaluate correctly because of the long-range nature of electrostatic
interactions: as a matter of fact, while there is no ambiguity in calculating the total
electrostatic energy of a finite set of point-charges, the same quantity becomes ill
defined for an infinite ionic crystal. The physical reason behind this is explained by a
simple argument: let us suppose we reach the limit of an infinite crystal by
assembling step-by-step increasingly large crystallites, each containing a finite
number of anion/cation pairs; it is easy to figure out that each crystallite can be
realised in many possible ways, each one with a different surface charge distribution.
The infinite crystal limit is, therefore, reached by following different growth paths,
resulting in a different total electrostatic potential energy because of the different
surface electrostatics of its building blocks21.
Methods mastering Coulomb lattice sums do nevertheless exist, among which
the Edwald summation technique is the most widely used in solid state physics.
Since it is anything but an elementary method, it is well beyond the level of our
treatment (for a detailed description see [21]). Here we limit ourselves to
outlining the main result of cohesion theory in ionic crystals. Let us consider a
binary ionic crystal and let Qβ be the charge of the ion in position Rβ . The
Coulomb potential VC(R α) felt by the ion in position R α due to all remaining ions
in the crystal is

e Qβ
VC(R α) =
4πϵ0
∑ R αβ
(2.28)
β≠α

where R αβ = ∣R α − Rβ∣ is the ion–ion distance. We now introduce the Madelung

constant Mα for the α-type ions by writing
e
VC(R α) = Mα, (2.29)
4πϵ0R nn

where Rnn is the equilibrium first nearest-neighbour distance and

R
Mα = ∑ Qβ R nn , (2.30)
β≠α αβ

a dimensionless constant which is calculated by some lattice summation method.

The Coulomb potential energy EC(R α) of the αth ion is eventually written as

e2
E C(R α) = QαeVC(R α) = Q Mα. (2.31)
4πϵ0R nn α

21
More formally, it is proved that the sum of Coulomb interactions among positive and negative charges
regularly distributed onto a translationally invariant lattice is only conditionally convergent: it provides any
number, depending on the order in which the terms in the sum are added [21].

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 Solid State Physics

There are as many Madelung constants as ions occupying different lattice sites. For
instance, for a typical binary ionic crystal we have two constants: one for each
chemical species; however, since anions and cations occupy lattice sites with the very
same symmetry, the two constants have opposite sign but same magnitude.
Furthermore, Mα is specific of any crystal structure and, in particular, its magnitude
is calculated to be 1.763 for the CsCl structure, 1.748 for the NaCl structure, and
1.638 for the zincblende structure.
In order to complete the model for the cohesion energy of ionic crystals, we must
add a contribution describing the short-range Pauli repulsion. Its is usually written
as an inverse power law of the distance between nearest-neighbour ions or, more
accurately, as a rapidly decaying exponential. In summary, the cohesive energy is
written as
Ecohesive = ∑ E C(R α ) + ∑ Bαβe−Rnn /Cαβ , (2.32)
α ion pairs

where the second term on the right-hand side of this equation is obtained by
summing over all the ion pairs in the crystal and the Bαβ and Cαβ terms are
phenomenological constants to be fitted on experimental information.

References
[1] Borchardt-Ott A 2012 Crystallography–An Introduction (Berlin: Springer)
[2] Hoffmann F 2020 Introduction to Crystallography 1 (Berlin: Springer)
[3] Glusker J P 2010 Crystal Structure Analysis: A Primer (Oxford: Oxford University Press)
[4] Hammond C 2015 The Basics of Crystallography and Diffraction (Oxford: Oxford Science
Publications)
[5] Evarestov R A 1997 Site Symmetry in Crystals: Theory and Applications (Berlin: Springer)
[6] Müller U 2017 Symmetry Relationships between Crystal Structures–Applications of
Crystallographic Group Theory in Crystal Chemistry (Oxford: Oxford Science Publications)
[7] Burns G and Glazer M 2013 Space Groups for Solid State Scientists 3rd edn (Waltham, MA:
Academic)
[8] Wright J D 1995 Molecular Crystals (Cambridge: Cambridge University Press)
[9] Kitaigorodsky A I 1973 Molecular Crystals and Molecules (New York: Academic)
[10] Dove M T 2003 Structure and Dynamics–An Atomic View of Materials (Oxford: Oxford
University Press)
[11] Kittel C 1996 Introduction to Solid State Physics 7th edn (Hoboken, NJ: Wiley)
[12] Ashcroft N W and Mermin N D 1976 Solid State Physics (London: Holt-Saunders)
[13] Kenyon I R 2008 The Light Fantastic–A Modern Introduction to Classical and Quantum
Optics (Oxford: Oxford Science Publications)
[14] Bransden B H and Joachain C J 1983 Physics of Atoms and Molecules (Harlow: Addison
Wesley)
[15] Hook J R and Hall H E 2010 Solid State Physics (Hoboken, NJ: Wiley)
[16] Tilley R J D 2008 Defects in Solids (New York: Wiley)
[17] Phillips R 2001 Crystals, Defects and Microstructures (Cambridge: Cambridge University
Press)

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 Solid State Physics

[18] Costantini S, Alippi P, Colombo L and Cleri F 2000 Phys. Rev. B 63 045302
[19] Cleri F, Keblinski P, Colombo L, Wolf D and Phillpot S R 1999 Europhys. Lett. 46 671
[20] Pauling L 1970 General chemistry (New York: Dover Publications Inc)
[21] Grosso G and Pastori Parravicini G 2014 Solid State Physics 2nd edn (Oxford: Academic)
[22] Finnis M 2003 Interatomic Forces (Oxford: Oxford University Press)

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 Part II
Vibrational, thermal, and elastic properties
 IOP Publishing

Solid State Physics

A primer
Luciano Colombo

Chapter 3
Lattice dynamics

Syllabus—The dynamics of a crystal lattice is developed under the leading adiabatic

and harmonic approximations. The main features are at first elaborated by studying
the model cases of a monoatomic and a diatomic linear chain: here, the concepts of
normal mode of vibration, dispersion relation, and acoustic or optical character of a
vibration are extensively discussed. The dynamics of realistic three-dimensional
crystals is formalised through the calculation of the dynamical matrix, which allows
for the standard representation of the vibrational dispersion relations. Next, by a
simple quantisation procedure of the ionic displacement field, we introduce the concept
of phonon and interpret the vibrational spectrum of a crystal as the physics of a gas of
such non-interacting pseudo-particles. Finally, we discuss the experimental determi-
nation of phonon dispersion relations by neutron spectroscopy and the vibrational
density of states.

3.1 Conceptual layout

The description of crystal structures developed in chapter 2 relies on an implicit (but
really very strong) assumption, namely: ions are clamped at their lattice positions.
This is, also, the situation assumed to define the eigenvalue problem for the total
electron wavefunction given in equation (1.15) within the framework developed
under the adiabatic approximation. While the assumption of static lattice is useful in
the above contexts, it is either conceptually wrong and inadequate to describe many
important solid state phenomena.
First of all, we recall that ions, although comparatively much more massive than
electrons, have in fact a finite mass: therefore, according to fundamental quantum
mechanics [1–4], they always (even at zero temperature) have a non-vanishing mean
square momentum. We can reconcile crystallography with the quantum uncertainty
principle by assuming that the mean position of an ion (obtained by averaging over
its zero-point motion) corresponds to Rl in Bravais lattices or to Rl + Rb for lattices
with a basis (see equations (2.1) and (2.3), respectively).

doi:10.1088/978-0-7503-2265-2ch3 3-1 ª IOP Publishing Ltd 2021

 Solid State Physics

This is not enough. Still considering ions at rest (although in some ‘average’
meaning) is inconsistent with a number of experimental evidences, including (but not
limited to): thermal expansion, melting, thermal conductivity, sound propagation,
inelastic scattering of electromagnetic waves or particles (electrons as well as
neutrons). All together these phenomena provide a robust body of experimental
evidence that lattice ions do undergo some kind of motion. The aim of this chapter is
to fully characterise the corresponding lattice dynamics.
We will accomplish this task at first under the leading adiabatic and classical
approximations (see sections 1.3.4 and 1.4.2, respectively). Non-classical dynamical
features will appear later, by a suitable quantisation procedure operated on the ionic
classical displacement field. In developing our classical phenomenological theory of
lattice dynamics, we will assume that there exists a many-body potential energy
U = U (R) governing the motion of the ions1. Basically, U contains the ion–ion
Coulomb interaction energy as well as their kinetic energy, as conceptualised in
section 1.3.4.
The next step of simplification consists in assuming that the displacement of each
ion from its mean equilibrium position2 is small as compared to the typical lattice
spacing. This assumption is proved by the very results of x-ray scattering: the crystal
structure appears cleanly clear in spite of the fact that ions undergo movement
during diffraction. In other words: ionic motion is neither diffusive nor so wide as to
alter the underlying lattice structure.
In order to proceed rigorously, we must define a more accurate notion for
ionic positions and displacements. To make it also compact, we agree to indicate
the lattice position of the generic ion by Rl + Rb and its displacement by u(lb ),
where it is understood that the indices (lb ) run, respectively, over the lattice
points and positions within the basis. Since electrons will not explicitly enter
our discussion, Latin indices i1, i2, … , in will be used to label the Cartesian
components.
The assumption of small displacements allows for the formal expansion of the
many-body potential energy U = U (R) into a power series of the ionic displace-
ments. By setting to zero the constant term3 U0 for further convenience, we can
write

∂U 1 ∂ 2U
U =∑ ui (lb) + ∑ u i1(lb)u i2(l ′b′) + ⋯
lb, i
∂ui (lb) 0
2 l1b1, i1
∂u i1(l1b1)∂u i2(l2b2 ) 0
l2b 2, i2
+∞
(3.1)
1
=∑ ∑ Ui1i2⋯i n(l1b1, l2b2 , … , lnbn) u i1(l1b1)u i2(l2b2 )⋯u i n(lnbn),
n=1
n! i1, i 2, … , i n
l1b1, l2b 2, … , lnb n

1
We recall that the symbol R indicates the full set of ionic positions.
2
Hereafter equivalently referred to as its lattice position.
3
This term represents the energy of a static lattice. Since it is constant, we can gauge the potential energy so
that U0 = 0 with no loss of generality.

3-2
 Solid State Physics

where we have used the compact notation

∂ nU
= Ui1i2…i n(l1b1, l2b2 , … , lnbn). (3.2)
∂u i1(l1b1)∂u i2(l2b2 )…∂u i n(lnbn) 0

As indicated, all derivatives are calculated with all ions at their rest positions; in
particular, this implies that
∂U
= 0, (3.3)
∂ui (lb) 0

which defines the rest condition for the crystal4. Since displacements are small, we
can truncate the expansion to the first non-vanishing term, thus obtaining the many-
body potential energy Uharmonic
classical
in harmonic approximation
1
classical
Uharmonic =
2
∑ Ui1i2(l1b1, l2b2 ) u i1(l1b1)u i2(l2b2 ).
(3.4)
l1b1, i1
l2b 2, i2

This approximation is named ‘harmonic’ since the physics described by equation (3.4)
is equivalent to that of a set of identical harmonic oscillators (as many as the ionic
dynamical degrees of freedom) whose force constants are given by the terms
Ui1i2(l1b1, l2b2 ). The analogy with classical mass-spring systems is further exploited by
naming the ion motions as vibrations. In short, in our picture ions harmonically vibrate
around their lattice position. This is the entry level for discussing lattice dynamics;
other physical features will be added later.
Before addressing the physics of ionic vibrations, it is important to remark that
the force constants are subject to formal restrictions [5, 6] which are imposed by some
true physical features. The Taylor expansion given in equation (3.1) makes use of the
full set of 3N vibrational degrees of freedom existing in a crystal made by N atoms.
The expansion coefficients there appearing, i.e. the force constants, must never-
theless guarantee that the crystal energy U is invariant under a rigid body translation
or rotation5. For instance, it is not difficult to prove that the invariance under a rigid
translation dictates

∑ Ui1i2…i n(l1b1, l2b2 , … , lnbn) = 0. (3.5)

l1b1, l 2, b 2, … l nbn

Instead of using the 3N Cartesian components of the ionic displacement to expand

U, we could rather use an alternative set of (3N -6) independent internal coordinates
(such as interatomic distances or bond angles), carefully chosen to be invariant
under rigid translations and rotations.

4
The equilibrium condition of an infinite crystal is more complicated: ions must be at rest so that equation
(3.3) is fulfilled and the corresponding configuration must be a stress-free one [5, 6]. The concept of stress will
be introduced in the next chapters.
5
These invariances reduce the number of degrees of freedom relevant for internal vibrations to (3N − 6), since
translations and rotations each involve three of them.

3-3
 Solid State Physics

The energy U must as well remain unchanged under any symmetry operation of
the given crystal we are dealing with. Group theory [7] allows studying the action of
any symmetry operation on each matrix whose elements are the 2nd-, 3rd-, 4th-, …
nth-order force constants defined in equation (3.2). In this way we can calculate
which are the independent non-zero elements of such matrices.
Finally, we remark that in the following force constants will be considered as
given parameters, based on whose knowledge we will build the theory of lattice
dynamics. This assumption is reasonable and adequate since, in general, force
constants can be independently calculated by guessing some interatomic force
field, whose parameters are typically fit on experimental vibrational frequencies
(and, sometimes, using ionic displacement patterns as well). In appendix D some
force constants models extensively used in literature are outlined. An alternative
approach consists in developing a fully quantum mechanical theory of U, from
which to derive the force constants; in general, they are much more accurate than
their empirically determined counterparts, but they are obtained at the cost of a
much higher computational burden. This more fundamental approach is based on
the knowledge of the total energy of the crystalline ground state configuration: a
task developed in chapter 10.

3.2 Dynamics of one-dimensional crystals

The most fundamental features of lattice dynamics can be brought out by
investigating a one-dimensional crystal. The main advantage in studying such a
simplified situation is that the heavy algebraic formalism indeed necessary to
manage the full set of indices occurring in three-dimensional crystals is largely
spared. A wire made by ions connected by effective harmonic springs is just enough
to catch the essential physics: we will refer to this model system as the harmonic
linear chain.

3.2.1 Monoatomic linear chain

Let us consider a linear chain where N identical ions of mass M are placed at
distance a when they are at rest in equilibrium positions. This corresponds to a one-
dimensional Bravais crystal with lattice spacing a; the primitive unit cell is obtained
by the Wigner–Seitz construction as a segment of length a with the ion placed at its
midpoint. By adopting Born–von Karman boundary conditions, the ionic positions
are indicated as Rl = la with l = 0, 1, 2, … , N − 1. Finally, following the force
constant approach discussed in the previous section, we represent the interactions
between nearest neighbouring ions as harmonic springs. The situation is sketched in
figure 3.1.
Let us consider a longitudinal vibration of the chain, that is a displacement pattern
in which the ions move along the chain direction. The classical equation of motion
for the lth ion is
Mul̈ = γ (L )(ul +1 + ul −1 − 2ul ), (3.6)

3-4
 Solid State Physics

uN = u0
uN −1 u1

u2
a
u3

Rl−1 Rl Rl+1

ul
Figure 3.1. A monoatomic spring-mass linear chain (left) under the Born–Von Karman boundary condition
(right); the unit cell is shaded in light-blue colour, with lattice constant a.

where γ (L ) is the force constant of the effective spring. Suggested by the elementary
mechanics of a vibrating wire, we seek a solution in the form
1
ul = ∣A q∣ cos[qRl − ω(q )t + φ(q )], (3.7)
NM
where the normalising factor (NM )−1/2 has been introduced for further convenience,
while ∣A q∣ and φ(q ) are the amplitude and the initial phase of the wave6. Of course, q
and ω(q ) are the wavenumber and the angular frequency of the travelling wave,
respectively. Replacing equation (3.7) into equation (3.6) leads to
⎛1 ⎞
Mω 2(q ) = 2γ (L ) [1 − cos(qa )] = 4γ (L ) sin2 ⎜ qa⎟ , (3.8)
⎝2 ⎠

which is known as the dispersion relation and it is shown in figure 3.2(top). This
representation is redundant since it ignores translational periodicity: it makes no
difference in the displacement ul by increasing q → q + G with G = 2mπ /a a
reciprocal lattice vector of the linear chain crystal (m is any positive or negative
integer number). It is therefore customary to adopt the reduced zone scheme: the
dispersion relation is represented only for q ∈1BZ or, equivalently, for
q ∈ [−π /a, +π /a ] as shown in figure 3.2(bottom). The actual number of allowed q
is determined by the imposed boundary conditions: since it must be u 0 = uN then
2π ξ
q= with ξ = 0, 1, 2, 3, … , N − 1 (3.9)
a N
This is a very important result: in an N-atom monoatomic chain there are only N
independent values of the wavevector q associated with as many independent
solutions of the equations of motion. In more physical terms: if we consider a
one-dimensional chain containing N identical ions, there are only N different ways in
which they can longitudinally oscillate around their equilibrium positions.

6
Their relation is A q = ∣A q∣ exp[iφ(q )].

3-5
 Solid State Physics

ω(q)

q
−2π/a −π/a 0 +π/a +2π/a
ω(q)

2 γ (L) /M

q
−π/a 0 +π/a

Figure 3.2. Top: dispersion relation ω = ω(q ) for the longitudinal vibrations of a monoatomic chain; the 1BZ
is shaded in light-blue colour. Bottom: the same dispersion represented in the reduced zone scheme. The ion
mass is M and the effective spring constant is γ (L ) .

The solutions of equation (3.8) are waves travelling along the chain with a group
velocity vg = dω(q )/dq . In the limit qa ≪ 1 the dispersion relation reduces to
ω(q ) = [γ (L ) /M ]1/2 qa , similarly to a wave propagating along a material wire with
mass density M/a and subject to a mechanical tension γ (L )a . In the long wavelength
limit the dispersion relation is, therefore, that of a compressional wave propagating
with group velocity vg = γ (L ) /M a . In other words, in the limit q → 0 (zone-centre
wavevector) the equation (3.8) describes the propagation of a longitudinal sound
wave. For this reason the ω = ω(q ) relation is referred to as the acoustic dispersion.
In the opposite situation of q = ±π /a (zone-boundary wavevector), the vibrational
frequency is found to reach the value ω(q = ±π /a ) = 2 γ (L ) /M which is understood
as the maximum frequency for longitudinal vibrations.
We complete the above discussion by observing that a linear chain can carry
transverse oscillations as well: in fact, ions can vibrate normally to the direction
along which they are initially aligned to form the chain itself. The mathematical
analysis of transverse oscillations is just the same as in the longitudinal case, with
one obvious difference: the force constants describing the transverse effective springs
may have a value γ (T) ≠ γ (L). So, at the mere cost of replacing in all the previous
equations the appropriate force constant value, we can draw similar conclusions. In
particular, we derive the transverse dispersion relations and get the speed of
transverse sound. In general, the transverse force constants are smaller than

3-6
 Solid State Physics

longitudinal ones. Furthermore, since for a linear chain there are two equivalent
transverse directions, the two force constants are equal γ (T1) = γ (T2 ) < γ (L ) and two
corresponding dispersion relations are degenerate.
Let us now move to consider displacements. In general, the oscillation paths of
the chain are obtained from equation (3.7) by considering the N allowed discrete
values of the wavenumber q, separated by the quantity 2π /Na within the interval
[−π /a, +π /a ]. In the presently discussed case of longitudinal oscillations the general
solution for the displacement of the lth ion is
1
ul = ∑∣A q∣ cos[qRl − ω(q)t + φ(q)], (3.10)
NM q

where the N possible amplitude-phase pairs7 [∣A q∣, φ(q )] are calculated by providing
the N initial positions and the N initial velocities of the ions. The picture should at
this point be clear: each ion vibrates around its equilibrium position as a harmonic
oscillator, while the chain as a whole is interested by N different normal vibrational
modes. The former describe single-particle motions, the latter provide information
on collective movements.
In figure 3.3 the displacement patterns for longitudinal and transverse vibrations
are shown, both at zone-centre and zone-boundary. The key point there illustrated is
that if we consider ions belonging to nearest neighbour unit cells they oscillate in
phase for zone-centre vibrational modes, while they oscillate in phase opposition for
zone-boundary vibrational modes. The case of zone-centre vibrations deserves a
further comment: figure 3.3 clearly shows that the travelling wave is a compressive
wave in both the longitudinal and transverse polarisation, although the compres-
sion, respectively, occurs along or perpendicularly to the travelling direction of the
wave. We can therefore distinguish between longitudinal sound and transverse sound.
Experimental measurements of the sound speed in a real specimen provide evidence
that the longitudinal sound (moving at speed vg(L ) = γ (L ) /M a ) is faster than the
transverse one (moving at speed vg(T ) = γ (T ) /M a ): this is a direct confirmation that
transverse force constants are smaller than longitudinal ones.

3.2.2 Diatomic linear chain

Let us now turn to consider the one-dimensional model of minimal complexity for a
lattice with a basis, namely a diatomic linear chain. We need to define two ion
masses M1 and M2 and two effective springs γ (L ) and ξ (L ), respectively, coupling ions
within the same unit cell or belonging to nearest neighbouring unit cells. Ion
positions are now indicated as Rl,1 = Rl + R1 and Rl,2 = Rl + R2 , where Rl labels the
lth unit cell, while R1 and R2 specify the ion within the basis. The situation is
sketched in figure 3.4 and once again we start by considering longitudinal
oscillations.

7
One pair for each allowed wavenumber.

3-7
 Solid State Physics

Figure 3.3. Zone-centre and zone-boundary displacement patterns of longitudinal and transverse vibrations in
a monoatomic linear chain whose unit cell is shaded in light-blue colour in the top panel. Blue arrows indicate
the direction and the orientation of the ionic displacement vector. The travelling direction of the wave is
towards left or right provided that its wavevector is, respectively, positive or negative.

γ (L) ξ (L)

M1 M2
Rl−1,2 Rl,1 Rl,2 Rl+1,1

Figure 3.4. A diatomic spring-mass linear chain; the unit cell is shaded in light-blue colour, with lattice
constant a. Ion masses M1 and M2 and force constants (for longitudinal oscillations) γ (L ) and ξ (L ) are indicated
as in the main text. Ion positions entering in equation (3.12) are shown as well.

3-8
 Solid State Physics

The equations of motion for the two ions in the lth unit cell form a system of two
differential equations
⎧ M1ul̈ ,1 = γ (L )(ul ,2 − ul ,1) + ξ (L )(ul −1, 2 − ul ,1)
⎪

⎨ (3.11)
⎩ M2ul̈ ,2 = ξ (L )(ul +1, 1 − ul ,2 ) + γ (L )(ul ,1 − ul ,2 ),
⎪

We seek solutions for this system of the same form given in equation (3.10).
However, for further convenience, it is useful to rewrite the amplitude as
∣A q∣ → ∣A q∣ ∣ai (q )∣ and the phase as φ(q ) → φ(q ) + ϕi (q ) (where i = 1, 2 labels the
ion within the basis) since, as we will prove soon, the terms ∣ai (q )∣ and ϕi (q ) are
determined by the very equations of motion, rather than by the boundary conditions
as instead ∣A q∣ and φ(q ). By this choice, we write

1
ul ,i = ∑∣A q∣ ∣ai (q)∣ cos ⎡⎣qRl ,i − ω(q)t + φ(q) + ϕi (q)⎤⎦, (3.12)
NMi q

which, if inserted in equations (3.11), leads to the following matrix equation

⎛ D11 D12 ⎞ a1 a
⎜ ⎟
⎝ D21 D22 ⎠ a2 ( )
= ω 2 a1 ,
2 ( ) (3.13)

where the square 2 × 2 Hermitean8 matrix {Dij}i,j =1,2 has elements

γ (L) + ξ (L)
D11 =
M1
γ (L) ξ (L)
D12 =− exp[iq(Rl ,2 − Rl ,1)] − exp[iq(Rl −1, 2 − Rl ,1)]
M1M2 M1M2
(3.14)
γ (L) ξ (L)
D21 = − exp[iq(Rl ,1 − Rl ,2 )] − exp[iq(Rl +1, 1 − Rl ,2 )]
M1M2 M1M2
γ (L) + ξ (L)
D22 = ,
M2

and it is referred to as the dynamical matrix of the system: its eigenvalues ω = ω±(q )
provide the dispersion relations for the diatomic chain, while its column eigenvector
⎛ a1(q ) ⎞
⎜ ⎟ provides information on the ionic displacements. It is now justified our
⎝a2 (q ) ⎠
choice of writing the oscillation amplitude as the product ∣A q∣ ∣ai (q )∣: the first term is
determined by the initial conditions imposed to solve equation (3.13) (that is, by
fixing the initial position and velocity of each ion), while the second term is
determined by the equations of motion (3.11). In particular, the quantities ai (q )
are named polarisation vectors.

8
It is laborious but not difficult to prove that D12 = D*21.

3-9
 Solid State Physics

By solving the matrix equation we obtain the eigenfrequencies

1 M + M2
ω±2(q ) = [γ (L) + ξ (L)] 1
2 M1M2
(3.15)
1 ⎛ M + M2 ⎞2 γ (L)ξ (L) ⎛ qa ⎞
± [γ (L) + ξ (L)]2 ⎜ 1 ⎟ − 16 sin2 ⎜ ⎟ .
2 ⎝ M1M2 ⎠ M1M2 ⎝2⎠

This result introduces a very important new feature: for each q wavenumber two
different vibrational frequencies ω±(q ) are found or, equivalently, in the diatomic
linear chain we have two different longitudinal dispersion relations, as shown in
figure 3.5. The lower dispersion ω−(q ) has similar characteristics as found in the case
of a monoatomic chain and for this reason is called the acoustic branch; once again,
its slope in the limit q → 0 corresponds to the speed of longitudinal sound. The
upper dispersion ω+(q ) is instead referred to as the optical branch for reasons that can
be easily understood by considering the displacement patterns in the unit cell. To this
aim we must preliminarily discuss an important feature. By replacing the solutions
ω±(q ) given by equation (3.15) into equation (3.13) we easily realise that we cannot
⎛a±(q ) ⎞
determine the absolute magnitude of the eigenvector elements ⎜ 1± ⎟ but only their
⎝ a 2 (q ) ⎠
ratio. The calculation is easy by assuming for simplicity that M1 = M2 and also
imposing the normalisation ∣a1±(q )∣2 + ∣a±2 (q )∣2 = 1. Under these conditions, after
some algebra we obtain

ω(q)

max
ω+ = ω+ (q = 0)
ω (q
)= min
ω+
(q ) ω+ = ω+ (q = π/a)
max
ω− = ω− (q = π/a)
(q )
ω−
)=
ω (q

q
−π/a 0 +π/a

Figure 3.5. Dispersion relations ω = ω±(q ) for the longitudinal vibrations of a diatomic chain in the reduced
zone scheme. The ion masses are M1 and M2 and the effective spring constants are γ (L) and ξ (L) . The ω±min,max
values are calculated by equation (3.15).

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 Solid State Physics

a1+(q ) A(q ) a1−(q )

= − = − , (3.16)
a+2 (q ) ∣A(q )∣ a−2 (q )
where A(q ) = [γ (L) + ξ (L)exp(iqa )] exp[iq(Rl,1 − Rl,2 )]. This result indicates that in
the limit of a vanishingly small wavevector the two ions of the unit cell always oscillate
in phase or in phase opposition in an acoustic or an optic vibrational mode,
respectively.
More in general, the relative ionic motions are affected by the q-vector through a
simple scheme: in vibrations of any kind (both acoustic and optical), nearest
neighbouring unit cells move in-phase or in phase opposition for zone-centre and
zone-boundary modes, respectively; on the other hand, in vibrations with any finite
q-vector, ions within the same unit cell vibrate in-phase or in phase opposition for
acoustic or optical modes, respectively. In figure 3.6 a graphical rendering of the
displacement patters for zone-centre and zone-boundary modes is reported.

Figure 3.6. Zone-centre and zone-boundary displacement patters of longitudinal acoustic and optical
vibrations in a diatomic linear chain. Blue arrows indicate the direction and the orientation of the ionic
displacement vector (not its amplitude).

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 Solid State Physics

We can now understand why optical modes are so named: in ionic crystals the
two ions have opposite charge and, therefore, long wavelength (zone-centre) optical
modes generate fluctuating electric dipoles which can interact with electromagnetic
radiation. Optical vibrational modes in fact dictate the overall optical behaviour of
an ionic crystal.
We conclude the discussion of the vibrational modes in a diatomic chain by
addressing its transverse oscillations. In analogy to the case of the monoatomic
chain, we say that for each q-vector there are two degenerate transverse dispersions,
both for the optical and the acoustic branch. This means that in a diatomic linear
chain, there are in total six dispersion relations: two transverse acoustic (TA1 and
TA2), one longitudinal acoustic (LA), two transverse optical (TO1 and TO2), and one
longitudinal optical (LO).

3.3 Dynamics of three-dimensional crystals

The dynamical properties of a three-dimensional solid with arbitrary crystalline
structure or basis configuration can only be described by means of a heavy
formalism [5, 6, 8], which somewhat hides the underlying physics. This is the
pedagogical reason why we have preliminarily treated the model system correspond-
ing to a linear chain: we will extensively make use of concepts developed in that
framework. As for the formal procedures, we will instead follow the same line of
action adopted in section 3.1. In particular, we will assume that a suitable force field
describing the interactions among ions is available (see appendix D); once for all,
therefore, the force constants defined in equation (3.2) are given as known. More
important, however, is the fact that we will rely on the harmonic approximation.
Before starting to develop our theory, we preliminarily remark that in a three-
dimensional crystal containing Natom atoms in its unit cell, we have 3Natom ionic
degrees of freedom (per unit cell) and, therefore, an equal number of branches in the
vibrational dispersion relations; among them we will find 3 acoustic and 3(Natom − 1)
optical branches.
In the harmonic approximation, the equations of motion are written as9
Mbuï (lb) = − ∑ Uij (lb , l ′b′)uj (l ′b′), (3.17)
jl ′ b ′

for which we guess solutions in the form

ai (b∣q) i q·Rl −iωt
ui (lb) = e e , (3.18)
Mb
where Mb is the mass of the bth ion in the basis10, q is the wavevector of the
vibrational wave and ai (b∣q) describes the amplitude of the corresponding ionic

9
Since in harmonic approximation derivatives appear just up to the second order, we indicate Cartesian
components by the two labels i, and j, instead of using the heavier notation of section 3.1.
10
Of course, in a perfect crystal the ion mass does not depend on the cell index l.

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 Solid State Physics

oscillations. By inserting this guessed displacement into the equations of motion we

get

∑ Dij(bb′∣q)aj (b′∣q) = ω 2ai (b∣q), (3.19)

jb ′

where the quantities

1
Dij(bb′∣q) = ∑ Uij(lb, l ′b′)e−iq·(R −R ),
l l′
(3.20)
MbMb ′ l′

define the dynamical matrix of the crystal. It is important to remark that in this
equation just a single summation of the cell index l ′ appears since the force constants
Uij (lb, l ′b′) of an ideal crystal depend on the pair (l , l ′) just through their difference11.
This also reflects the choice of the guessed solution for the ionic displacements in the
form of a Bloch wave.
Equation (3.20) is the three-dimensional counterpart of equation (3.13) and,
similarly to the linear chain case, the 3Natom × 3Natom dynamical matrix is once again
found to be Hermitean: Dij(bb′∣q) = D *ji(b′b∣q). Accordingly, the lattice dynamics
problem of a three-dimensional crystal is reduced to the diagonalisation of the
dynamical matrix written in equation (3.19). Nowadays this task is solved numeri-
cally, once the matrix has been built using either an empirical force field or by
performing a full quantum mechanical calculation of U (R) and its second-order
derivatives.
Let us now discuss some model cases corresponding to metallic, ionic, and
covalent crystals. We will follow the standard convention to represent the vibra-
tional dispersion relations by choosing q along the borders of the irreducible part of
the 1BZ.
In figure 3.7(left) we report the dispersion relations of copper, a prototypical fcc
metal. Since Cu is a Bravais crystal (one atom in its unit cell), just three dispersion
branches are found. As predicted by the elementary theory developed in section 3.2,
in the limit of a vanishingly small wavevector the dispersion relations are linear: this
allows one to predict the speed of longitudinal and transverse sound. LA vibrations
have in general the higher frequency, although for some q -values along the Σ
direction the TA2 branch overcomes the LA one. Interestingly enough, along the Δ
and Λ directions the two transverse branches are degenerated, as we commented in
the simple case of a linear chain. However, along the Σ direction such a degeneracy is
removed because of a reduced symmetry. This is a general feature found in other
crystals as well.
Another case of metallic system is shown in figure 3.7(right) where the dispersions
of lead, another fcc crystal structure, are reported. While the main characteristics of
the vibrational spectrum are similar to the previous case, we note a lowering of both
the longitudinal and the transverse phonon frequency occurring at X point of the

11
This is related to the fact that the total classical energy of a harmonic ideal crystal is invariant upon rigid
body translations and rotations.

3-13
 Solid State Physics

Figure 3.7. Vibrational dispersion relations of copper (left) and lead (right). The calculations have been
performed by using the QUANTUM ESPRESSO integrated suite of Open-Source computer codes for
materials modelling; see website https://www.quantum-espresso.org (by courtesy of Aleandro Antidormi).

Brillouin zone. This feature is named Kohn anomaly and it is the signature of a
strong electron–phonon interaction effect: in fact, its full explanation requires one to
develop a theory beyond the adiabatic approximation. A Kohn anomaly occurs
whenever it is observed that the dielectric screening12 of the Coulomb interactions
among bare nuclei abruptly changes: this affects the ionic motion or, equivalently,
their vibrational frequencies. The change is theoretically explained by the Lindhard
theory for the dielectric function of a free electron gas [9–11] which is found to
diverge for certain wavevectors: exactly those ones at which the anomaly is found in
the dispersion relations. The modes with lowered frequency are named soft modes.
Sometimes the Kohn anomaly can be so huge that the vibrational frequency is
lowered to zero: in this case a static distortion of the crystal lattice is generated.
We now move to consider the case of two covalent solids, namely silicon and
gallium arsenide, shown in figure 3.8. They both are fcc crystals with a two-atom
basis and a diamond-like or zincblende-like structure, respectively. Because of that,
both acoustic and optical dispersions are found, summing up to six branches, as
shown along the low-symmetry Σ direction (while along the Δ and Λ ones vibrations
with transverse polarisation are degenerate). Once again, the acoustic branches are
correctly linear as q → 0. A very meaningful difference between Si and GaAs is
found in the zone-centre optical modes: while transverse and longitudinal vibrations
are degenerate in Si, they are not in GaAs. In this latter case one observes the so
called LO–TO zone-centre splitting which is basically due to the fact that the two
atoms in the unit cell are not the same and, therefore, the crystal is partly ionic. In
this situation, an electric dipole moment occurs when the two ions are displaced
from their equilibrium positions; in other words, the ionic motion causes an electric
polarisation, giving rise to a Coulomb field. On its part, the field affects the motion
of the ions in two ways: (i) it acts as an additional external force (thus modifying the
force constants entering in the dynamical matrix); and (ii) it polarises the ions, thus
enhancing their dipole moments. The microscopic theory needed to deal with these

12
The screening is provided by the system of the valence electrons.

3-14
 Solid State Physics

Figure 3.8. Vibrational dispersion relations of silicon (left) and gallium arsenide (right). The calculations have
been performed by using the QUANTUM ESPRESSO integrated suite of open-source computer codes for
materials modelling; see website https://www.quantum-espresso.org (by courtesy of Aleandro Antidormi).

subtle phenomena [6, 8] goes far beyond the level of treatment we are developing.
However, by following a more phenomenological approach, it possible to prove that
ϵ0
ω LO(q = 0) = ω TO(q = 0), (3.21)
ϵ∞
a very general result referred to as the Lyddane–Sachs–Teller relation (LST relation),
which is derived in the next section. In this equation ϵ0 and ϵ∞ are the static and high-
frequency13 dielectric constant, respectively: they rule over the polarisation phe-
nomena mentioned above. Since ϵ0 > ϵ∞, we have ωLO(q = 0) > ω TO(q = 0). In
silicon this effect is not observed since it is an elemental solid and, therefore, no first-
order dipole moments are generated by optical modes.
For sake of completeness, we finally consider the case of a pure ionic crystal like
KBr whose dispersions are shown in figure 3.9. The main features previously
discussed are found even in this case, including the expected large LO–TO zone-
centre splitting: the fingerprint of a strong ionic character.
In concluding this section we observe that if we consider the whole crystal, instead
of just its unit cell, the total number of vibrational degrees of freedom must be
appropriately counted. If the crystal is made of Ncell unit cells, each containing Natom
atoms, in total we have Ncell × 3Natom vibrational modes. We describe the overall
dynamics as the superposition of Ncell × 3Natom non-interacting harmonic vibrational
modes labelled by a branch index s and a q wavevector. The branch index runs over
the full set of TA, TO, LA and LO modes, while q takes those values within the 1BZ
which are allowed by the periodic boundary conditions.

3.4 The physical origin of the LO–TO splitting

The derivation of the LST relation anticipated in equation (3.21) is rigorously framed
only within the theory of the dielectric properties of crystalline solids [9, 10], indeed an

13
High-frequency here means a frequency of the order of optical vibrational frequencies or beyond.

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 Solid State Physics

Figure 3.9. Vibrational dispersion relations of KBr. The calculations have been performed by using the
QUANTUM ESPRESSO integrated suite of open-source computer codes for materials modelling; see website
https://www.quantum-espresso.org (by courtesy of Aleandro Antidormi).

advanced topic of solid state physics. Good for us, it is possible to elaborate a
phenomenological model which, although derived under some important simplifying
assumptions, nevertheless leads to a result of general validity. More specifically, we are
going to consider a dielectric ionic crystal containing just two atoms in its unit cell.
Let the dielectric crystal be subject to the action of an external macroscopic
electric field E. Because of polarisation phenomena, the local electric field Eloc found
at any position r within the crystal differs from the applied one: the theory of the
dielectric properties of crystalline solids displays exactly here. We are not developing
this calculation; rather, we assume that the local field is known. The electrostatic
action on the two ions within the unit cell causes their displacements, but since such
a perturbation occurs on a length scale much longer that the typical interatomic
distances, we can assume that equally charged ions move as a whole. Accordingly, in
harmonic approximation we can guess the ionic equations of motion in the form
⎧ m+ü + = −K (u + − u −) + e E loc
⎨ (3.22)
⎩ m−ü − = +K (u + − u −) − e E loc ,

where for sake of simplicity we have assumed just one force constant K for any
interaction and indicated by m± and u ± respectively the mass and the displacement
of the positive (+) and negative (−) ion. By further setting w = (u + − u−) and
1/m = 1/m+ + 1/m− we derive a forced oscillator equation
e K
ẅ = E loc − w, (3.23)
m m

3-16
 Solid State Physics

which describes the relative motion within the unit cell. If we assume a harmonic
time dependence of the electric field E loc = E(0) loc exp( −iωt ) we calculate
w(t ) = w0 exp( −iωt ) with
e 1
w0 = 2
E(0)
loc , (3.24)
m ω0 − ω2
with ω0 = K /m . Since the oppositely charged ions are unequally displaced, an
induced displacement electric dipole moment p(t ) = e w(t ) = p0 exp( −iωt ) is gener-
ated within the unit cell with
e2
p0 = e w0 = E(0)
loc , (3.25)
m(ω02 − ω2 )
and we can accordingly define
e2
αdispl = , (3.26)
m(ω 02 − ω 2 )
as its displacement polarisability.
So far we have implicitly assumed a sort of rigid ion approximation, that is, we
have neglected atomic polarisation phenomena. Atomic physics in fact provides us
with the fundamental notion that the atomic electronic clouds can be distorted [12],
that is, atoms are polarisable. By assuming that ionic polarisabilities α± are known,
we can refine the above model by introducing the ultimate expression for the unit cell
polarisability αtot as
αtot = αdispl + α+ + α−. (3.27)
This expression can be safely used in the Clausius–Mossotti relation of macroscopic
electrostatics [13, 14]

ϵr(ω) − 1 1 1 ⎡ e2 ⎤
= αtot = ⎢ 2 2
+ α+ + α−⎥ , (3.28)
ϵr(ω) + 2 3ϵ0 3ϵ0 ⎣ m(ω0 − ω ) ⎦

and is valid for ionic crystals. In this equation ϵr(ω ) is the frequency-dependent
relative dielectric constant (or permittivity) of the material. It is easy to calculate the
limiting behaviour of equation (3.28) at very low frequencies ω ≪ ω0, a condition
where we can replace ϵr(ω ) with the static dielectric constant ϵr(0) and write
ϵr(0) − 1 1 ⎡ e2 ⎤
= ⎢ + α+ + α −⎥ . (3.29)
ϵr(0) + 2 3ϵ0 ⎣ m(ω02 − ω2 ) ⎦
Similarly, we can evaluate the high-frequency ω → ∞ behaviour14 for which we can
replace ϵr(ω ) with the ϵr(∞) and write

14
In this discussion an ‘infinite’ frequency means a frequency much larger than the lattice vibrational ones, but
still lower that the typical frequencies associated with electronic transitions.

3-17
 Solid State Physics

ϵr(∞) − 1 1
= (α+ + α−), (3.30)
ϵr(∞) + 2 3ϵ0
where ϵr(∞) is of course the dielectric constant of the material at optical frequencies.
By combining equations (3.29) and (3.30) we get
ϵr(∞) − ϵr(0)
ϵr(ω) = ϵr(∞) + , (3.31)
ω2 / ω¯ 2 − 1
where ω̄ is at present just a shortcut for
ϵr(∞) + 2 2
ω¯ 2 = ω0 . (3.32)
ϵr(0) + 2

Let us now introduce the lattice dynamics by considering a long-wavelength

optical mode: ions in the same unit cell move in phase opposition, thus giving rise to
an electric dipole moment. This in turn implies the onset of a non-zero polarisation
P within the crystal. By adding the assumption that the material is homogeneous
and isotropic, the resulting electrostatics can be summarised in the equation
D = ϵ0E + P = ϵ0E + ϵ0χ E = ϵ0(1 + χ )E = ϵ0ϵr E, (3.33)
where D is the electric displacement field and χ the electric susceptibility of the
material. We now add three more simplifying assumptions, namely: (i) there are no
free charges within the crystal and, therefore, ∇ · D = 0; (ii) we neglect retardation
effects or, equivalently we set the electrostatic approximation ∇ × E = 0; and (iii) we
impose the same space variation C = C0 exp(i k · r), for the three relevant vectors
C = E, D or P . These assumptions lead to the noteworthy conclusions that:
whenever k · D0 = 0 then either D0 = 0 or D0 , E0, P0⊥k (condition 1); on the
other hand, whenever k × E0 = 0 then either E0 = 0 or D0 , E0, P0 k (condition 2).
These constraints have different, but equally meaningful, consequences for longi-
tudinal and transverse modes. In a longitudinal mode the polarisation vector is
parallel to k and, therefore, condition 1 imposes that the electric displacement
field must vanish. Since E ≠ 0, equation (3.33) dictates that ϵr = 0. On the other
hand, in a transverse mode the polarisation vector is normal to k and, therefore,
condition 2 imposes that the electric field must vanish: this time equation (3.33)
imposes that ϵr = ∞. This situation is only found if ω = ω¯ in equation (3.31).
Therefore, we can identify ω̄ with the zone-center frequency of a transverse optical
mode and accordingly set ω¯ = ω TO. Finally, the condition ϵr = 0 valid for a
longitudinal optical mode leads, once again through equation (3.31), to the LST
relation anticipated in equation (3.21).
Interestingly enough, this analysis also provides a valuable model for the
frequency dependent dielectric constant of a diatomic ionic crystal, as reported in
figure 3.10, left. It is interesting to observe that for ω TO < ω < ωLO the dielectric
constant is negative and, therefore, no electromagnetic mode can propagate within the
crystal in this frequency range, as indeed found experimentally. The dispersion
relation ω = kc/ ϵ of electromagnetic modes outside such a forbidden gap is

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 Solid State Physics

ω
r (ω) kc/ r (∞)

e
lik
n-
oto
ph
kc/ r (0)

r (0) ωLO phonon-like

r (∞)
polariton

ωTO
ωTO ωLO ω phonon-like
e
n -lik
oto
ph

k
Figure 3.10. Left: frequency-dependent dielectric constant of a diatomic ionic crystal. Right: dispersion
relation for transverse electromagnetic modes in a diatomic ionic crystal.

reported in figure 3.10, right. It must be understood that the plot only refers to the
long wavelength region (that is, the range of photon k-values reported here is very
small as compared with the typical phonon wavevectors). The picture suggests that
the coupling between the transverse electromagnetic wave and the vibrational
optical modes of the crystal has determined a dramatic variation of the linear ω
versus k dependence observed in free space: the character smoothly changes from
phonon-like to photon-like or vice versa according to the actual branch considered.
In the region where the character changeover occurs, the excitation is better defined
as a polariton, a sort of combination of a photon and a phonon.

3.5 Quantum theory of harmonic crystals

Moving to a quantum description, as simple as it may appear, represents a major
conceptual step forward in our search for a truly fundamental description of lattice
dynamics. To appreciate its relevance, we anticipate a result more extensively
discussed in the next chapter. The specific heat of a crystal described as an assembly
of classical harmonic oscillators is calculated to be independent of temperature
(Dulong–Petit law). Contrary to this prediction, experimental measurements pro-
vide evidence that the specific heat becomes vanishingly small as T → 0, thus
proving that it is in fact temperature-dependent. Only a full quantum treatment is
able to reconcile theoretical predictions to measurements.
Based on the theory developed in the previous section, we will agree to describe each
classical (sq) vibrational mode as a quantum one-dimensional harmonic oscillator [1–3]
whose energy is restricted to the values (ns q + 1/2)ℏωs (q) where nsq = 0, 1, 2, … is the
vibrational quantum number and ωs (q) is obtained by diagonalising the dynamical
matrix. Since the vibrational energy levels are equally spaced, we can look at the state
with energy (ns q + 1/2)ℏωs (q) as a single nsq th excited state or, equivalently, as the
state obtained by adding nsq identical energy quanta ℏωs(q). We will adopt this second
approach since it is especially effective in describing the dynamical and thermal

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 Solid State Physics

characteristics of a crystal lattice through the properties of a gas of pseudo-particles,

hereafter named phonons. This choice introduces a corpuscular description of lattice
dynamics, where phonons are the energy quanta of the ionic displacement field15.
Let us now consider in some detail the physics of phonons. First of all, we clarify
that phonons, similarly to photons, are named pseudo-particles since they do not have
a mass. Furthermore, in addition to an energy ℏωs(q), they also carry a momentum
ℏq . Since such a momentum is exact, the uncertainty principle imposes that the
phonon position is totally undetermined and, therefore, they must be understood as
delocalised pseudo-particles. This is consistent with the fact that their corresponding
non-interacting classical vibrational modes extend throughout the system16.
The phonon momentum must be treated with some care. We know that, because
of translational invariance, each classical vibrational mode remains unaffected by
replacing q with q + G , where G is a reciprocal lattice vector. This implies that ℏq
does not actually represent the kinematic momentum of the phonon since ℏq and
ℏ(q + G) are equivalently good choices for it. For this reason the phonon
momentum is better referred to as its crystal momentum. An elastic phonon
scattering event will conserve the crystal momentum unless an ℏG term, as discussed
in the next section.
Similarly to the photon case, the total number of phonons is not conserved: they can
be created or annihilated by increasing/decreasing temperature or by phonon–phonon
scattering events. This urges us to understand their statistics. Let us consider a gas of
( )
phonons with energies ns q + 12 ℏωs(q) in thermal equilibrium at temperature T. The
average energy 〈U 〉 of the gas is calculated according to statistical physics [15] as
⎡ (nsq + 1/2)ℏωs(q) ⎤⎥
1
〈U 〉 =
Z
∑(nsq + 1/2)ℏωs(q) exp⎢⎢− kBT ⎥⎦
, (3.34)
sq ⎣
where
⎡ (nsq + 1/2)ℏωs(q) ⎤⎥
Z= ∑ exp⎢− , (3.35)
sq
⎢⎣ kBT ⎥⎦

is the partition function17 for the canonical (i.e. constant-temperature) ensemble.

This energy is the quantum counterpart of equation (3.4). After some algebra18 the
series are summed obtaining the quantum vibrational energy of a harmonic crystal

15
It could be helpful to consider the analogy with the case of classical electromagnetic waves described by the
Maxwell equations: the first step toward the quantisation of the electromagnetic field is to describe it as a flux
of radiation quanta, referred to as photons.
16
We remark that we are considering an ideal and harmonic crystal. The situation looks very different when
considering the role of anharmonicity and crystal defects.
17
We adopt a widely used abuse of notation. For quantum systems, as the phonon gas indeed is, the partition
function is more rigorously defined as the trace of the Boltzmann factor exp(−Ĥ /kBT ), where Ĥ is the
Hamiltonian operator (in the present case: the total energy operator of an assembly of quantum harmonic
oscillators). By assuming that we are working with a basis set for which the matrix representation of Ĥ is
diagonal, the above notation for Z is correct.
18
We remember that ∑n exp(−nx ) = [1 − exp(−x )]−1 and ∑n n exp( −nx ) = exp(x )[exp(x ) − 1]−2 .

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 Solid State Physics

quantum
〈U 〉 = Uharmonic = ∑[nBE(s q, T ) + 1/2] ℏωs(q), (3.36)
sq

where
1
n BE(s q , T ) = ,
⎡ ℏωs(q) ⎤ (3.37)
exp ⎢ ⎥−1
⎣ kBT ⎦

represents the average number of phonons with energy ℏωs(q) (also referred to as the
phonon population) found in the crystal at temperature T. As indicated, this average
number is provided by the Bose–Einstein statistics, proving that phonons are bosons
as consistent with the fact that they are spinless pseudo-particles (see appendix E).
In summary, in the harmonic approximation a crystal is quantum mechanically
described as a gas of non-interacting phonons of different character (s = TA, LA, TO,
LO) and wavevector q . Their number varies upon temperature according to the
Bose–Einstein statistics given in equation (3.37). Interesting enough, in the limit
quantum
T → 0 we find Uharmonic = 1/2 ∑s q ℏωs (q) representing the crystalline zero-point
energy: in the quantum description, therefore, the vibrational energy of a crystal
is never zero. Finally, we remark that the dispersion relations ω = ωs(q) discussed in
section 3.3 are usually referred to as the phonon dispersion relations.

3.6 Experimental measurement of phonon dispersion relations

The experimental determination of the ω = ωs(q) dispersion relations over the entire
1BZ needs a probe fulfilling two conditions: (i) its wavelength must be comparable
with the typical interatomic distances in the crystal structure and (ii) its energy must
be of the same order of typical phonon quanta ℏωs(q), which range mostly in the
interval [1, 102] meV. Optical probes are unsuitable: x-rays have the right wave-
length, but a too high energy of the order O(10 4eV); other kinds of photons, instead,
can only explore the q ∼ 0 region of the Brillouin zone, i.e. they can only detect
(some) zone-centre phonons. A probe consisting in a flux of electrons is also
impractical for a twofold reason: (i) their surface scattering is very strong and,
therefore, they are unable to probe the bulk region of the crystal; (ii) multiple
scattering is likely to occur in the case of electrons and this makes the analysis of the
experiment a very challenging task. In contrast, both requirements of
suitable wavelength and energy are guaranteed by a flux of thermal neutrons which
have typical wavelengths of the order of just a few Å and energies in between a few
and a few tens of meV. Accordingly, neutron spectroscopy is the most powerful
technique for measuring the phonon dispersion relations [16, 17].
The typical experimental setup for neutron spectroscopy is shown in figure 3.11.
A continuous flux of neutrons is emitted by a source, typically a fission reactor; the
beam is thermalised by collisions within the reactor moderator and, therefore,
neutrons are produced with a Maxwell–Boltzmann distribution of velocities
corresponding to the room temperature (for this reason they are referred to as
‘thermal neutrons’). The thermalised beam emerging from the source is Bragg

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 Solid State Physics

Figure 3.11. Schematic representation of a triple-axis neutron spectrometer.

reflected by a single-crystal monochromator which selects neutrons of same

momentum ℏpin and energy Ein = ℏ2pin2 /2m n where mn is the neutron mass. The
resulting monochromatic beam is now focussed by a collimator and then it falls on a
sample with known crystalline orientation. The momentum vector ℏpout of the
scattered neutrons is defined by letting the beam pass through a second collimator
and eventually arrive at the analyser. Here the magnitude pout and the corresponding
energy Eout = ℏ2pout
2
/2m n of the detected neutrons are determined by measuring the
Bragg reflection angles from the planes of the crystalline analyser.
Two possible physical mechanisms occur in a scattering event. First of all,
neutrons are scattered by very strong nuclear forces: basically, the nuclei of the
atoms in the sample scatter neutrons as they were hard spheres. In addition,
provided that such atoms have a non-zero magnetic moment19, neutrons are
scattered by magnetic interactions between their intrinsic magnetic dipole moment
and the atomic magnetic moments. While this latter interaction is helpful to
investigate magnetic ordering in solids, we will take into consideration only the
first mechanism which is active in any (even non-magnetic) material.
To the aim of investigating a scattering event, we will keep things easy by
considering just a single vibrational mode of wavevector q and frequency ω in the
sample hit by the monochromatic beam of neutrons. We will also use a simplified
version of the notation introduced in section 3.2 according to which the position R(t )
of an atom sitting at rest in the equilibrium site R eq is written as
R(t ) = R eq + u cos(q · R eq − ωt ), (3.38)
where u contains information on both the amplitude and the polarisation of the
selected vibrational mode. In addition, to take profit from our previous knowledge it
is useful to develop our arguments in analogy with the discussion of x-ray scattering
(see section 2.4.1). Therefore, we will at first consider the neutron beam as an
incoming wave whose quanta (neutrons) have energy ℏΩ in ; next, we will interpret the
scattering outcome using the phonon corpuscular language.
Similarly to equation (2.9) the amplitude of the scattered wave is written as

19
This situation is found in atoms with an electronic configuration giving a net magnetic moment or with an
uncompensated electron spin condition [12].

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 Solid State Physics

A out = ∑ bR exp[−i(P · R + Ω int )], (3.39)

R

where b R is the so called scattering length20. In this case the scattering vector is
defined as P = (pout − pin ). By inserting equation (3.38) into equation (3.39), after
some algebra we get

A out = ∑ b R exp[ −i (P · R eq + Ω int ]

R eq

i
− (P · R eq)∑ b R exp{ −i [(P + q) · R eq + (Ω in − ω)t ]}
2 R (3.40)
eq

i
− (P · R eq)∑ b R exp{ −i [(P − q) · R eq + (Ω in + ω)t ]}
2 R eq

2
+ O(u ),

where, by virtue of the fact that we are in harmonic approximation, the underlying
assumption is that atomic displacements are small and, therefore, the development
has been safely truncated at the linear terms in u . The maximum conditions for A out
are found in three occurrences: the first term imposes that the scattering vector is
equal to a translational vector of the reciprocal lattice of the sample, that is: P = G;
in turn, the second and third terms provide the conditions P ± q = G . In the first
case the maximum peak of the scattered wave is found at the frequency Ω out = Ω in
and, therefore, the scattered neutrons have the same energy of the incident ones: this
is the condition of elastic scattering. In the two remaining cases the scattered
neutrons have instead a different energy since the maximum peak is, respectively,
found at frequency Ωout = Ω in ∓ ω , corresponding to the condition of inelastic
scattering. The analysis is completed by also considering the momentum exchange
described in equation (3.40): by passing to the corpuscular picture this leads us to
identify two unalike scattering conditions
⎧ ℏΩout = ℏΩ in − ℏω ⎧ ℏΩout = ℏΩ in + ℏω
⎨ or ⎨ (3.41)
⎩ ℏpout = ℏpin − ℏq + ℏG ⎩ ℏpout = ℏpin + ℏq + ℏG ,

which are easily interpreted as conservation laws. A neutron with energy ℏΩ in can be
inelastically scattered by emitting or absorbing a phonon of energy ℏω. The
corresponding conservation of the momentum is more subtle: in fact, inelastic
scattering events conserve momentum to within a reciprocal lattice vector21. This

20
In the theory of neutron scattering it is customary to write the total cross section for a scattering event as
4πb R2 . It plays a role similar to the atomic form factor.
21
In our treatment this important result has been obtained phenomenologically. It is, however, possible to
formally prove it by invoking the general result of quantum theory which makes a correspondence between the
symmetry properties of the Hamiltonian operator and some conservation laws of physical observables. More
specifically, the crystal Hamiltonian is invariant upon discrete translations and this implies that momenta are
conserved to within a translational vector of the reciprocal lattice [9, 10].

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 Solid State Physics

result experimentally confirms what we anticipated in section 3.5 when the phonon
momentum was better referred to as ‘crystal momentum’: while in an inelastic
scattering event the crystalline sample does absorb a momentum ℏP, it is just a
matter of convention to assign the part ℏq to a phonon and the part ℏG to the crystal
as a whole; equivalently, we could totally assign it to the phonon with momentum
ℏ(q + G). There is no difference in physics, since the two phonons with wavevector q
or (q + G) represent exactly the same vibrational mode.
This state of affairs suggests that the neutron scattering spectrum consists in
peaks, each corresponding to a possible scattering event. In a real experiment it is
possible to vary pin and/or pout still keeping the scattering vector P = pout − pin at the
exact value of the phonon wavevector q whose energy we want to measure. This
situation fulfils the conservation laws (3.41) and, therefore, each peak appearing in
the spectrum corresponds to a different existing phonon of wavevector q . By
repeating the measurement for a different scattering vector we can move along
the edges of the irreducible part of the 1BZ (see section 2.4.4) and, therefore, we
obtain the complete phonon dispersion relations.

3.7 The vibrational density of states

In many frameworks it is necessary to explicitly take into account the distribution of
phonon frequencies over the full vibrational spectrum. To this aim it is useful to define
the quantity G(ω ), hereafter referred to as the vibrational density of states (vDOS),
defined such that G (ω )dω represents the number of phonons (that is: vibrational
modes) with frequency in the interval [ω, ω + dω ]. In order to calculate a general
expression for G(ω ) we will proceed by two steps of increasing generality.
Let us preliminarily consider the model case of a monoatomic sc crystal with first
nearest neighbours distance a and subject to Born–von Karman periodic boundary
conditions. For convenience we choose the crystal in the form of a cube with edge
length L and assume that there is only one dispersion relation: basically, this is the
three-dimensional counterpart of the monoatomic linear chain. Accordingly, by
generalising equation (3.9) we understand that the allowed phonon wavevectors q
have Cartesian components qi = 2πξi /L with L = Na , while i = x , y, z and
ξi = 0, 1, 2, …(N − 1) if N is the total number of atoms in the crystal. Their
number density in the reciprocal space is straightforwardly calculated as
L3 /(2π )3 = V /(2π )3 where V = L3 is the volume of the sample. The number of
allowed wavevectors corresponding to vibrational modes with a frequency in
between ω and ω + dω is given by the product between their number density and
the infinitesimal volume 4πq 2dq of the reciprocal space. Since we have just one
phonon branch, this number equals the number of vibrational frequencies in the
selected interval. We accordingly write
V
G ( ω ) dω = 4πq 2dq . (3.42)
(2π )3
We remark that 4πq 2dq is the volume of the spherical shell precisely bounded by the
two constant-frequency surfaces corresponding to ω and ω + dω, respectively.

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 Solid State Physics

Therefore, equation (3.42) is strictly true only if we assume that the phonon
properties depend on their magnitude but not their direction, indeed a simplification
we have implicitly used.
This constraint must be removed if we seek for a general theory. To this aim we
consider the more general case of a three-dimensional crystal and we focus at first
just on its sth dispersion ω = ωs(q): in this case, the vibrational frequencies are not
only determined by the magnitude of the wavevector q , but also by its orientation
within the 1BZ. This calls for some care since we are forced to use vector formalism.
Similarly to the simple model previously discussed, the number of allowed wave-
vectors in the sth branch for which the frequency is in between ωs and ωs + dω is
once again given by the product between their number density and an appropriate
infinitesimal volume d q of the reciprocal space. Since we are considering just the sth
phonon branch, this number equals the number of its allowed modes in the selected
frequency interval. The vDOS we are looking for is
V
Gs(ω)dω = d q, (3.43)
(2π )3
where we made use the same number density V /(2π )3 as before; the subscript s
reminds us that we are only considering this dispersion branch. The difficult task is
to calculate d q which now corresponds to the generic volume of the reciprocal space
between the two surfaces with ωs(q) = constant and ωs(q) + dω = constant. We
remark that their shape is not known and it can be in principle very complex; in
other words, the calculation of d q cannot be traced back to that of a simple known
volume: we need a more general formalism.
If we consider an element dSωs on the inner surface ωs (q) = constant, we can name
dq⊥(q) its distance to the outer surface ωs (q) + dω = constant; as shown in figure 3.12,
such a distance is q -dependent since it varies according to the selected surface
element dSωs . This procedure allows us to write the volume element d q as

dq = ∫S ωs
dSωs dq⊥(q), (3.44)

where it is understood that the integral is extended to all points of the surface Sωs
where we know that ωs(q) = constant. The infinitesimal difference in frequency dω
between the two surfaces can be written as
dω = ∇qω(q) dq⊥(q), (3.45)

ωs (q) + dω = constant

dq⊥ (q) dq
dSωs

ωs (q) = constant

Figure 3.12. The red and blue lines represent two constant-frequency surfaces embedding the reciprocal space
volume d q . The two surfaces are separated by the distance dq⊥(q) which is shown for the element dSωs on the
inner surface.

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 Solid State Physics

where ∇qωs(q) is the gradient of ωs(q). By using this result in equation (3.44) we get
for the volume element in the reciprocal space the following general expression
dSωs
d q = dω ∫S ωs ∇qω(q)
, (3.46)

which is used in equation (3.43) to obtain the vDOS of the sth vibrational branch
V dSωs
Gs(ω) =
(2π )3
∫S ωs ∇qωs(q)
, (3.47)

where it is understood that the integral is extended to all points of the surface Sωs
defined by ωs(q) = ω. This result is used to calculate the total vibrational density of
states simply by summing over all branches
V dSωs
G (ω ) = ∑ Gs(ω) = ∑∫ , (3.48)
s
(2π )3 s
Sωs ∇qωs(q)

where q ∈ 1BZ .
It is important to remark an intriguing feature of the vDOS as formalised in
equation (3.48): the denominator appearing in the integrated function is nothing else
than the magnitude of the group velocity vg,s(q) = ∇qωs(q) of the (s, q) phonon.
This allows us to predict that whenever such velocity is zero, a singularity must
appear in the vDOS: they are referred to as van Hove singularities. This situation
occurs whenever a phonon branch flattens, as typically found for zone-centre optical
modes and zone-boundary acoustic ones. The van Hove singularities are clearly
shown in figure 3.13, where we report the calculated vDOS for metallic Cu and
covalent Si. By comparing this figure with the corresponding phonon dispersion
relations reported in figures 3.7 and 3.8, respectively, it is easy to spot the
singularities.

Figure 3.13. The vibrational density of states (vDOS) of Cu (left) and Si (right). The calculations have been
performed by using the QUANTUM ESPRESSO integrated suite of open-source computer codes for materials
modelling; see website https://www.quantum-espresso.org (by courtesy of Aleandro Antidormi).

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 Solid State Physics

Another interesting characteristics emerging from figure 3.13 (but not easy to see
in equation (3.48)) is that G(ω ) ∼ ω 2 in the small frequency range. This behaviour is
explicitly predicted in the Debye theory of the specific heat, as extensively discussed
in section 4.1.2.
We finally remark that equation (3.48), although providing the most general
theoretical formulation for the vibrational density of states, is not that easy to handle
in practice. However, nowadays solid state theory makes extensive use of computa-
tional methods and, therefore, the ωs(q) frequencies are in most cases determined
numerically. This means that a more practical formulation for the vDOS is the
following
V
G (ω ) = ∑∫ δ(ω − ωs(q))d q , (3.49)
(2π )3 s

where the integral is over the full 1BZ.

The vDOS is also referred to as the phonon density of states, as allowed by the
direct correspondence between the classical and quantum pictures, respectively,
formulated in terms of vibrational modes and phonons.

References
[1] Miller D A B 2008 Quantum Mechanics for Scientists and Engineers (New York: Cambridge
University Press)
[2] Bransden B H and Joachain C J 2000 Quantum Mechanics (Upper Saddle River, NJ:
Prentice Hall)
[3] Sakurai J J and Napolitano J 2011 Modern Quantum Mechanics 2nd edn (Reading, MA:
Addison-Wesley)
[4] Griffiths D J and Schroeter D F 2018 Introduction to Quantum Mechanics 3rd edn
(Cambridge: Cambridge University Press)
[5] Born M and Huang K 1954 Dynamical Theory of Crystal Lattices (Oxford: Oxford
University Press)
[6] Böttger H 1983 Principles of the Theory of Lattice Dynamics (Berlin: Akademie)
[7] Burns G and Glazer M 2013 Space Groups for Solid State Scientists 3rd edn (Waltham, MA:
Academic)
[8] Maradudin A A, Montroll E W, Weiss G H and Ipatova I P 1971 Theory of Lattice
Dynamics in the Harmonic approximation (New York: Academic)
[9] Ashcroft N W and Mermin N D 1976 Solid State Physics (London: Holt-Saunders)
[10] Grosso G and Pastori Parravicini G 2014 Solid State Physics 2nd edn (Oxford: Academic)
[11] Martin R M 2012 Electronic Structure–Basic Theory and Practical Methods (Cambridge:
Cambridge University Press)
[12] Colombo L 2019 Atomic and Molecular Physics: A Primer (Bristol: IOP Publishing)
[13] Feynman R P, Leighton R B and Sands M 1963 The Feynman Lectures on Physics (Reading:
Addison-Wesley)
[14] Panofsky W K H and Phillips M 1990 Classical Electricity and Magnetism (New York:
Dover)

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 Solid State Physics

[15] Glazer M and Wark J 2001 Statistical Mechanics–A Survival Guide (Oxford: Oxford
University Press)
[16] Hook J R and Hall H E 2010 Solid State Physics (Hoboken, NJ: Wiley)
[17] Dove M T 2003 Structure and Dynamics–An Atomic View of Materials (Oxford: Oxford
University Press)

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 IOP Publishing

Solid State Physics

A primer
Luciano Colombo

Chapter 4
Thermal properties

Syllabus—The need of a more fundamental theory of lattice dynamics going beyond

the approximations adopted in the previous chapter is discussed by investigating the
thermal properties of a crystalline solid. The calculation of the heat needed to increase
the temperature of insulating materials is developed at different levels of increasing
complexity, eventually providing the general quantum theory of the heat capacity.
Next, the anharmonic features affecting the ionic oscillations are extensively discussed
by considering at first the thermal expansion phenomenon. A full quantum picture of
anharmonicity is then worked out, based on the phonon language which allows one to
quantitatively describe their scattering mechanisms. A rigorous treatment of thermal
transport in non-metallic crystals is eventually presented, based on the fundamental
Boltzmann transport equation.

4.1 The lattice heat capacity

4.1.1 Historical background
The energy content of a physical system is thermodynamically accounted for by its
internal energy U (see appendix C) whose derivative with respect to temperature
dU
CV = , (4.1)
dT V

is known as the heat capacity at constant volume V: it represents the amount of heat
we need to quasi-statically provide in order to increase the system temperature by
one degree.
The first attempts to derive a microscopic theory for CV in crystalline solids were
developed at the dawn of the XXth century. In many respects, we can consider these
investigations as the beginning of quantum solid state physics [1]. Developing a
microscopic theory was certainly worthy of effort since the classical theory of CV is
contradicted by the experimental evidence. In order to outline this theory, outdated

doi:10.1088/978-0-7503-2265-2ch4 4-1 ª IOP Publishing Ltd 2021

 Solid State Physics

but still valuable for our pedagogical approach to the thermal properties, we
preliminarily remark that there are three main contributions to the heat capacity
of a crystal, respectively, deriving from lattice vibrations, conduction electrons, and
magnetic ordering. In non-magnetic insulators the first one is by far the leading one
and in this chapter we focus just on it1.
Classically the internal energy U of a crystal containing N atoms corresponds to
the vibrational energy of 3N one-dimensional harmonic oscillators, as calculated by
means of the equipartition theorem: if the crystal is in equilibrium at temperature T,
an average energy kBT /2 is attributed to each energy contribution which is quadratic
either in general coordinates or momenta. Therefore, the average energy of each
atomic oscillator is estimated to be 〈u(T )〉 = kBT so that

U = 3N 〈u(T )〉 = 3NkBT = 3RT , (4.2)

where we have hereafter assumed that N = NA (i.e. we have an Avogadro number of

atoms in the crystal), while R = 8.314 J K−1 mol−1 is the universal gas constant. The
corresponding classical prediction for the heat capacity CV = 3R is known as the
Dulong–Petit law. Contrary to this, experimental measurements provide evidence
that CV → 0 for T → 0. More specifically, it is found that CV ∼ T 3 in the range of
vanishingly small temperatures. The measured CV approaches the predicted value
only at very high temperature. In conclusion, the classical theory is unable to justify
the observed CV = CV (T ) trend over the full range of temperatures.
A major step forward to a more fundamental theory was taken in 1907 by A
Einstein who replaced the classical harmonic oscillators by quantum ones further
assuming that all oscillators vibrate at the same frequency ωE , known as the Einstein
frequency. Following the arguments developed in section 3.5, we write the average
energy of the single one-dimensional Einstein oscillator at temperature T as
⎡ 1⎤
〈u E(T )〉 = ⎢n BE(ω E ) + ⎥ℏω E , (4.3)
⎣ 2⎦

where nBE(ω ) is the Bose–Einstein distribution introduced in equation (3.37) and

more formally discussed in appendix E. In figure 4.1 a comparison is shown between
the energy of a classical and a quantum Einstein oscillator as a function of
temperature. We accordingly calculate the heat capacity as

⎛ ℏω E ⎞2 exp(ℏω E / kBT )
CVEinstein(T ) = 3R ⎜ ⎟ , (4.4)
⎝ kBT ⎠ [exp(ℏω E / kBT ) − 1]2

which fulfils the Dulong–Petit law at high temperature, while becoming vanishingly
small at zero temperature. However, it is easy to prove that that
CVEinstein(T ) ∼ exp( −1/T ) as T → 0; in other words, the Einstein model is qualitatively
correct, but it provides a wrong behaviour for the heat capacity at small temperatures.

1
The other contributions to CV will be discussed in the following chapters.

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 Solid State Physics

u(T ) 1
nBE (ωE ) + ωE
2

kB T

1
ωE
2

Figure 4.1. The average energy 〈u(T )〉 of a classical (red dashed line) and a quantum Einstein (blue full line)
oscillator as a function of the temperature T.

ω(q)

qD q

Figure 4.2. Definition of the effective acoustic Debye branch (red) and corresponding Debye wavevector qD .

A more refined model is actually needed; since the theory must necessarily be
quantum, we will refer to vibrational modes as phonons, as explained in section 3.5.

4.1.2 The Debye model for the heat capacity

The Einstein model is correct in treating atomic vibrations as quantum oscillators, but
it fails in attributing the same frequency to all of them: simply, this is inconsistent with
the knowledge of the dispersion relations we developed in chapter 3. We must
therefore introduce in the theory the fundamental notion that atomic oscillators can
vibrate at different frequencies. Within the Debye model this notion is developed in a
simplified way which allows us to carry on a clean analytical calculation of the heat
capacity.
According to Debye, all phonon dispersion relations are effectively described by
only three effective acoustic branches whose extension in wavevector, however,
exceeds the boundary of the 1BZ. This is shown in figure 4.2: the low and high
q-values of the effective branch, respectively, describe an acoustic and an optical

4-3
 Solid State Physics

vibration of the real crystal. Furthermore, since for any direction there are in fact
three possible phonon polarisations, the linearisation of their dispersions must
properly take care to distinguish between one effective longitudinal and two effective
transverse branches with slope vg(L ) and vg(T ), respectively (see section 3.2.1). To this
aim it is useful to introduce the effective speed of sound veff defined as
3 1 2
3
= 3
+ . (4.5)
veff ⎡v ⎤
(L ) ⎡v T )⎤3
(
⎣ g ⎦ ⎣ g ⎦
We can now calculate the density of vibrational states in the Debye model G D(ω ) by
making use of equation (3.42) elaborated for a single branch so that
⎧ ⎫
V 2 1 V ω2 V ⎪ 1 2 ⎪ 2
G D(ω) = 3 q =3 2 3 = ⎨ + ⎬ω , (4.6)
2π 2 dω / dq 2π veff 2π 2 ⎪ ⎡⎣vg(L )⎤⎦3 ⎡⎣v (T )⎤⎦3 ⎪
⎩ g ⎭
where we set ω = veff q consistently with the linearisation procedure; the factor 3
takes into account the three possible polarisations. Interesting enough, we find a
quadratic dependence of the VDOS upon the frequency: this feature is indeed found in
real materials in the acoustic region of the vibrational spectrum, that is, exactly
where the phonon dispersion relations are linear as supposed in the Debye model
(see section 3.7). Since G D(ω )dω is the number of vibrational modes with frequency
in the range [ω, ω + dω ], it is straightforward to impose the following normalisation
condition to a crystal containing N atoms
ωD
∫0 G D(ω) dω = 3N , (4.7)

where ω D, known as the Debye frequency, dictates that no phonons with higher
frequency are found in the crystal. By using the expression given in equation (4.6) for
the vibrational density of states we easily obtain
⎧ ⎫−3
6π 2N 18π 2N ⎪ 1 2 ⎪
ωD = 3 veff = 3 ⎨ + ⎬ , (4.8)
V V ⎪ ⎡⎣vg(L )⎤⎦3 ⎡v (T )⎤3 ⎪
⎣ ⎦
⎩ g ⎭
which in turn defines the maximum Debye wavevector introduced above since the
two quantities are related as ω D = veff qD.
The full quantum expression for the internal energy2 is eventually written as
ωD
U= ∫0 [n BE(ℏω, T ) + 1/2] ℏω G D(ω) dω
9N ℏ ωD
=
ω D3 0
∫
[n BE(ℏω, T ) + 1/2]ω3dω (4.9)
9 9N ℏ ωD
= N ℏω D +
8 ω D3 0
∫
n BE(ℏω, T ) ω3dω,

2
We recall once more that this is just the lattice contribution, the leading one for non-magnetic insulators.

4-4
 Solid State Physics

where nBE(ℏω, T ) is the Bose–Einstein distribution law (see appendix E). This result
is the Debye counterpart of the more general expression given in equation (3.36),
where the sum over the index s labelling the discrete vibrational modes has been
replaced by the integral over a continuum of frequencies: this is certainly valid for a
large enough N. The first term on the right-hand side is the Debye estimate for the
quantum zero-point energy: it is independent of temperature and, therefore, gives no
contribution to the heat capacity; accordingly we calculate
⎛ T ⎞3 TD /T
x 4 exp(x ) ℏω
CVDebye(T ) = 9R ⎜ ⎟
⎝ TD ⎠
∫0 [exp(x ) − 1] 2
dx with x =
kBT
, (4.10)

where has been introduced the Debye temperature TD = ℏω D /kB and, as before, it has
been assumed N = NA . It should be clear from its very definition that the Debye
temperature is an empirical parameter as much as the Debye frequency or wave-
vector: it can be determined by fitting experimental data with equation (4.10). In this
respect, the Debye model we worked out should be more rigorously looked at as a
simple interpolation scheme. Nevertheless, we can attribute to TD a very intuitive
physical meaning by calculating CVDebye(T ) in the limit of a vanishingly small or of a
very high temperature, respectively corresponding to T ≪ TD or T ≫ TD. It is easy to
prove that3
⎧ 12Rπ 4 3
⎪ limT →0 CVDebye(T ) = T
⎨ 5TD3 (4.11)
⎪ Debye
⎩ limT →+∞ CV (T ) = 3R ,

while the trend for intermediate temperatures is reported in figure 4.3. The predicted
behaviour exactly corresponds to the experimental observation that CVexpt(T ) ∼ T 3 at
low temperature, thus making the Debye interpolation scheme superior to the
simpler Einstein model which, as already commented, failed in this temperature
regime predicting CVEinstein(T ) ∼ exp( −1/T ). On the other hand, in the high-temper-
ature limit the Debye formula recovers the classical model and, once again, it is in
perfect agreement with measurements.
An important remark must be added in commenting on figure 4.3 where the
CVEinstein
(T ) is also reported: the Einstein and the Debye models use a different temperature
scale which we have so far defined only in the latter case by introducing TD. In contrast,
the only way to define the notion of temperature in the Einstein model is setting
ℏωE = kBTE , where TE will be hereafter addressed as the Einstein temperature: the
problem is finding its relationship with TD. The sole vibrational frequency defined in
the Einstein model should necessarily be linked to the speed of sound, if we imagine the
sound propagation within the crystal as a sequence of hits between oscillating ions.
Accordingly, the relationship between the effective speed veff defined in equation (4.5)
and the Einstein frequency is

3
We remark that if T ≪ TD then TD /T → +∞ and the integral appearing in equation (4.10) is tabulated to be
+∞ 4
∫0 x exp(x )/[exp(x ) − 1]2 dx = 4π 4 /15.

4-5
 Solid State Physics

Figure 4.3. The lattice heat capacity calculated according to the Debye model (full blue line) for a crystalline
non-magnetic insulator containing N = NA atoms. The same quantity calculated according to the Einstein
model (red dashed line) is shown for comparison. Temperatures are measured in units of the Debye
temperature TD and heat capacities in units of 3R .

ωE
veff = λE , (4.12)
2π
where λE corresponds to the minimum wavelength of the ionic oscillations sustained
by the crystal. For a system in the form of a cube with edge L and containing N
atoms we have λE /2 = L / 3 N and therefore

h N
TE = 3 veff , (4.13)
2kB V

where V = L3 for the adopted hypotheses. It easy to prove that

TE π
= 3 , (4.14)
TD 6

which allows for a direct comparison between the Einstein and the Debye model, as
shown in figure 4.3.
In conclusion, the performances of the Debye model allow us to interpret TD as
the threshold temperature below which the crystal behaves quantum mechanically,
while above it a classical picture is reasonably adequate. In other words, we can say
that if T > TD then all vibrational modes have been excited, while if T < TD some of
them (just a few or very many, according to the Bose–Einstein distribution) are still
not populated. The Debye temperature for some crystalline solids is reported in
table 4.1.

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 Solid State Physics

Table 4.1. The calculated Debye temperature TD (in Kelvin units) for some crystalline solids with metallic,
covalent, or ionic bonding.

Metallic Covalent Ionic

a
Al Cu Au Fe Pb C Si Ge NaCl LiF KBr
TD 428 343 170 470 105 2230 645 374 321 730 173
a
Diamond crystalline structure.

4.1.3 The general quantum theory for the heat capacity

Although the Debye expression for the lattice heat capacity is rather accurate over a
wide range of temperatures for most materials, deviations from laboratory measure-
ments are nevertheless found. The most effective way to develop the comparison is
to fit experimental data taken at different temperatures by means of equation (4.10),
while keeping TD as the only calibration parameter for the fitting. For many systems
this procedure returns a Debye temperature varying within few tens of Kelvin
degrees: this is the fingerprint of some failure of the interpolation scheme, which is
conceptually based on the existence of a unique Debye temperature. Good for us,
these deviations are small for many practical applications and, therefore, the Debye
model can be used as a very good approximation.
If, however, a high degree of accuracy is needed, then there is no better solution
than using a full quantum theory where the lattice contribution to the internal
energy U is calculated according to equation (3.36) so that
U = U0 + ∑[nBE(s q, T ) + 1/2] ℏωs(q), (4.15)
sq

where U0 is the total energy content of the static lattice. We accordingly calculate4
∂ ℏωs(q)
CVquantum(T ) =
∂T
∑ exp[ℏ
sq
ωs(q)/ kBT ] − 1
(4.16)
∂n (s q , T )
= ∑ ℏωs(q) BE = ∑ CV ,sq(T ),
sq
∂T sq

where we used equation (3.37) for the phonon population nBE(s q, T ) and we
introduced the specific contributions CV ,sq(T ) of each (s, q) mode to the heat capacity.
This expression is more easily handled in the limit of a very large crystal: the
density of allowed q wavevectors in the reciprocal space is so high that we can treat
them as a continuum
V ∂ ℏωs(q)
CVquantum(T ) =
(2π )3 ∂T
∑∫ exp[ℏωs(q)/ kBT ] − 1
d q, (4.17)
s

where the integral is taken for q ∈ 1BZ.

4
The zero-point energy contribution has been ignored since it is independent of temperature.

4-7
 Solid State Physics

The quantum expression CVquantum(T ) (both in discrete or continuum form)

approaches the Dulong–Petit law at high temperature, while going to zero as ∼T 3
for a vanishingly small temperature. In the intermediate-temperature regime it
provides the correct behaviour for the lattice heat capacity without any simplifying
assumption. The price to pay is a complete (numerical) calculation of all phonon
frequencies.

4.2 Anharmonic effects

The crystal lattice dynamics has been so far described under the harmonic
approximation which allowed us to understand many fundamental intrinsic proper-
ties of solids. It is, however, just an approximation, as emerged from the discussion
developed in section 3.1 where it has been presented as a convenient truncation of a
Taylor expansion of the total ionic potential energy U = U (R) (see equation (3.1)
and relative discussion). Beyond this formal argument, robust experimental evidences
suggest that a real system is in fact not purely harmonic; they are mostly related to
thermal properties like:
• if kBT /ℏ is much larger than typical phonon frequencies, deviations of the
predicted heat capacity from the experimental data are actually observed:
they are the onset of anharmonic effects, not yet explicitly included in the
theory leading to equation (4.17);
• a crystalline solid differently resists to positive or negative strains of identical
magnitude; since any volume variation reflects a change in the lattice spacing,
this suggests that ions are confined nearby their equilibrium positions by a
non-parabolic (that is, non harmonic) potential;
• real solids undergo thermal expansion; this would not be possible if the ions
thermally oscillate under the action of a perfectly parabolic potential since the
average ion–ion distance would not increase upon temperature;
• finally, a beam of phonons travelling along a given direction within an infinite
defect-free crystal would propagate with no damping if anharmonic effects
were not included (harmonic vibrational modes overlap without interference);
this would imply an infinite lattice thermal conductivity.

In the following we are going to treat separately thermal expansion and thermal
conduction in the next subsections.

4.2.1 Thermal expansion

Thermal expansion is due to the dependence of vibrational frequencies on the crystal
volume. To exploit this notion, we will make use of some fundamental thermody-
namic definitions reported in appendix C.
Our first goal is to work out an equation of state P = P (V , T ) relating the
pressure P acting on the system to its volume V and temperature T. To this aim, we
use the Helmholtz free energy F , since we assume that our solid is coupled to a heat
reservoir, that is T = constant. We also understand that no matter is added to or
removed from the system and, therefore, the numbers of moles of any chemical

4-8
 Solid State Physics

species are also constant. Under these assumptions, we can write (see equations (C.4)
and (C.8))
d F = d (U − TS ) = −PdV − SdT , (4.18)
so that the equation of state for the pressure is cast in the form
∂F
P=− , (4.19)
∂V T

which is conveniently developed as follows

∂(U − TS )
P =−
∂V T

∂ ⎛ T
∂S ⎞
=− ⎜U − T
∂V ⎝
∫0 ∂T ′
dT ′⎟
⎠T
(4.20)
V

∂ ⎛ T
1 ∂U ⎞
=− ⎜U − T
∂V ⎝
∫0 T ′ ∂T ′
dT ′⎟ ,
⎠T
V

where we used the identity T (∂S /∂T )V = (∂U/∂T )V . In order to proceed we need an
explicit expression for the internal energy: by using equation (4.15) in the equation of
state, after some non trivial algebra we obtain
∂U 0 1 ∂ℏωs(q)
P (V , T ) = − − ∑
∂V T 2 s q ∂V T

T = 0 contribution
(4.21)
⎡ ∂ℏωs(q) ⎤
+ ∑ ⎢− ⎥n BE(s q , T ) ,
sq
⎣ ∂V T ⎦
 
T > 0 contribution

indicating that the pressure depends on V and T through the variations of the phonon
frequencies upon volume changes and through the Bose–Einstein phonon population,
respectively.
In a purely harmonic crystal we have (∂ℏωs(q)/∂V )T = 0 for any (s, q) mode, since
the vibrational frequencies do not depend on the crystal volume5: therefore, the
pressure turns out to be independent of temperature and we accordingly conclude
that in the harmonic approximation the equation of state reduces to a simple law
P = P (V ). Consistently, the thermal expansion coefficient β of the harmonic crystal
(see appendix C) is calculated to be
1 ∂P
β= = 0, (4.22)
B ∂T V

5
To make it simple: the frequencies of an array of harmonic atomic oscillators just depend on the force
constants of the springs connecting them and on their masses, while they are independent of the interatomic
separations, as explicitly shown in section 3.2.

4-9
 Solid State Physics

where B is the bulk modulus of the system. In short: a harmonic crystal does not
undergo thermal expansion, contrary to experience. Another unphysical consequence
of the harmonic approximation is that by combining equation (C.14) with equation
(4.22) we immediately obtain that the constant-pressure and constant-volume heat
capacities should be equal, contrary to experimental evidence. We need to do better.
The most obvious option to improve our physical picture would be to explicitly
introduce the anharmonic terms in the evaluation of the vibrational crystal energy,
as explained in equation (3.1). This can be done either classically or quantum-
mechanically, but it requires some non-trivial formal developments [2]. We will
rather follow a more phenomenological approach, nevertheless leading to mean-
ingful conclusions.
Let us assume that (∂ℏωs(q)/∂V )T ≠ 0 for any (s, q) mode: basically, we are
assuming that the vibrational frequencies are no longer purely harmonic or, equiv-
alently, that ions oscillate in non-parabolic potential wells: their frequencies now
depend on the crystal volume. For further convenience, it is useful to define the
mode-specific Grüneisen parameter γsq as

∂ ln ωs(q) V ∂ωs(q)
γs q = − =− , (4.23)
∂ ln V ωs(q) ∂V
so that it is possible to recast the equation of state (4.21) in the new form
∂U 0 1 ⎡1 ⎤
P (V , T ) = −
∂V
+
V
∑ γs q ℏωs(q) ⎢ + n BE(s q , T )⎥ ,
⎣2 ⎦ (4.24)
T sq

from which we calculate the thermal expansion coefficient of an anharmonic crystal as

1 ∂P 1 ∂n BE(s q , T )
β=
B ∂T
=
VB
∑ γs q ℏωs(q)
∂T
V sq
1
= ∑ γ CV ,sq(T )
VB s q s q
(4.25)

γ
= CV (T ),
VB
where we have introduced the weighted Grüneisen parameter

∑s,q CV ,sq(T )γsq

γ= . (4.26)
CV (T )
If we consider γ as an empirical material-specific parameter6, we can use equation
(4.25) as a meaningful phenomenological model, usually referred to as the Grüneisen
model, for the thermal expansion coefficient, properly taking into account the
volume-dependence of the vibrational modes: it effectively explains the experimental

6
For instance, we could calculate γ within the Debye model by setting γ = −∂ ln ωD /∂ ln V .

4-10
 Solid State Physics

evidence that real crystals expand upon heating. In table 4.2 we report the thermal
expansion coefficient and the bulk modulus of some selected crystals.
The Grüneisen model predicts that the temperature dependence of β is dominated
by the heat capacity term, since the temperature dependence of the volume and bulk
modulus is much weaker in most materials. This implies that the ratio BVβ /CV = γ
should be almost constant in temperature. Both features7 are roughly found in many
materials, although deviations have been predicted in some cases [8]. This is a first
evidence that the Grüneisen model, although very useful, can be improved: as a
matter of fact, the most accurate way to proceed is to calculate β as reported in the
first row of equation (4.25). Another limitation of the model is highlighted by the
direct evidence that anharmonicity really affects each mode in a different way [2], as
shown in in table 4.3: there we report some calculated mode-specific Grüneisen
parameter in covalent materials. Not only do we observe large variations in the
absolute value of γsq , but this parameter is even found either positive and negative.
Finally, the complex role played by anharmonicity is especially evident in ferro-
electric crystals which undergo displacive transitions [8, 9]: at a particular temper-
ature, the frequency of some vibrational modes (known as soft modes) is lowered
down to zero by anharmonicity. Accordingly, the ion displacements assume a

Table 4.2. Room temperature bulk modulus B (in units of 102 GPa) and the thermal expansion coefficient β
(units of 10−6 K−1) for some crystalline solids with metallic, covalent, or ionic bonding.

Metallic Covalent Ionic

a
Al Cu Au Fe Pb C Si Ge NaCl LiF KBr
Bb 0.72 1.37 1.73 1.68 0.43 4.43 0.99 0.77 0.24 0.67 0.15
βc 70.8 51.0 41.7 35.1 86.4 2.4 6.9 18.3 118.5 98.7 115.5
a
Diamond bulk modulus is measured at T = 4 K.
b
Data taken from: [3, 4].
c
Data taken from: [5, 6, 7].

Table 4.3. Some calculated Grüneisen parameter for selected zone-centre (Γ) and zone-boundary (X)
vibrational modes in covalent crystals.

TO(Γ) LO(Γ) LA(X) LO(X) TO(X) TA(X)

a
Si 0.90 0.90 1.30 1.30 0.90 −1.50
Gea 0.90 0.90 1.40 1.40 1.00 −1.50
GaAsb 1.42 1.11 0.91 1.56 −3.48
a
Data taken from: [10].
b
Data taken from: [11].

7
Namely: lim β (T ) = lim CV (T ) and γ ∼ constant.
T →0 T →0

4-11
 Solid State Physics

permanent character which causes a variation of the lattice parameters and the
crystal is faced with a structural phase transition8.

4.2.2 Phonon–phonon interactions

The Grüneisen model has phenomenologically treated anharmonic effects in lattice
dynamics through volume-dependent vibrational frequencies. The most fundamen-
tal description of anharmonicity is, however, based on quantum theory and exploits
the phonon language: while in the harmonic approximation phonons are described
as a gas of free pseudo-particles, in a most realistic anharmonic crystal they actually
undergo mutual interactions.
Phonon–phonon interactions are not so strong to fully invalidate the harmonic
picture: this is proved by the true existence of well-resolved peaks in neutron
scattering spectra (see section 3.6), each peak being the fingerprint of a specific
harmonic phonon mode. Therefore, anharmonicity can be treated as a perturbation
on the quantum states of the harmonic crystal: while its energy spectrum remains
basically unaffected by phonon–phonon interactions (that is, we can still speak
about phonon frequencies and vibrational modes with different character s and
wavevector q ), anharmonicity causes transitions between different states of quantum
harmonic oscillator.
The formal treatment of such a perturbation is non trivial and falls beyond the
present level of discussion [2, 12, 13], but we can assimilate the underlying physical
concept by means of an analogy with atomic physics: the energy spectrum of, say, an
isolated hydrogen atom remains unaffected by a low-intensity electromagnetic field9,
whose perturbative effect is only to promote electronic transitions between the
discrete stationary-state levels of the atom [14]. We can say that, for both absorption
or emission transitions, the occupation of the initial and final stationary state has
been varied by −1 and +1, respectively, while a photon has been annihilated
(absorption) or created (emission). This is a three-particle event involving two-
electron and one-photon populations.
Similarly, any anharmonic term of the vibrational Hamiltonian appearing in
equation (3.1) causes transitions among harmonic eigenstates, correspondingly
affecting their phonon populations. The physical picture is simple: we can say
that the nth order term (with n ⩾ 3) in the Taylor expansion of the lattice potential
energy activates interactions among n phonons, which we will refer to as n-phonon
scattering events. Since the phonon number is not conserved, during a scattering
event phonons of some harmonic mode are annihilated (their population is
decreased), while other phonons of different modes are created (their population
is increased).
Let us consider the specific case of a three-phonon scattering event: the phonon
populations of any vibrational mode remain unchanged except for the three modes

8
Ferroelectric crystals are especially prone to this kind of displacive transitions since the soft modes generate a
permanent electric dipole moment.
9
This is strictly true only in the semi-classical picture, where the atomic energy spectrum is treated by quantum
mechanics, while the electromagnetic field is described by Maxwell equations.

4-12
 Solid State Physics

(s1, q1), (s2, q2), and (s3, q)3 which are coupled by a cubic term of the Hamiltonian.
This may occur in two different ways, namely
⎧ n s1q → n s1q − 1 ⎧ n s1q → n s1q − 1
1 1 1 1
⎪ ⎪
⎨ n s 2q 2 → n s 2q 2 + 1 and ⎨ n s 2q 2 → n s 2q 2 − 1 (4.27)
⎪ ⎪
⎩ n s 3q 3 → n s 3q 3 + 1 ⎩ n s3q3 → n s3q3 + 1,

which correspond to a phonon generation event (left) or to a phonon annihilation event

(right). The two events are shown in figure 4.4 (left).
The rate of occurrence per unit time Pi(3)→f of a three-phonon event can be
calculated according to first-order quantum perturbation theory by means of the
Fermi golden rule [15–17] as
2π
〈 f ∣Vˆ3∣i〉 δ(ℏω s1q1 ∓ ℏω s 2q 2 − ℏω s3q3),
2
Pi(3)
→f = (4.28)
ℏ
where V̂3 is the quantum operator describing the cubic anharmonic perturbation [2, 12, 13].
For a phonon creation event, the initial ∣i 〉 and final ∣ f 〉 states correspond to the quantum
states with phonon populations (n s1q1, n s2q2 , n s3q3 ) and (n s1q1 − 1, n s2q2 + 1, n s3q3 + 1),
respectively. On the other hand, for a phonon annihilation event, the final state is rather
corresponding to (n s1q1 − 1, n s2q2 − 1, n s3q3 + 1). The quantum calculation appearing
in equation (4.28) is rather complicated; we skip it and just focus on the δ-term which
imposes that the total energy of the phonon system is conserved during the scattering
event; more specifically, we get
phonon creation: ℏω s1q1 = ℏω s 2q 2 + ℏω s3q3
(4.29)
phonon annihilation: ℏω s1q1 + ℏω s 2q 2 = ℏω s3q3,

s1 q1

s1 q1 s2 q2
s2 q2

s3 q3
s3 q3 s1 q1

s2 q2
s1 q1 s3 q3

s3 q3
s2 q2

Figure 4.4. Pictorial representation of allowed (left) and forbidden (right) three-phonon scattering events. As
usual sn q n are the branch index and wavevector of the three n = 1, 2, 3 harmonic vibrational modes
undergoing mutual interaction.

4-13
 Solid State Physics

where it is understood that the energy contribution of all remaining modes not of
interest to the scattering event remains unaffected. This explains why the three-
phonon events shown in figure 4.4 (right) are never observed: simply, they are
forbidden since they do not conserve energy. If we still retain the approximation that
ionic oscillations have small amplitudes, the dominant terms ruling over anharmo-
nicity are expected to be the cubic and quartic ones. These latter are treated similarly
to the discussion above, provided that a suitable expression V̂4 for the quantum
operator describing the quartic anharmonic perturbation is worked out. However, in
the attempt to keep our development simple, we will only consider three-phonon
events in the following.
Another important anharmonic feature is that a phonon scattering also conserves
the crystal momentum. In particular, with reference to figure 4.4 a three-phonon
event must obey the following rules
phonon creation: ℏq1 = ℏq 2 + ℏq 3 ± ℏG
(4.30)
phonon annihilation: ℏq1 + ℏq 2 = ℏq 3 ± ℏG ,

where G is a reciprocal lattice vector. We can read these conservation laws by stating
that q1, q2 , and q3 directly involve three specific vibrational modes, while G is
absorbed or provided by the crystal as a whole. In studying three-phonon scattering
events it is customary to distinguish between normal processes (or N-processes) with
G = 0 and Umklapp processes10 (or U-processes) with G ≠ 0. They are represented
in figure 4.5 in the case of a phonon annihilation process occurring in a two-
dimensional crystal for which the graphics is easier to draw and to understand (but,
of course, the basic notion is also valid in one- and three-dimensions). For further
convenience, it is important to remark here that the true phonon momentum is not
conserved in a U-process, because of the ±G term appearing in equations (4.30). For
this reason, it is often said that Umklapp processes destroy (or generate) momentum.
qy qy

G
q3 q2
q2 q′3 = q1 + q2
q3 = q′3 − G
q1 q1
qx qx

Figure 4.5. Pictorial representation of a normal (left) and an Umklapp (right) three-phonon scattering event in
a two-dimensional square lattice. The grey-shaded area represents the 1BZ.

10
The German word ‘umklapp’ means ‘flipped over’: this wording will be explained soon.

4-14
 Solid State Physics

We conclude this section by remarking that the rigorous perturbative treatment of

phonon–phonon scattering [2, 12] proves that harmonic vibrations are slightly
shifted in frequency ωs(q) → ωs(q) + Δsq and damped in amplitude. This second
feature is translated in the corpuscular phonon language by saying that because of
anharmonic effects each phonon mode has a finite lifetime τsq . Our simplified
description of phonon–phonon interactions is consistent with this notion: for
instance, if we consider a three-phonon annihilation process, then the inverse of
its scattering rate per unit time given in equation (4.28) is related to the lifetime of
the decaying phonon. This concept will be fully exploited in the theory of thermal
transport developed in the next section where phonons will be treated as the
microscopic heat carriers. Each carrier will be treated as a ℏωs(q) quantum of energy
travelling with velocity vg(s q) = dωs(q)/d q = λs qτs q , where λsq is its average mean free
path (that is, the average distance covered by the phonon between two successive
scattering events).

4.3 Thermal transport

In order to set up the investigation on thermal transport phenomena in a solid state
system, let us consider the situation represented in figure 4.6, where (i) a small
temperature gradient is imposed along the x direction of (ii) a homogeneous
insulating crystal. This is the minimal complexity framework containing all the
most relevant physical features which rule over the thermal energy transport. Under
these constitutive hypotheses we can assume that the microscopic heat carriers are the
lattice vibrations11 and, therefore, the most appropriate approach is based on the
phonon language. Furthermore, under a small imposed thermal gradient dT/dx
experiments provide evidence that, after a transient time, a steady state thermal

T (x)

Thot

Jh,x

Tcold
x

Figure 4.6. A homogeneous crystalline specimen is coupled to a hot (left) and cold (right) thermostat,
respectively, at temperature Thot and Tcold < Thot . The white full line represents the temperature profile across
the sample in the steady-state condition. The indicated volume element is used in equation (4.34) for the
continuity equation of the thermal current described by the heat flux Jh,x .

11
In metals there is an additional important contribution due to conduction electrons. It will be investigated in
section 7.2. For this reason the parameter κ l introduced just below will be referred to as the lattice thermal
conductivity.

4-15
 Solid State Physics

conduction regime is established, which is pretty well described by the phenomeno-

logical linear Fourier law [18, 19]
dT
Jh,x = −κl , (4.31)
dx
where Jh,x is the heat flux along x, namely the amount of thermal energy crossing a
unit area normal to the x direction per unit time (it is measured in units J m−2 s−1).
The key physical parameter is the lattice thermal conductivity κ l : it is a material-
specific quantity making the difference between thermal insulators (low κ l values)
and good thermal conductors (large κ l vaules). While the non-equilibrium thermo-
dynamics fundamentals of thermal transport can be found elsewhere [20], here we
aim at developing a microscopic theory of κ l based on the phonon language.
The heat flux Jh,x can be calculated by summing over all phonons (s, q) the
product [energy carried] × [number of phonons] × [speed of propagation] × [1/V]
(where V is the volume crossed by the thermal energy flux) or equivalently12
1
Jh,x = ∑ ℏωs(q) n¯(s q, T ) vg,x(s q), (4.32)
V s, q

where it must be noted that the phonon population n̄(s q, T ) is not given by the Bose–
Einstein distribution since the condition represented in figure 4.6 does not correspond
to an equilibrium situation13. Our next task is, therefore, to calculate the non-
equilibrium population n̄(s q, T ) for the steady state condition we are interested in.
To this aim we resume the full phonon picture including their mutual interactions
developed in the previous section and follow a simple argument: if we assume that
the temperature gradient is removed, then we expect the system to recover the
equilibrium situation after a suitable time. This implies that during such a transient
regime each single-mode phonon population n̄(s q, T ) must relax back to its
equilibrium value nBE(s q, T ). The simplest situation we can guess is that such
relaxation processes are independent or, equivalently, we can assume that the
evolution of each single phonon population is unaffected by the relaxation of all the
other populations. This is the essence of the so-called single-mode relaxation time
approximation (SM-RTA) which allows us to write
∂n¯(s q , T ) n¯(s q , T ) − n BE(s q , T )
=− , (4.33)
∂t τsq

where the single-mode relaxation time τsq is identified with the lifetime of the
corresponding vibrational mode, as discussed in the previous section. We remark

12
We incidentally remark that equation (4.32) explains why in equilibrium conditions no thermal current
occurs: if the temperature is uniform throughout the whole system, then Jh,x = 0 since the phonon group
velocities are isotropically distributed.
13
We will nevertheless rely on the local equilibrium hypothesis [20] which allows us to define anywhere within
the system a local temperature and to use standard thermodynamics equations (although extensive quantities
must be replaced by their volume densities).

4-16
 Solid State Physics

that at this stage only anharmonicity (i.e. phonon–phonon interactions) has been
included in our analysis. This will be soon critically readdressed in the following.
Let us now consider the unit volume element V shown in figure 4.6 by a dashed
line. In steady-state conditions we can write a continuity equation for the thermal
current
1 ∂U ∂Jh,x
= , (4.34)
V ∂t ∂x
where the internal energy U is calculated by means of equation (4.15) with the non-
equilibrium population n̄(s q, T ); in equation (4.34) no term describing the rate of
energy generation/removal appears since, consistently with the assumed homoge-
neity of the system, the selected unit volume does not contain any energy source or
sink. This equation basically proclaims energy conservation within the selected
volume element, properly taking into account that the incoming and the outcoming
thermal energy fluxes are not equal because of the space variation of n̄(s q, T ) due to
the temperature profile established across the system. By inserting equations (4.15)
and (4.34) into equation (4.34) we easily obtain
∂n¯(s q , T ) ∂n¯(s q , T ) ∂T
− vg,x(s q) = 0. (4.35)
∂t ∂T ∂x
In order to proceed we will further assume that the temperature-dependence in the
non-equilibrium population is the same as in the Bose–Einstein distribution, that is
∂n¯(s q , T ) ∂n (s q , T )
= BE . (4.36)
∂T ∂T
This procedure eventually leads to the linearised Boltzmann transport equation (BTE)
∂n BE(s q , T ) ∂T
n¯(s q , T ) = n BE(s q , T ) − τs q vg,x(s q) , (4.37)
∂T ∂x
where we explicitly made use of the SM-RTA expression given in equation (4.33).
The Boltzmann transport equation allows for a straightforward calculation of the
non-equilibrium phonon population n̄(s q, T ) once:
• the temperature gradient ∂T /∂x is assigned;
• the phonon dispersion relations are known, so that the phonon group
velocities vg,x(sq) are easily calculated;
• the relaxation times τsq are calculated by quantum perturbation theory, as
outlined in the previous section.

We duly remark that so far we have considered just one single physical
mechanism for the determination of the phonon lifetimes, namely phonon–phonon
interactions. As a matter of fact, they are many more: phonons are also scattered by
lattice point defects (like isotope impurities, native defects, contaminant or doping
impurities), by extended surface (grain boundaries) or line (dislocations) defects, as

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 Solid State Physics

well as by the boundaries of the crystal14. If each scattering mechanism were to act
alone, we could define a different scattering rate for each mechanism or, equiv-
alently, a different relaxation time. If we assume that the scattering rates are
additive, we can calculate the total relaxation time τsqtot for the (sq) mode according to
the Matthiessen rule [2]:
1 1
= ∑ , (4.38)
τstot
q i
τs q,i

where the label i runs over all possible phonon scattering mechanisms. The
Matthiessen rule is similarly applied to the physics of electron scattering; while
violations of this rule are in fact observed [13], it is in practice rather well satisfied in
many circumstances and, therefore, we will fully rely on it.
We can now calculate the heat flux
1
Jh,x = ∑ ℏωs(q) n BE(s q , T ) vg,x(s q)
V sq
(4.39)
1 ∂n BE(s q , T ) ∂T
−
V
∑ ℏωs(q)τstot 2
q vg , x (s q)
∂T ∂x
,
sq

where we have inserted the non-equilibrium population provided by the Boltzmann

transport equation into equation (4.32). The first term on the right-hand side
obviously does not contribute to the heat current since it only depends on
equilibrium quantities. Therefore, by comparing equations (4.31) and (4.39) we
eventually obtain a microscopic equation for the lattice thermal conductivity15
1 ∂n BE(s q , T )
κl(T ) =
V
∑ ℏωs(q) τstot 2
q vg , x (s q)
∂T
sq
1
=
3V
∑ τstot 2
q 〈vg , s q 〉 CV ,s q(T ) (4.40)
sq
1
= 〈τ〉 〈vg2〉 cV (T ),
3
which we have written in the three more commonly used forms. More specifically,
the second form is obtained by using equation (4.16) and by defining a mode-specific
mean square group velocity 〈vg2, s q〉/3 = vg2, x(s q) = vg2, y(s q) = vg2, z (s q), while the third
form is found by further attributing the same mean value of relaxation time 〈τ 〉 and
square group velocity 〈vg2〉 to all phonon modes, as well as by using the constant-
volume specific heat cV (T ) = (1/V )∑s q CV ,s q(T ) = CV (T )/V . Interestingly enough,

14
This latter scattering mechanism plays an important role whenever the phonon average mean free paths are
longer than or comparable to the system dimension.
15
This value of κ is referred to as ‘lattice thermal conductivity’ to highlight the fact that the contribution from
the valence electron system is not accounted for.

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 Solid State Physics

this third form for κ l(T ) is the same provided by elementary kinetic theory, where
the heat carriers are treated as an interacting phonon gas [13].
In figure 4.7 we report the SM-RTA-BTE calculated thermal conductivity of
crystalline silicon. In this case the system has been considered isotopically pure,
totally defect-free, and large enough to exclude boundary scattering. Therefore, the
resulting picture can be considered as representative of the general κ = κ l(T ) trend
observed in most non-metallic crystals when only phonon–phonon scattering events
(that is, anharmonic effects) are at work. The agreement with experimental data (red
dots) is impressive, proving that the adopted hierarchy of approximations is rather
physically sound. We can distinguish three regimes:
• very low temperature: the lattice thermal conductivity increases with temper-
ature simply because the higher the temperature the larger are the phonon
populations; in this temperature range, however, phonon scattering plays a
minor role, since vibrational modes are still poorly populated; overall the
temperature-dependence is dictated by the specific heat and, therefore, we
have κ l(T ) ∼ T 3, reflecting into a linear trend in the reported bi-logarithmic
plot;
• intermediate temperatures: a larger and larger number of phonons is gen-
erated and, therefore, scattering plays an increasingly important role in this
temperature range; the maximum lattice thermal conductivity value is found
at the temperature where the rate of generation of new phonons per unit
temperature increase equals the rate of annihilation because of scattering
events; above this temperature, the thermal conductivity begins to decrease as
κ l(T ) ∼ exp[(1/b )TD /T ] where it is typically found that 2 < b < 3 according
to the selected material;

Figure 4.7. Full blue line: the lattice thermal conductivity of crystalline silicon calculated by the SM-RTA-
BTE approach. The calculations have been performed by using the QUANTUM ESPRESSO integrated suite
of open-source computer codes for materials modelling; see website https://http://www.quantum-espresso.org
(by courtesy of Giorgia Fugallo). Red dots: experimental data taken from [21].

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 Solid State Physics

• high temperatures: phonon annihilation mechanisms are more efficient than

generation ones and the lattice thermal conductivity is found to decrease
according to a power law κ l(T ) ∼ T −d , where for most materials it is found
that 1 < d < 2.

Finally, it is interesting to observe that in the harmonic approximation there are

no phonon–phonon interactions and, therefore, the lifetime τsq of any vibrational
mode is infinite. According to equation (4.40), this would suggest an infinite lattice
thermal conductivity: another indeed striking failure of the purely harmonic crystal
picture.
We now address more in detail the different role played by normal and Umklapp
processes (see section 4.2.2) in affecting lattice thermal conductivity. We begin by
considering a general feature which we elaborate by a simple order-of-magnitude
estimate of the energies and momenta of phonons which undergo three-phonon
scattering events. Let us assume a system temperature below the Debye temperature:
then, the frequency of the three involved phonons must necessarily be much smaller
than the Debye frequency. This is in fact a direct consequence of the Bose–Einsten
distribution which phonons must obey: at any temperature T the only populated
phonon modes are those with vibrational energy ℏω ⩽ kBT . Accordingly, at T < TD
the only incoming phonons that can undergo scattering events have energy
ℏωin ≪ ℏω D. Because of energy conservation, the outcoming phonons emerging
from the scattering event must similarly have ℏωout ≪ ℏω D or, equivalently, the
magnitude of their wavevector must be ∣q out∣ ≪ qD. This is possible only if G = 0 in
equation (4.30). In conclusion, at low temperature mainly N-processes occur; on the
other hand, U-processes must be duly considered at moderate or high temperature
where scattering events are compatible with the ±G exchange of a reciprocal lattice
vector with the crystal as a whole.
Next, we address the different effect due to U- and N-processes on the heat
current and phonon populations; to make things easy, let us refer to figure 4.5 and
assume a situation such that the thermal transport occurs along the positive
orientation of the qx-axis (that is, we assume a rightward thermal current). It is
evident that the three-phonon Umklapp process shown on the right flips over the
momentum of the outcoming scattered phonon with respect to the direction of the
heat current: this deprives the net thermal energy current of its energy contribution.
Because of this, U-processes are said to be resistive, meaning that their main
contribution is to decrease lattice thermal conductivity or, equivalently, to increase
thermal resistivity. On the other hand, N-processes have the main role to repopulate
each single phonon mode, which is continuously affected by opposite annihilation
events.
By combining all the results discussed so far, we classify different thermal
conduction regimes. When the size of the crystal is comparable with the phonon
mean free path, the dominant scattering mechanisms are due to dislocations, grain
boundaries and surfaces; this is referred to as the ballistic regime since phonons

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 Solid State Physics

freely propagate, until they meet some extrinsic16 scattering source. This typically
occurs at very low temperatures (no intrinsic17 resistive scattering processes are
active) and, as commented above, in the ballistic regime we have κ (T ) ∼ T 3. By
increasing temperature we enter the Poiseuille regime where the phonon gas flows
under the action of the temperature gradient within the system, whose boundaries
work as a friction source; this situation is similar to that found in fluid hydro-
dynamics. Resistive processes of any kind (that is both intrinsic U- or extrinsic ones)
occur at higher temperatures, giving rise to the Ziman regime; here N-processes are
still dominant, but they are increasingly contrasted by resistive ones (although
boundary mechanisms are no longer important). Finally, in the high-temperature
kinetic regime thermal transport is dominated by anharmonic resistive processes.
The Ziman and kinetic regimes are together referred to as diffusive thermal transport;
here it is found that κ (T ) ∼ T −d (with 1 < d < 2).
We conclude the investigation on thermal transport by considering a very special
situation referred to as second sound. Ordinary sound is the propagation of a
compressive wave of frequency ωs within a gas made of massive particles [22]. This
phenomenon is observed provided that ωs ≪ 1/τ , where τ is the average relaxation
time for the collisions among gas particles (in other words, 1/τ is the rate of particle–
particle collisions). This condition is necessary to guarantee that local thermody-
namical equilibrium is always reached by the gas at each instant of the sound wave
oscillation. It is important to stress that collisions among gas particles conserve
energy and momentum. We now turn to look at phonons as a gas: we must
preliminarily admit that there are two average relaxation times, namely τN and τU for
normal and Umklapp processes, respectively. In a U-process the true phonon
momentum is actually not conserved (see equations (4.30) and related discussion)
and, therefore, their occurrence should be very rare to observe a sound-like
phenomenon in the phonon gas. On the other hand, normal processes should
obey the condition valid for ordinary sound. Accordingly, provided that the
condition 1/τU ≪ ωss ≪ 1/τN is fulfilled, the analogue of mechanical sound is found
in the phonon gas: the ‘second sound’ is a wavelike oscillation in temperature18
occurring at frequency ωss. Since the necessary condition is that τN ≫ τU , second
sound is typically observed at very low temperature.

References
[1] Eisberg R and Resnick R 1985 Quantum Physics of Atoms, Molecules, Solids, Nuclei, and
Particles 2nd edn (Hoboken, NJ: Wiley)
[2] Srivastava G P 1990 The Physics of Phonons (Bristol: Adam Higler)

16
In this framework ‘extrinsic’ means not related to anharmonicity. Extrinsic processes are due to isotope,
point-defect, grain boundary, dislocation, and surface scattering.
17
In this framework ‘intrinsic’ means related to anharmonicity. Intrinsic processes are only due to direct
phonon–phonon interactions.
18
An oscillating density of phonons reflects in an oscillation of their energy density; since their energy is
thermal, this corresponds to a temperature oscillation.

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 Solid State Physics

[3] Birch F 1966 Handbook of Physical ConstantsGSA Memoirs vol 97 (McLean, VA: The
Geological Society of America) p 97
[4] Tosi M P 1964 Solid State Physicsvol 16 (New York: Pergamon) p 44
[5] Pearson W B 1958 A Handbook of Lattice Spacings and Structures of Metals and Alloys (New
York: Pergamon)
[6] White G K 1965 Proc. R. Soc. London A286 204
[7] Wolfe C M, Holonyak N and Stillman G E 1989 Physical Properties of Semiconductors
(Englewood Cliffs, NJ: Prentice-Hall)
[8] Dove M T 2003 Structure and Dynamics–An Atomic View of Materials (Oxford: Oxford
University Press)
[9] Kittel C 1996 Introduction to Solid State Physics 7th edn (Hoboken, NJ: Wiley)
[10] Yin M T and Cohen M L 1982 Phys. Rev. B 26 3259
[11] Kunc K and Martin R M 1981 Phys. Rev. B 24 2311
[12] Böttger H 1983 Principles of the Theory of Lattice Dynamics (Berlin: Akademie)
[13] Ashcroft N W and Mermin N D 1976 Solid State Physics (London: Holt-Saunders)
[14] Colombo L 2019 Atomic and Molecular Physics: A Primer (Bristol: IOP Publishing)
[15] Sakurai J J and Napolitano J 2011 Modern Quantum Mechanics 2th edn (Reading, MA:
Addison-Wesley)
[16] Miller D A B 2008 Quantum Mechanics for Scientists and Engineers (New York: Cambridge
University Press)
[17] Griffiths D J and Schroeter D F 2018 Introduction to Quantum Mechanics 3rd edn
(Cambridge: Cambridge University Press)
[18] Incoprera F P and Dewitt D P 2011 Fundamentals of Heat and Mass Transfer (New York:
Wiley)
[19] Lienhard H H IV and Lienhard J H V 2006 A Heat Transfer Textbook (Cambridge, MA:
Phlogiston)
[20] Kjelstrup S and Bedeaux D 2008 Non-equilibrium Thermodynamics of Heterogeneous
Systems (Singapore: World Scientific)
[21] Inyushin A V, Taldenkov A N, Gibin A M, Gusev A V and Pohl H J 2004 Phys. Status
Solidi C 1 2995
[22] Feynman R P, Leighton R B and Sands M 1963 The Feynman Lectures on Physics (Reading,
MA: Addison-Wesley)

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 IOP Publishing

Solid State Physics

A primer
Luciano Colombo

Chapter 5
Elastic properties

Syllabus—We model the discrete atomistic structure of a solid by a continuous

distribution of matter and describe its deformations occurring on a length scale much
larger than the interatomic spacings by means of the strain and stress tensors. Within
this continuum picture, the constitutive equation for the elastic behaviour of a solid
body is derived under the assumption of small deformations, defining the core of linear
elasticity. The constitutive stress–strain relation is derived in the paradigmatic case of
a homogeneous and isotropic solid, whose thermoelastic behaviour is thoroughly
discussed.

5.1 Basic definitions

5.1.1 The continuum picture
We have so far developed our theory of the crystalline state on the basis of the
atomistic picture (see section 1.1). We are now about to consider two situations that
are more effectively described by looking at a solid in a rather different way.
Let us start by considering ionic vibrations in the very long wavelength limit: in
this case, the deformation of the crystal lattice with respect to its ideal crystallo-
graphic configuration occurs on a length scale which is much larger than the typical
interatomic spacings. This corresponds to the propagation of elastic waves (sound)
in the material. The situation is even more marked if we suppose that the solid is
subject to some external action of mechanical, electric or magnetic nature. For
instance, this happens, respectively, when a compressive/tensile or bending load is
applied to the system, when the onset of a coupling between a state of charge and a
state of deformation is observed in the material response to some mechanical stress
(a phenomenon known as piezoelectricity), or when during the magnetisation
process of a material its shape or dimension are changed (a phenomenon known
as magnetostriction). Once again, in all cases the resulting deformation typically
unfolds on a macroscopic scale that, although it may result much shorter than the

doi:10.1088/978-0-7503-2265-2ch5 5-1 ª IOP Publishing Ltd 2021

 Solid State Physics

specimen dimensions, is nevertheless definitely much longer than interatomic

distances.
In order to formally describe this kind of situations on a general ground, it is
convenient to switch to a new conceptual paradigm: the discrete atomistic structure of
the crystal is now replaced by a continuous distribution of matter, and its deforma-
tions of any origin are better described by a continuous displacement field. We will
refer to this approach as the continuum picture which lies at the foundation of solid
mechanics, a huge and independent scientific discipline with many applications in
physics and engineering [1–3]. In this chapter we will just outline the fundamentals of
elasticity theory which represents the sub-field of solid mechanics which mostly
overlaps to solid state physics.
In the continuum picture we no longer deal with atomic positions onto a lattice
but, rather, with a continuous position variable r representing the location of each
material point within the solid. Its components are calculated with respect to a
Cartesian frame of reference and, contrary to the notation adopted elsewhere in this
volume, they will be labelled by Latin indices1 r = {ri}i =1,2,3. For consistency, the
Cartesian axes will be indicated by the set (x1, x2, x3). The continuous displacement
vector u(r) is introduced by the following relation
r → r′ = r + u(r), (5.1)
where the change r → r′ is caused by some external action. When the solid is
described by the r positions is said to be in its reference configuration, while the set of
r′ positions is usually referred to as the deformed configuration. The Jacobian matrix
 = {Jij}i,j =1,2,3 describing the change from the reference configuration r to the
deformed one r′ can be written as
∂ui
Jij =
∂xj
1 ⎛ ∂u ∂uj ⎞ 1 ⎛ ∂u ∂uj ⎞ (5.2)
= ⎜ i + ⎟ + ⎜ i − ⎟,
2 ⎝ ∂xj ∂xi ⎠ 2 ⎝ ∂xj ∂xi ⎠
 
symmetric part antisymmetric part

where the symmetric and antisymmetric part of  describe deformations of rather

different nature. In order to provide a simple explanation of this statement2 we
consider a pure infinitesimal local rotation described by the matrix  . In this case, the
Jacobian matrix for any point r in the volume element under rotation is easily
calculated:
r ′ =  r = r + u( r ) → u( r ) = (  −  ) r →  =  − , (5.3)

1
This choice is forced by the fact that high-order tensorial quantities appear in the elasticity theory and,
therefore, the use of explicit Cartesian indices (x, y, z ) is unpractical. In any case, the new notation will not
generate confusion since electrons, elsewhere in this volume labelled by Latin indices, will never appear in the
discussion we are going to develop.
2
In solid mechanics this result is known as the Cauchy polar decomposition theorem.

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 Solid State Physics

where  is the identity matrix. The product between the Jacobian matrix and its
transpose matrix t is vanishingly small since the rotation is infinitesimal and,
therefore, to a good approximation we can write
0 = t = ( −  )(t −  ) = − − t →  = −t , (5.4)
or, equivalently, we state that the Jacobian matrix of a pure rotational deformation is
antisymmetric. Combining this result with equation (5.2) we conclude that in general
the deformation of any volume element consists in the combination of a local rotation
and a local volume variation, which can be either compressive or tensile, depending
on the actual deformation considered.
This important result allows us to define the domain of linear elasticity theory we
are developing in the present chapter: we will study the physics of small volume
variations of a solid body on the continuum scale under the action of sole deformation
forces, while pure rotations will not be considered. The reference to ‘small’
deformations is fully specified by the condition Tr(t ) ≪ 1.

5.1.2 The strain tensor

In the approximation of small deformations, the stretching, compression, bending or
shearing of a solid body is entirely described by the symmetric part of the Jacobian
matrix given in equation (5.2)

1 ⎛ ∂ui ∂uj ⎞
ϵij = ⎜ + ⎟, (5.5)
2 ⎝ ∂xj ∂xi ⎠

which is referred to as the strain tensor and, by its very definition, is symmetric
ϵij = ϵji . Its physical meaning is revealed by the two examples shown in figure 5.1
where a simple tensile deformation (left) and a pure shear deformation (right) is
applied to a homogeneous solid.
In the first case, a total length variation ΔL along the x1 direction is applied
without affecting the cross section. An infinitesimal slab originally placed in position
x1 is displaced by an amount3 u = (x1(ΔL /L ), 0, 0); we accordingly calculate the
only non-zero component of the strain tensor du1/dx1 = ΔL /L and we get the

x2 x2

x1 ΔL
x1
L ΔL L

Figure 5.1. A simple tensile deformation (left) or pure shear deformation (right) is applied to a homogeneous
solid. The deformation ΔL is applied along (left) or normal (right) to the x1-axis. The third x3-axis is
perpendicular to the plane of the figure.

3
This guarantees that the right terminal end of the sample is eventually found at position x1 = L + ΔL as
indicated in figure 5.1.

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 Solid State Physics

⎛ΔL / L 0 0 ⎞
strain tensor for a simple uniaxial tensile deformation = ⎜⎜ 0 0 0 ⎟⎟ , (5.6)
⎝ 0 0 0⎠
This result holds also for the simple compressive deformation, but in this case
obviously ΔL < 0 so that L + ΔL < L .
Let us now consider the case of the pure shear deformation: the face of the solid
which lies at distance L for the origin is displaced along x2 by an amount ΔL , while
the total thickness along x1 remains unchanged from its initial value L. In this case
an infinitesimal slab originally placed in position x1 is displaced by
u = (0, (ΔL /L )x1, 0). Accordingly, only two off-diagonal elements of the strain
tensor, namely ϵ12 and ϵ21, are found to be non-zero and we get
⎛ 0 ΔL /2L 0 ⎞
strain tensor for a pure shear deformation = ⎜⎜ΔL /2L 0 0 ⎟⎟ . (5.7)
⎝ 0 0 0⎠
Equations (5.6) and (5.7) provide the most general form of a tensile/compressive and
shear strain tensor, respectively. They can be used to describe any general
deformation; for instance, if we non-isotropically disfigure a cubic solid with initial
edge length L into an orthogonal prism, the strain tensor defining this deformation is
⎛ΔL1/ L 0 0 ⎞
⎜ ⎟
cube ⟶ orthogonal prism = ⎜ 0 ΔL 2 / L 0 ⎟, (5.8)
⎜ ⎟
⎝ 0 0 ΔL 3/ L ⎠

where ΔL1 ≠ ΔL 2 ≠ ΔL3 are the three unalike length variations along x1, x2, and x3,
respectively. This example illustrates an important property of the strain tensor
ΔV
∑ ϵii = V
, (5.9)
i

or equivalently: the trace of the strain tensor describes the fractional volume change of
the system. The strain tensor also describes the case of simple tensile deformation
along x1 with allowed lateral contractions/extensions (if the system is isotropic we
have ΔL 2 = ΔL3).

5.1.3 The stress tensor

So far we have discussed the mathematical entity in charge of describing any kind of
small deformation, namely the strain tensor. We now need to address the forces
acting on a material continuum and causing such deformation. It is customary to
distinguish between two different categories: body forces and surface forces.
Body forces act throughout the volume of the solid and are typically due to an
external field like a gravitational, electric, or magnetic one. They are formally
described by means of their volume density fv(r) so that the body force d Fv acting on
the infinitesimal volume dV centred around the point r is d Fv = fv(r)dV .

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 Solid State Physics

Surface forces, instead, are due to the mechanical actions that each portion of the
solid undergoes because of the remaining part of the continuum4; they act across a
surface within the solid body which ideally separates the selected portion from the
surrounding distribution of continuum matter. If we identify the orientation of an
infinitesimal portion dS of such a surface by means of its outward-pointing normal
unit vector nˆ = (n1, n2 , n3), then we understand that compressive and tensile surface
forces are, respectively, negative and positive since they are antiparallel and parallel
to n̂, respectively. The infinitesimal surface force d Fs acting on dS is written as
d Fs = fs dS , (5.10)
where fs has the physical meaning of an internal pressure and plays a key role in
elasticity theory. It is often referred to as tension.
In order to calculate the internal force per unit area fs , let us consider the
tetrahedrally shaped5 infinitesimal volume dV shown in figure 5.2. In particular, we
observe that the four faces of the infinitesimal tetrahedron have area dA1, dA2 , dA3 e
dAn and outward-pointing normal unit vectors −î, −ĵ, −k̂, and n̂, respectively. The
total force acting on the infinitesimal volume is
ρm dV a = d Fs + d Fv
(5.11)
= fs,n dAn − fs,1dA1 − fs,2dA2 − fs,3dA3 + fvdV ,
where a is its acceleration, ρm the mass density of the solid body, and fs,1,2,3 are the three
surface forces acting on the surfaces dA1,2,3, respectively. Similarly, fs,n acts on dAn (here
dh is the distance between the origin and the face dAn ). By dividing any term of this
equation by dAn and considering that dV = dAn dh/3 we get6

x3
dA1
n̂ = (n1 , n2 , n3 )
dA2
dAn

k̂

ĵ x2
î

dA3
x1

Figure 5.2. A tetrahedrally shaped infinitesimal volume. Its faces with area dA1, dA2 , dA3 and dAn are,
respectively, orthogonal to the unit vectors iˆ , j
ˆ , kˆ and nˆ = (n1, n2 , n3).

4
This is clearly a macroscopic wording for the full set of ion–ion, electron–electron, and ion–electron
interactions we have classified in the chapter 1.
5
The final result will not depend on this arbitrary (but very convenient) assumption.
6
By elementary geometry we have that dAi = ni dAn for any i = 1, 2, 3.

5-5
 Solid State Physics

1 1
fs,n − fs,1 n1 − fs,2 n2 − fs,3 n3 + fv dh = ρm a dh , (5.12)
3 3
which in limit dh → 0 leads to
fs = fs,1 n1 + fs,2 n2 + fs,3 n3 (5.13)
This is a result of paramount importance: we proved that the surface force fs referred
to any generic plane passing through a point inside the solid is fully determined by a set
of three mutually orthogonal vector tensions. This formal result, known as Cauchy
theorem, states the existence anywhere within a solid body of a stress tensor  such
that
fs =  nˆ , (5.14)
where  = (fs,1, fs,2, fs,3) or equivalently

fs,i = ∑ Tij nj , (5.15)

j

which corresponds to the mathematical entity in charge of describing the surface

forces causing small deformations.
We illustrate this important achievement with two simple examples. Let us first
suppose that the stress tensor is diagonal Tij = Pδij ; then, it is easy to prove that
fs,i = P ni and we therefore understand that the matrix element P is nothing other
than the hydrostatic pressure applied to the solid. On the other hand, if the stress
tensor has the form
⎛0 τ 0⎞
 = ⎜⎜ τ 0 0 ⎟⎟ , (5.16)
⎝0 0 0⎠

then we get
fs,1 = τ n2 fs,2 = τ n1 fs,3 = 0, (5.17)

which corresponds to a shear stress with two non-zero vector components acting
along the x1- and x2-axis, respectively. We understand this physical situation by
looking at figure 5.3(left): a force per unit area of magnitude τ is applied to the face 2
along the x1 direction and a similar action is applied to face 1 along the x2 direction.
These tangential forces transform the square face 4 into a rhombus.
We conclude that the physical meaning of the stress tensor is that of a vector
pressure whose geometrical representation is reported in figure 5.3(right) following
the Voigt notation: Tij represents the pressure applied to the jth face along the ith
direction. It can be proved that the stress tensor is symmetric or, equivalently,
Tij = Tji . While the formal details can be found elsewhere [3, 4], we can understand
this symmetry property by developing a very simple argument with reference to
figure 5.3(right): if we had T12 ≠ T21 then the volume element would undergo a

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 Solid State Physics

x3 x3

T33
T23
T13
T31 T32

T11 T21 T12 T22

x1 x2 x1 x2

Figure 5.3. Left: the faces of an infinitesimal volume element are labelled according to the Voigt notation.
Right: the corresponding nine components of the stress tensor Tij with i , j = 1, 2, 3.

rotation around the x3-axis; this would violate the local equilibrium condition and,
therefore, it must be T12 = T21.
We finally remark that, since the stress tensor is symmetric, it is always possible to
define a frame of reference where it assumes a diagonal form. In order to distinguish
its diagonal representation from the general one valid in any other frame of
reference, we will indicate the matrix entries of a stress tensor in its diagonal form
as Tij*.

5.2 Linear elasticity

5.2.1 The constitutive equation
So far we have prepared the formal environment to describe force actions and
corresponding deformations. The next point is to look for the mathematical
relationship linking the strain and stress tensors, which is usually referred to as
the constitutive equation of elasticity.
In this regard, it must be preliminarily observed that elasticity theory is unable to
provide this relationship which, instead, must be assumed ‘a priori’ of the problem we
aim at investigating. Accordingly, any result of continuum elasticity will specifically
depend on the adopted constitutive equation. This is the level at which the atomistic
theory plays a major role since it provides the needed fundamental knowledge. In
fact, once assigned the most appropriate model for the lattice many-body potential
energy U = U (R) governing the ion displacements (see sections 1.3.4 and 3.1), the
constitutive stress–strain relationship is there contained, even if not always imme-
diately apparent.
Consistently with the hypothesis of small deformations, we guess that they are
linearly dependent on their causing action7 and formally write
Tij = ∑ Cijkh ϵkh , (5.18)
kh

7
Elasticity theory can be formally developed under the more general assumption of arbitrary deformations.
This is the realm of nonlinear elasticity [5] which requires much more sophisticated mathematical apparatus
than here developed.

5-7
 Solid State Physics

which can be simply looked at as the most general form of the Hooke law. By this
linear elastic constitutive equation we introduce the fourth rank elastic tensor Cijkh. In
general, among its 34 = 81 components only 21 are independent as determined by the
symmetric character of the strain and stress tensor which imposes
Cijkh = Cjikh Cijkh = Cijhk (5.19)
Another symmetry is imposed by the guessed constitutive equation (5.18) which
in fact represents the macroscopic counterpart of the harmonic crystal model
developed in chapter 3. Therefore, by analogy with equation (3.4) we can surely
state that there exists a formal dependence of the elastic energy density u = u(ϵij )
on the strain tensor which, within the adopted constitutive model, is cast in the
harmonic form
1
u=
2
∑ Cijkh ϵij ϵkh . (5.20)
ijkh

This allows us to identify the matrix entries of the elastic tensor as the macroscopic
counterparts of the microscopic force constants appearing in equation (3.4) and
accordingly write
∂ 2u
Cijkh = , (5.21)
∂ϵij ∂ϵkh

which leads to the third symmetry property

Cijkh = Ckhij , (5.22)
and
⎡ ⎤
∂u ∂ ⎢1
∂ϵij
=
∂ϵij ⎢⎣ 2
∑ ijkh ij kh⎥⎥ =
C ϵ ϵ ∑ Cijkh ϵkh = Tij . (5.23)
ijkh ⎦ kh

This result eventually establishes the full set of relations linking the most important
quantities entering the linear elasticity theory.

5.2.2 The elastic tensor

The matrix entries of the elastic tensor are specific of any material. They are usually
referred to as its elastic (or stiffness) constants: they determine, within the
approximation of linear elasticity, the overall response of a solid to (small)
deformation actions.
The symmetry properties of the strain and stress tensors, as well as equations
(5.19) and (5.22), suggest we use the Voigt compact notation replacing second-rank
and fourth-rank tensors with a vector and a square matrix, respectively. More

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 Solid State Physics

specifically, by considering the six independent components of the strain and stress,
the pairs of Cartesian indices i , j = 1, 2, 3 are grouped as8
T11 → T1 T22 → T2 T33 → T3 T12 → T4 T23 → T5 T13 → T6
(5.24)
ϵ11 → ϵ1 ϵ22 → ϵ2 ϵ33 → ϵ3 2ϵ12 → ϵ4 2ϵ23 → ϵ5 2ϵ13 → ϵ6.

In this way, the stress–strain equation (5.18) is written in the compact matrix form
⎛T1 ⎞ ⎛C11 C12 C13 C14 C15 C16 ⎞
⎜T ⎟ ⎜C ⎟ ⎛ ϵ1 ⎞
⎜ 2 ⎟ ⎜ 12 C22 C23 C24 C25 C26 ⎟ ⎜ ϵ2 ⎟
⎜T3 ⎟ ⎜C13 C23 C33 C34 C35 C36 ⎟ ⎜ ϵ3 ⎟
⎜T4 ⎟ = ⎜C14 C24 C34 C 44 C 45 C 46 ⎟ ⎜ ϵ4 ⎟ , (5.25)
⎜ ⎟ ⎜ ⎟ ⎜ϵ ⎟
⎜T5 ⎟ ⎜⎜C15 C25 C35 C 45 C55 C56 ⎟ ⎜ 5⎟
⎟ ⎝ ϵ6 ⎠
⎝T6 ⎠ ⎝C16 C26 C36 C 46 C56 C66 ⎠

which is a very effective way to highlight the fact that there are only 21 independent
components of the elastic tensor, once its symmetry properties are properly taken into
account.
Their number is further reduced if we take into consideration the symmetry of the
underlying crystal lattice [3, 4]. In other words, the number of independent elastic
constants varies according to the crystal system actually considered: the higher the
symmetry of the lattice, the fewer of them are independent. For instance, it is found
that the triclinic, monoclinic, orthorombic, and cubic lattice (order of increasing
symmetry) have, respectively, 21, 13, 9, and 3 independent stiffness constants. In this
latter case we write their matrix as
⎛C11 C12 C12 0 0 0 ⎞
⎜ ⎟
⎜C12 C11 C12 0 0 0 ⎟
⎜C C C 0 0 0 ⎟
Voigt = ⎜ 12 12 11 , (5.26)
0 0 0 C 44 0 0 ⎟
⎜ ⎟
⎜⎜ 0 0 0 0 C 44 0 ⎟
⎟
⎝ 0 0 0 0 0 C 44 ⎠

where we added the label ‘Voigt’ to make it clear that this matrix representation is
only valid in the compact notation.
Because of its linearity, the stress–strain equation (5.18) can be straightforwardly
inverted
ϵij = ∑ Dijkh Tkh, (5.27)
kh

where the elastic compliance constants Dijkh are the matrix entries of the inverse
elastic tensor. For a cubic crystal, it is easy to prove that

8
Unfortunately, the compact notation is not always defined by the same grouping convention. This calls for
some care when comparing the elastic equations reported in different textbooks.

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 Solid State Physics

Table 5.1. Room temperature stiffness constants (in 1011 N m−2 units) for some cubic crystalline solids with
metallic, covalent, or ionic bonding.

Metallic Covalent Ionic

a
Al Cu Au Fe Pb C Si Ge NaCl LiF KBr

C11 1.07 1.68 1.92 2.34 0.49 10.76 1.66 1.28 0.49 1.12 0.34
C12 0.61 1.21 1.61 1.36 0.42 1.25 0.64 0.48 0.12 0.46 0.05
C44 0.28 0.75 0.42 1.18 0.15 5.76 0.80 0.68 0.13 0.63 0.05
a
Diamond crystalline structure.

1
D44 =
C 44
C11 + C12
D11 = (5.28)
(C11 − C12 )(C11 + 2C12 )
C12
D12 = − .
(C11 − C12 )(C11 + 2C12 )

In table 5.1 we report the elastic constants for some cubic crystalline systems.

5.2.3 Elasticity of homogeneous and isotropic media

We are now going to investigate the elasticity of a homogeneous and isotropic solid,
that is a system where (i) the elastic constants are just the same everywhere and (ii) its
elastic response is just the same along any direction. More formally, homogeneity
and isotropicity are found whenever the elastic tensor is invariant upon translations
and rotations. Although these features define a somewhat idealised situation, this
case study is paradigmatically important and leads to very general results which, in
fact, can be widely applied in practice.
By choosing a frame of reference where the stress tensor is diagonal (see section
5.1.3) and considering a uniaxial traction along the x1 axis, it is empirically found
that the system response is twofold: (i) it stretches along the x1 direction and (ii) it
shrinks in the (x2, x3) plane. Since the only non-zero stress component is T11* , we can
formalise the observed phenomenology by defining the following strain tensor
1 ν ν
ϵ11* = + T11* * = − T*
ϵ22 11
* = − T*
ϵ33 11 ϵ12* = ϵ23
* = ϵ * = 0,
31 (5.29)
E E E
where the two E and ν constants are introduced in the proposed combination for
further convenience. It is important to remark that just two constants are in fact
needed to fully accomplish with this elastic problem since we must only describe the
observed material stretching along x1 and its corresponding shrinking in a normal
plane. We summarise the physical situation by saying that a homogeneous and
isotropic linear elastic medium has only two independent elastic moduli, namely E and ν
which are known as the Young modulus and the Poisson ratio, respectively. In table 5.2
we report their value for some elemental crystalline system.

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 Solid State Physics

Table 5.2. The Young modulus E, Poisson ratio ν, first λ and second μ Lamé coefficients for some elemental
metallic solids.

Al Fe Cu Zn Sn W Pb

E [GPa] 70 211 130 108 50 411 16

ν 0.35 0.29 0.34 0.25 0.36 0.28 0.44
λ [GPa] 34.3 82.4 28.3 57.1 98.9 192.9 11.7
μ [GPa] 26.9 82.4 46.2 38.1 16.1 151.6 6.0

If a uniaxial stress is similarly applied along the three axes we straightforwardly

generalise equation (5.29) as follows

* =
1
ϵkk [(1 + ν )Tkk
* − ν T* + T* + T* ]
( 11 22 33)
E (5.30)
ϵij* = 0 with i ≠ j ,

which is convenient to write in matrix formalism

1
* = [(1 + ν ) * −ν Tr ( *)  ], (5.31)
E
where we have introduced the notation * = {ϵij*}i, j = 1,2,3 . In the attempt to generalise
this stress–strain equation, let us introduce the rotation matrix  transforming the
frame of reference where the strain and stress tensors are diagonal into a generic one;
accordingly, since
 = t *  and  = t  *  , (5.32)
we calculate that9

 = t { 1
E
[(1 + ν ) * −ν Tr ( *)  ]  }
1 (5.33)
= [(1 + ν ) t * − ν Tr ( *) t  ]
E
1
= [(1 + ν )  − ν Tr ( *)  ].
E
We must invert this equation in order to cast the stress–strain relation in the typical
form given in linear elasticity by equation (5.18); this leads to
E ν
 = + Tr ( )  . (5.34)
1+ν 1+ν

9
In this derivation we made of the identity Tr( *) = Tr( ) which is easily proved by considering the following
results of matrix algebra: t =  and Tr( ) = Tr(  ) for any pair of matrices  and  .

5-11
 Solid State Physics

The trace of the stress tensor is easily calculated from equation (5.33)
E
Tr ( ) = Tr ( ), (5.35)
1 − 2ν
so that
E νE
 = + Tr ( )  . (5.36)
1+ν (1 + ν )(1 − 2ν )
By defining the two Lamé coefficients (whose values for some materials are reported
in table 5.2)
E
μ= (5.37)
2(1 + ν )

νE
λ= , (5.38)
(1 + ν )(1 − 2ν )
we eventually write
⎛ ⎞
 = 2μ + λTr( )  and Tij = 2μϵij + λ⎜⎜∑ϵkk⎟⎟δij , (5.39)
⎝ k ⎠

which represent the linear elastic stress–strain equation for a homogeneous and
isotropic medium in matrix (left) or scalar (right) form. We derived it by a (robust)
phenomenological approach, but it can be derived even more formally [3, 4].

5.3 Elastic moduli

We have so far introduced four different elastic parameters, namely the Young
modulus, the Poisson ratio and the two Lamé coefficients. We previously introduced
also the bulk modulus B when we investigated thermal expansion in section 4.2.1.
Since B was defined as the inverse of the isothermal compressibility (see appendix C),
that is it deals with volume variations, it is expected to be related to some elastic
parameter. In order to elucidate this issue, let us consider a hydrostatic stress
Tij = Pδij (where P is the macroscopic hydrostatic pressure) and insert it into the
constitutive equation (5.39) so as to get
1 1
= P .
3 λ + 2μ (5.40)
3
The connection with equation (C.11) is established by defining
2
B=λ+ μ, (5.41)
3

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 Solid State Physics

so that
1 ΔV P
=
3B
P  → Tr ( ) = ∑ ϵii = V
= ,
B
(5.42)
i

which leads to the following definition

1 1 ΔV
= , (5.43)
B V P
representing the finite difference counterpart of equation (C.11). This result
reconciles the thermodynamical and elastic treatment of deformations affecting
the system volume and it allows us to recast the stress–strain relation of a
homogeneous and isotropic linear elastic medium in the form
⎛ 2 ⎞
 = 2μ + ⎜B − μ⎟ Tr( ) 
⎝ 3 ⎠
(5.44)
⎡1 ⎤ ⎡ 1 ⎤
= 3B⎢ Tr( )  ⎥ + 2μ⎢ − Tr( )  ⎥ ,
⎣3 ⎦ ⎣ 3 ⎦

where the first and second term on the right-hand side are, respectively, named
spherical part and deviatoric part of the stress tensor: they describe the hydrostatic
volume variation and the change in shape of the solid body subject to  .
If we represent equation (5.44) in its matrix form and use the compact Voigt
notation we immediately get
⎛B + 4μ/3 B − 2μ/3 B − 2μ/3 0 0 0⎞
⎜ ⎟
⎜B − 2μ/3 B + 4μ/3 B − 2μ/3 0 0 0⎟
⎜B − 2μ/3 B − 2μ/3 B + 4μ/3 0 0 0⎟
 = Voigt =⎜ ⎟ , (5.45)
⎜ 0 0 0 μ 0 0⎟
⎜ 0 0 0 0 μ 0⎟
⎜ ⎟
⎝ 0 0 0 0 0 μ⎠

which, by comparison to equation (5.26), leads to the equation

2C 44 = C11 − C12 (5.46)
known as the isotropy condition, indeed a very useful result to assess to what extent a
cubic crystal can be modelled as an isotropic elastic continuum. By inspection of
table 5.1 we understand that a cubic crystal can be considered as elastically isotropic
only in a first approximation. Instead, amorphous solids and polycrystals (see
appendix C) fulfil equation (5.46) much more accurately and, therefore, they can be
considered as the real counterpart of the ideal case of isotropic medium.
The stress–strain equation (5.44) has been written in terms of the bulk modulus
and second Lamé coefficient. We can, however, write it in several equivalent ways by
using any pair of elastic coefficients. In table 5.3 we report how to express any elastic
coefficient in terms of any pair of the remaining parameters.

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 Solid State Physics

Table 5.3. Correspondence among the elastic moduli of linear elasticity.

(λ , μ ) (B , μ ) (μ , ν ) (E , ν ) (E , μ )
2 2μν νE μ(E − 2μ)
λ B− 3
μ 1 − 2ν (1 + ν )(1 − 2ν ) 3μ − E

μ E
2(1 + ν )

3λ + 2μ 2μ(1 + ν ) E Eμ
B 3(1 − 2ν ) 3(3μ − E )
3 3(1 − 2ν )

E μ(3λ + 2μ) 9Bμ 2(1 + ν )μ

λ+μ 3B + μ

λ 3B − 2μ E − 2μ
ν 2(λ + μ) 2(3B + μ) 2μ

5.4 Thermoelasticity
We have so far implicitly assumed that the deformations are imposed to the system
at zero temperature. While this assumption was useful to define a clean purely-elastic
problem, we must duly generalise our theory to include stress actions applied at
T > 0 K as well [6].
The starting point is of course the energy balance stated by the first law of
thermodynamics (see equation (C.5)) which for a system with volume V in
equilibrium at temperature T under some elastic action is written as
dU = V ∑ Tijdϵij + T dS, (5.47)
ij

where the mechanical work contributing to the internal energy U has been written in
terms of the stress tensor since we know that this latter describes any possible kind of
volume and shape variation of the system. It is easy to reconcile equation (5.47) with
the standard thermodynamical formulation by simply considering the case of a
hydrostatic stress Tij = −P δij , where P is the applied pressure whose negative sign
indicates that the mechanical action is compressive. We assume it to operate quasi-
statically, like anywhere else in the remaining of this chapter. By inserting the
hydrostatic stress into equation (5.47) we get

dU =V ∑ ( −Pδij )dϵij + T dS
ij

= − PV ∑ dϵii + T dS
i (5.48)
dV
= − PV + T dS
V
= − PdV + TdS ,

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 Solid State Physics

consistently with the first law of thermodynamics. Equation (5.47) is valid for any
arbitrary deformation and, therefore, it allows for a thermodynamical definition of
the stress tensor

1 ∂U 1 ∂F
Tij = = , (5.49)
V ∂ϵij S
V ∂ϵij T

where F = U − TS is the Helmholtz free energy, corresponding to the work

exchanged quasi-statically during the constant-temperature deformation (see
appendix C). This result defines the next task to accomplish, namely: deriving
the explicit dependence F = F(ϵij ).
Under the adopted assumption of small deformations we can expand the
Helmholtz free energy in powers of the strain

∂F 1 ∂ 2F
F = F0 + ∑ ∂ϵij
ϵij +
2
∑ ∂ϵij ∂ϵkh
ϵij ϵkh + ⋯
ij T ijkh T
(5.50)
1 ∂ 2F
=
2
∑ ∂ϵij ∂ϵkh
ϵij ϵkh ,
ijkh T

where F0 is the free energy of the unstrained system at temperature T. As already

done in many frameworks, we stopped the expansion to the second order10 and
gauged the free energy such that F0 = 0 with no loss of generality. Furthermore, we
made use of the fact that at equilibrium there are no deformations or, equivalently,
we set to zero the linear term in the power expansion thanks to equation (5.18).
We have come to an important result: in the small deformation approximation, the
Helmholtz free energy is quadratic in the strain tensor. Since such an energy is of
course a scalar quantity, we can easily guess its strain dependence: the mathematical
expression for F can only contain terms depending on [Tr()]2 or [Tr(2 )]. We
accordingly write
λ
F(ϵij ) = ∑ ϵii ϵjj + μ ∑ ϵij ϵij , (5.51)
2 ij ij

where the two Lamé coefficients are conveniently used to define the arbitrary weights of
the two terms What it is really remarkable, however, is that we have formally proved that
a linear elastic homogeneous and isotropic medium has only two independent elastic
moduli, as anticipated by a phenomenological argument in section 5.2.3.
Equation (5.51) contains some far-reaching results. In the first instance, it dictates
that the elastic moduli must obey some fundamental conditions. We known that the
Helmholtz free energy is minimum at equilibrium (see appendix C); on the other
hand, if we consider the zero-strain configuration (that is we set ϵij = 0 for any i , j in
equation (5.51)) we find F = 0. By combining these elementary observations, we

10
Once again, this is basically equivalent to the harmonic approximation of lattice dynamics.

5-15
 Solid State Physics

conclude that F is a positive-definite function. Accordingly, if we consider two

specific deformations such that, respectively, ϵii = 0 and ϵij = ϵδij (ϵ any real
number), we easily get two conditions on the Lamé coefficients, namely: μ > 0
and 3λ + 2μ > 0. By using the correspondences reported in table 5.3 we immediately
obtain the noteworthy constraints on the bulk modulus B > 0 and Young modulus
E > 0 which have an obvious physical interpretation: under a hydrostatic compres-
sion or an uniaxial tensile stress the volume and the length of a solid decreases or
increases, respectively. The case of the Poisson ratio is more intriguing: since
3B − 2μ
ν= , (5.52)
2(3B + μ)
we easily obtain that −1 < ν < 1/2. A negative Poisson ratio value implies that upon
stretching a solid body along the x1 direction, its section on the (x2, x3) plane
increases. While this is not observed in most natural materials, this behaviour is
observed in auxetic systems because of their internal structure and the way it replies
to an uniaxial load: if stretched, they become thicker perpendicular to the tensile
force. The auxetic behaviour is observed in macroscopically-engineered structures
such as that shown in figure 5.4, as well as in some molecules (like paraffin, an
acyclic chain hydrocarbon) or solid state systems (like cristobalite, a SiO2 poly-
morph generated at very high temperature).
Next we observe that equation (5.51) provides the thermodynamical foundation of
the linear elastic constitutive relation. In order to prove this, let us more conveniently
rewrite the Helmholtz free energy as (the correspondences reported in table 5.3 are
used)
1 ⎛ 1 ⎞
(ϵij ) = B ∑ ϵiiϵjj + μ ∑⎜⎝ϵij ϵij − ϵiiϵjj ⎟ , (5.53)
2 3 ⎠
ij ij

Figure 5.4. A mass-spring auxetic structure made with negative Poisson ratio: upon uniaxial loading (red), a
transverse expansion (blue) is observed.

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 Solid State Physics

and use the stress definition given in equation (5.49)

1 ∂
Tkh =
V ∂ϵkh
T

1 ⎛ ∂ϵ ∂ϵjj ⎞
= B ∑⎜ ii ϵjj + ϵii ⎟
2 ij ⎝ ∂ϵkh ∂ϵkh ⎠
(5.54)
⎛ ∂ϵij 1 ∂ϵii 1 ∂ϵjj ⎞
+ μ ∑⎜2 ϵij − ϵjj − ϵii ⎟
ij ⎝ ∂ϵkh 3 ∂ϵkh 3 ∂ϵkh ⎠

⎛ 2 ⎞⎛ ⎞
=2μϵkh + ⎜B − μ⎟⎜⎜∑ ϵii⎟⎟ δkh,
⎝ 3 ⎠⎝ i ⎠

which, as anticipated, represents the counterpart of the stress–strain relation given in

equation (5.44).
In concluding this section, we aim at defining the general framework to deal with
deformations occurring at variable temperature. This class of phenomena includes
two different situations: either a deformation induced by a temperature variation11
or a change in temperature generated by a deformation caused by some mechanical
action. Their formal theory is not trivial at all and it is developed in more advanced
textbooks. Here we rather follow a phenomenological argument. The key point is to
work out a model describing the coupling between temperature and strain: to this
aim, we will rely on the general assumption of small deformations and, furthermore,
we will consider just small temperature variations ΔT . These approximations make
it physically sound to consider a coupling term linear in both temperature and strain.
By following the same reasoning that led to equation (5.51), we understand that such
a coupling term can only depend on strain through its trace12. All together these
arguments suggest writing the Helmholtz free energy as13

1 ⎛ 1 ⎞ ⎛ ⎞
F= B ∑ ϵii ϵjj + μ ∑ ⎜⎝ϵijϵij − ϵii ϵjj ⎟ − Bβ ⎜⎜∑ ϵii ⎟⎟ ΔT (5.55)
2 ij 3 ⎠ ⎝ i ⎠
ij

where for further convenience we replaced in equation (5.51) the two Lamé
coefficients with the (B, μ) pair (see table 5.3). The last term on the right-hand
side represents the combined effect of the strain field and temperature variation. The
corresponding coupling coefficient has been conveniently written as the product
between the bulk modulus B and the thermal expansion coefficient β, as proved by
the following simple argument. Let us consider a free thermal expansion, occurring
in absence of any applied stress: when the system has equilibrated after expansion, it

11
This is the thermal expansion phenomenon, already discussed in section 4.2.1.
12
More formally: since the strain tensor is a second-rank symmetric tensor, we can build out of its elements
only one scalar term, namely Tr( ).
13
As previously, we gauged to zero the free energy of the unstrained system at temperature T.

5-17
 Solid State Physics

reaches a new stress-free Tij = 0 configuration for which, according to equation

(5.49), we can write
⎛ ⎞ ⎡ ⎛ ⎞ ⎤
∂F 1
Tij = 0 = = B ⎜⎜∑ ϵii ⎟⎟δij + 2μ ⎢ϵij − ⎜⎜∑ ϵii ⎟⎟δij ⎥ − Bβ ΔT δij , (5.56)
∂ϵij ⎝ i ⎠ ⎣⎢ 3 ⎝ i ⎠ ⎦⎥

and by considering the diagonal element i = j we easily obtain14

⎛ ⎞
0 = B ⎜⎜∑ ϵii ⎟⎟ − B β ΔT , (5.57)
⎝ i ⎠

or equivalently

1 ⎛ ⎞ 1 ΔV
β= ⎜⎜∑ ϵii ⎟⎟ = , (5.58)
ΔT ⎝ i ⎠ V ΔT

which in fact represents the finite difference counterpart15 of equation (C.9).

References
[1] Gurtin N E 1981 An Introduction to Continuum Mechanics (New York: Academic)
[2] Asaro R and Lubarda V 2006 Mechanics of Solids and Materials (Cambridge: Cambridge
University Press)
[3] Love A E H 2002 A Treatise on the Mathematical Theory of Elasticity (New York: Dover)
[4] Atkin R J and Fox N 1980 An Introduction to the Theory of Elasticity (New York: Dover)
[5] Novozhilov V V 1999 Foundations of the Nonlinear Theory of Elasticity (New York: Dover)
[6] Weiner J H 2002 Statistical Mechanics of Elasticity (New York: Dover)

14
Consistently with the assumed isotropic elastic behaviour, we have ϵii − (∑i ϵii )/3 = 0 .
15
In this case the constant-pressure condition required by the thermodynamic definition of the thermal
expansion coefficient is rigorously fulfilled, since the system is supposed to be stress-free before and after the
expansion.

5-18
 Part III
Electronic structure
 IOP Publishing

Solid State Physics

A primer
Luciano Colombo

Chapter 6
Electrons in crystals: general features

Syllabus—We establish the conceptual framework and some general results useful in
addressing the electronic structure of crystalline solids. In particular, based on the
general discussion developed in chapter 1, we outline three different approaches based
on the single-particle approximation, namely the free electron theory, the tight-binding
theory and the density functional theory, which will be thoroughly discussed in the
following chapters. Next, we elaborate the Fermi–Dirac distribution law (describing
the probability that a given electron energy level is occupied), the Bloch theorem
(defining the most general form of the one-electron crystalline wavefunction), and the
general features of the energy spectrum of an electron which is subject to a periodic
potential. This will naturally lead to the concept of allowed energy bands and forbidden
energy gaps which is the essence of the band theory.

6.1 The conceptual framework

In developing an atomistic and quantum picture of the solid state we had to admit
that it is necessary to adopt approximations, even very drastic ones as thoroughly
reported in section 1.3. As far as electrons are concerned, this has led to equation
(1.15) which defines the constitutive eigenvalue problem for the total crystalline
electron wavefunction1 Ψ (eR)(r): by solving it, we ultimately define the electronic
structure of the crystal.
Despite the remarkable number of adopted simplifications already introduced2,
equation (1.15) still represents a formidably complicated many-body quantum
problem. The search for general methods able to solve it, either analytically or
numerically, represents an advanced topic in theoretical condensed matter physics,
ranging from single-particle theories to many-body formalisms to quantum field

1
Nomenclature, notation, and symbols hereafter used have been previously defined in chapter 1.
2
Namely: the semi-classical, frozen-core, non-magnetic, non-relativistic, and adiabatic approximations which
have been introduced, motivated, and discussed in section 1.3.

doi:10.1088/978-0-7503-2265-2ch6 6-1 ª IOP Publishing Ltd 2021

 Solid State Physics

theories [1–6]. These methods are currently the topic of active research and mostly fall
well beyond the level (and the scope) of this introductory textbook. In our attempt to
elaborate a more elementary theory we will follow a simplified approach, which
basically consists in two logical steps: first, we will root our theory in the single-particle
approximation; second, we will take profit from some materials-specific characteristics
so as to further reduce the mathematical complexity of the physical problem.
The first step was already introduced in section 1.4.1: electron–ion and electron–
electron interactions are replaced by the crystal field potential Vcfp(r), namely a local
one-electron potential with the same periodicity of the underlying crystal lattice and
effectively describing the most relevant many-body features. Hereafter, we will
assume that such a potential is known, for instance through a self-consistent
calculation, as outlined in section 1.4.1. Its practical implementation is an advanced
topic of computational solid state theory [1–3].
The second step is by its nature less general and takes a different form depending on
which class of solids, either metals or insulators, is considered. Let us start by
considering metals since, while most of ordinary solids are insulators, they deserve
special attention because of their unique physical properties. In addressing a metal we
will draw the very drastic approximation to completely overlook ions. In other words,
we will neglect any effect due to the lattice on the electron system. This is indeed a
much stronger approximation than the adiabatic one, since it basically states that
electrons do not undergo the action of any external potential3: they are considered as
free particles. In addition, the problem is further simplified by neglecting as well the
internal interactions among electrons: they are further treated as independent particles.
There is no need, therefore, to introduce a suitable crystal field: electron–electron
Coulomb interactions as well as exchange and correlation effects will not be explicitly
taken into account and, accordingly, we will set Vcfp(r) = 0 anywhere within the metal.
The resulting picture is referred to as the free electron theory4. We will make use of the
free electron theory in chapter 7 to describe the main features of metals: basically, their
system of conduction electrons will be treated as a gas of free and independent
quantum particles, confined within the volume of the solid. The validity of the model
will be heuristically proved by the very good agreement between its predictions on
thermal and electric transport in metals and the experimental data. Also, through this
model we will introduce in a very simple way some fundamental concepts largely used
in electronic structure theory. In this respect, the free electron theory is still a mostly
useful pedagogical tool.
In the opposite case of insulating solids, valence electrons will be treated as a
system of single-particles, individually subject to the action of the external crystal field
potential Vcfp(r) and the corresponding eigenvalue problem will be formulated as in
equation (1.22). The true existence of such a periodic potential allows the internal

3
In this case the external potential would be due to the electron–ion Coulomb interactions.
4
Historically, this was the first model developed by scientists in addressing the electronic structure of
crystalline solids [7, 8]: it was originally elaborated by Drude in 1900 in order to explain the electrical
conduction in metals and it was entirely based on the classical kinetic theory of gases; next, it was generalised
by Sommerfeld so as to include quantum features.

6-2
 Solid State Physics

volume of the crystal to be periodically separated into core regions and interstitial
regions. With reference to figure 6.1, the former ones correspond to the crystal
portions nearest to the ion positions5: here Vcfp(r) is expected to be very similar to the
potential generated by a single isolated atom. Correspondingly, within each core
region the crystalline wavefunction is expected to be very similar to an atomic
wavefunction centred on the selected lattice site. On the other hand, in the interstitial
regions Vcfp(r) is substantially different from the atomic case6 and, therefore, it is
quite reasonable to state that here the crystalline wavefunction is no longer atomic-
like: as a matter of fact, in these crystal portions there is naturally expected to be
some overlap among atomic wavefunctions centred on neighbouring lattice sites. In
brief, we can summarise the physical situation by saying that (i) some overlap
between atomic wavefunctions does occur, but (ii) it is not strong enough to fully
invalidate the atomic description of electron quantum states. This argument, as
qualitative as nevertheless firmly rooted in some fundamental quantum features of
the single-particle picture, suggests that in a non-metallic solid7 the electron
wavefunction can be cast in the form of a linear combination of localised atomic
orbitals. This is the essence of the tight-binding theory, which for obvious reasons is
alternatively referred to as the linear combination of atomic orbitals (LCAO) method.
We will make use of the tight-binding theory in chapter 8 to describe the main
features of insulators; in particular, we will rely on it in investigating the so-called
‘band structure’ of semiconductor materials (chapter 9): a class of non-metallic
systems with several intriguing physical properties, making them of paramount
importance for technological applications.

Figure 6.1. Schematic representation of the crystal field potential Vcfp(r) acting on a valence electron in a one-
dimensional solid. Core and interstitial regions are shown: the darker the grey shadowing, the more atomic-like
is the character of the electron wavefunction.

5
For further convenience, we have gauged to zero the topmost value of Vcfp(r) in the interstitial regions. This
convention will be fully exploited in section 6.4.
6
In these regions the net potential is given by the overlap of the tails of the single-atom potentials centred on
the various lattice sites.
7
In fact, it is also valid to describe the electronic structure of transition metals, like Ni or Cu, with partially
filled d-subshells [9].

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 Solid State Physics

A third completely different approach is indeed possible, no longer based on the

knowledge of the single-electron wavefunctions (either free-particle-like or atomic-
like) but rather on the electron density [10]. This approach, named density functional
theory, entirely rests on two fundamental quantum mechanical theorems, respec-
tively, stating that (i) the ground-state energy of an electron system is a unique
functional of the electron density and that (ii) the density minimising the total energy
of the system is actually the true electron density than one would calculate by solving
the full Schrödinger problem. This theory fully incorporates any kind of interactions
ruling over the physics of crystalline electrons, also including exchange and
correlation effects. Once the electron density is known, most of the ground-state
properties of a crystal can be calculated, sometimes straightforwardly and some-
times by more sophisticated methodologies. The density functional theory is
considered as the ‘standard model’ of modern solid state physics since it is not
only more fundamental and superior to the free electron or tight-binding theories,
but also because it can be implemented in a totally parameter-free way and it
provides the most accurate predictions on the electronic structure properties. This
theory, which is in fact an advanced topic of solid state physics, will be outlined in
chapter 10.
In the remaining part of this chapter we discuss some general results that will
represent as many cornerstones for the electronic structure theory of crystalline
solids. We will not refer to any specific kind of solids and we will instead develop our
arguments at a very general level.

6.2 The Fermi–Dirac distribution function

Let us consider a gas of N electrons, confined within a volume V in equilibrium at
temperature T. Quantum mechanics provides the energy spectrum of this system, as
we will extensively discuss in chapters 7 and 8; we label by Ei with i = 1, 2, 3, … the
energies of the single-electron levels8. Hereafter in this section we assume that such
energies are known and ask ourselves how likely each level is to be occupied.
Let us assume that two energy levels E1 e E2 are occupied with probability
nFD(E1, T ) and nFD(E2, T ), respectively. Since the system is in equilibrium, on
average the number of electrons undergoing the transition E1 → E2 in the unit time
equals the number of electrons undergoing the inverse transition E2 → E1. This
statement represents a specific formulation of the more general principle of micro-
reversibility9. Such transitions could be generated by any kind of mechanism, like
electron–electron or electron-defect scattering.

8
We remark that this simplified labelling convention is only valid to count the different states; more rigorously,
they should be indicated as E mi as discussed in section 1.4.1. However, for the present discussion the simplified
notation is sufficient and will not generate any ambiguity.
9
This is tantamount to stating that in the guessed equilibrium condition, each elementary electron process is as
equally probable as its reverse process [11]: since microscopic physical laws are ultimately invariant under time
reversal, a transition between two microscopic quantum states and its inverse transition are equally probable.
Microreversibility is, in turn, the foundation of the detailed balance principle [11, 12] which is widely used in
statistical mechanics.

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 Solid State Physics

The number of electrons undergoing the transition E1 → E2 per unit time is

calculated as the product between the number of electrons initially occupying the
level E1 and the rate R1→2 of occurrence of such transition. The initial number of
electrons is in turn given by the product between the total number of particles N and
the probability that the initial energy level is occupied. Similar definitions hold for
the inverse transition. If electrons were classical particles, the occupation probability
of any given level with energy E would be proportional to the Boltzmann factor
exp( −E /kBT ) [11] and, therefore, we could write the classical version of micro-
reversibility as
exp( −E1/ kBT ) R1classical
→2 = exp( −E2 / kBT ) R 2classical
→1 , (6.1)
where R1classical
→2 and R1classical
→2 are the transition rates calculated according to classical
physics. Since, however, electrons are identical and indistinguishable quantum
particles they must obey the Pauli principle (see appendix E): two electrons cannot
have the same set of quantum numbers. This suggests the most general way to switch
from classical to quantum transition rates
R1quantum
→2 = R1classical
→2 [1 − nFD(E2, T )] and R 2quantum
→1 = R 2classical
→1 [1 − nFD(E1, T )], (6.2)

where the terms in square parenthesis [⋯] define the probability that the final state is
not previously occupied. Quantum microreversibility is straightforwardly cast in the
form
nFD(E1, T ) R1quantum
→2 = nFD(E2, T ) R 2quantum
→1 , (6.3)
which, by inserting equation (6.1), leads to
R1classical
→2 exp( −E2 / kBT ) n (E , T ) 1 − nFD(E1, T )
classical
= = FD 2 , (6.4)
R 2→1 exp( −E1/ kBT ) nFD(E1, T ) 1 − nFD(E2, T )
or equivalently
1 − nFD(E1, T ) 1 − nFD(E2, T )
exp( −E1/ kBT ) = exp( −E2 / kBT ). (6.5)
nFD(E1, T ) nFD(E2, T )
Since this result applies to each arbitrary pair of energies E1 and E2, we logically
draw the conclusion that the two expressions appearing on the left and right side of
this equation must be equal to an identical function, which only depends on
temperature. Following the standard convention [11, 13, 14], we name this function
exp( −μc /kBT ) where μc is the chemical potential. For any arbitrary value of the
electron energy E we obtain
1 − nFD(E , T )
exp( −E / kBT ) = exp( −μc / kBT ), (6.6)
nFD(E , T )
which leads to
1
nFD(E , T ) = , (6.7)
1 + exp[(E − μc )/ kBT ]

6-5
 Solid State Physics

describing the probability that the quantum level E is occupied when the electron
system is in equilibrium at temperature T. As indicated by the subscript FD this
probability corresponds to the Fermi–Dirac distribution law obtained in appendix E
by means of a very general statistical argument: this provides evidence that electrons
are fermions, consistently with the fact that they have half odd spin ℏ/2.
By comparing equation (6.7) with equation (E.11), we realise that apparently only
the latter contains information about the degeneracy of the single-particle quantum
level. This could be confusing at first glance and, therefore, it deserves a further
comment. In non-magnetic approximation, spin-related interactions are strictly
speaking neglected: electron energies are independent of the spin (in the absence of
an external magnetic field) and the Pauli principle allows up to two electrons to
occupy the same single-particle quantum state, provided that they have opposite
spin. This corresponds to a twofold degeneracy. On the other hand, if spin-related
interactions are included, each single-particle level can only be occupied by a sole
electron. In conclusion, equation (6.7) must be understood as the occupation
probability of spin-resolved single-particle energy levels. Instead, a factor 2 should
be added in case the spin is fully neglected, in order to properly take into account
that each level with energy E actually represents two possible states, one for each
electron spin polarisation.
It is easy to recognise that the chemical potential μc appearing in equation (6.7)
corresponds to the Lagrange multiplier introduced in appendix E to enforce the
conservation of the total number of electrons; by the arguments developed in
appendix C, μc is therefore understood as the thermodynamical parameter ruling
over the chemical equilibrium condition between systems of electrons: if two electron
reservoirs are put into contact, then some electrons will flow from one to the other
until the chemical potentials of the two systems are the same. Its physical meaning is
so clarified by thermodynamics: μc provides the work needed to add one electron to the
system at constant temperature, or equivalently:
μc (T ) = F(T , V , N + 1) − F(T , V , N ), (6.8)

where F(T , V , N ) is the free energy of the system of N electrons.

The discussion so far developed clearly indicates that both the Fermi–Dirac
distribution and the chemical potential depend on temperature. More specifically,
μc (T ) must be adjusted10 so that at any temperature T the total number of electrons
remains constant (for an isolated system). Also, it is easy to see from equation (6.7)
that
lim nFD(E , T ) = 1 and lim nFD(E , T ) = 0, (6.9)
E →−∞ E →+∞

at any non-zero temperature; the width of the energy range over which the Fermi–
Dirac distribution is non-constant is of the order of ∼kBT . Furthermore, by
calculating the derivative

10
This procedure will be fully exploited in section 7.3.2.

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 Solid State Physics

∂nFD(E , T ) 1 exp⎡⎣(E − μc )/ kBT ⎤⎦

=− , (6.10)
∂E kBT {1 + exp[(E − μc )/ kBT ]}2

we get the maximum value of the slope of the Fermi–Dirac distribution law

∂nFD(E , T ) 1
=− , (6.11)
∂E E =μc 4kBT

proving that as the temperature decreases, nFD(E , T ) gets closer and closer to a step-
like function centred at energy E = μc . In particular, at zero temperature we have

⎧ 1 for E < μc
⎪
nFD(E , 0) = ⎨1/2 for E = μc (6.12)
⎪
⎩ 0 for E > μc ,

We can generalise this result by observing that at any finite temperature it is

always found that nFD(μc , T ) = 1/2: in other words, the chemical potential corre-
sponds to that energy value which selects the electron state occupied with 50%
probability. Since nFD(E , T ) is smoothed by increasing the temperature, the
chemical potential is correspondingly shifted, as shown in figure 6.2. The specific
temperature-dependence of μc (T ) will be addressed in chapter 7 in the framework of
the free electron theory and in chapter 9 when we will discuss the case of extrinsic
semiconductors.
In concluding this section we anticipate a nomenclature widely used in the
following: the zero-temperature value of the chemical potential is usually referred to
as the Fermi energy EF which is therefore formally defined as EF = limT →0 μc (T ).

Figure 6.2. The Fermi–Dirac distribution law nFD(E , T ) calculated at four different temperatures
T1 < T2 < T3 < T4 . The 50% occupation probability (corresponding to nFD = 1/2 ) is shown by a black dashed
line: its intercepts with the curves nFD(E , T ) define the positions of the chemical potential μc (T ) at the
different temperatures.

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 Solid State Physics

6.3 The Bloch theorem

We now derive a formal result due to the translational invariance of any crystal
lattice discussed in chapter 2.
Let us start from the single electron approximation developed in section 1.4.1,
where we proved that the Schrödinger problem given by equation (1.22) must be
solved for each crystalline electron. The electron Hamiltonian operator Ĥ (r)

ℏ2 2
Hˆ (r) = − ∇ + Vˆcfp(r), (6.13)
2m e

is obviously depending upon the position r of the particle within the crystal and,
because of the property of translational invariance of the lattice, we have

Hˆ (r) = Hˆ (r + Rl), (6.14)

as shown in figure 2.3. In order to formally treat such an invariance, it is useful to

introduce the translation operator T̂Rl whose action on a generic space function f (r) is
defined as

TˆRl f (r) = f (r + Rl). (6.15)

The translational invariance is revealed by stating that the one-electron Hamiltonian

operator commutes with the translation operator, that is: [Hˆ (r), TˆRl ] = 0. Therefore,
the solutions ψ (r) of equation (1.22) are also eigenfunctions of the translation
operator
TˆRlψ (r) = t(Rl) ψ (r), (6.16)

where the number t (Rl) is the eigenvalue of T̂Rl; it is intuitive to figure out that,
according to equation (2.1), t (Rl) depends on the set {n1, n2 , n3}. Furthermore, by
composing two translations

TˆRl ′TˆRlψ (r) = TˆRl +Rl ′ψ (r) = t(Rl + Rl ′) ψ (r), (6.17)

we understand that t (Rl + Rl ′) = t (Rl) t (Rl ′).

We assume that the electron wavefunctions have been properly normalised

∫V ∣ψ (r)∣2 d r = 1, (6.18)

over the finite volume V of the crystalline sample we are studying. Since such a
normalisation condition is not affected by translations, we conclude that ∣t (Rl)∣2 = 1
or, equivalently, that t (Rl) = exp(iα ) where α is a still undetermined real number. By
combining this result with our knowledge about the composition of t (Rl) numbers,
we understand that α must proportional to Rl and, therefore, we can write
α = k · Rl , (6.19)

6-8
 Solid State Physics

where it is straightforward to identify k with the wavevector of the matter wave that
quantum mechanically describes the crystalline electron. In solid state physics k is
looked at as the counterpart of the quantum numbers used in atomic physics to label
electronic states.
By combining the results we obtained, we can eventually state that the wave-
function of a crystalline electron must always have the following property
ψ (r + Rl) = e ik·Rl ψ (r), (6.20)
a fundamental result known as the Bloch theorem. It is really hard to underestimate
the importance of such a result: under the sole single-particle approximation and by
only exploiting the inherent symmetry property of any crystal systems (namely, its
translational invariance), we have derived a general common feature of any electron
wavefunction: a translation only affects the eigenfunction of the single-particle
Hamiltonian by a phase change.
The Bloch theorem can be cast in a different and similarly useful form. Let us
introduce a new function u k(r) such that
u k(r) = e−i k·r ψ (r). (6.21)
The translation operator transforms such a function as
TˆRlu k(r) = u k(r + Rl) = e−i k·(r+Rl) ψ (r + Rl). (6.22)

If we now make use of equation (6.20) we can write

TˆRlu k(r) = e−i k·(r+Rl) ψ (r + Rl)

= e−i k·(r+Rl) e i k·Rlψ (r)
(6.23)
= e−i k·rψ (r)
= u k(r),

which leads to the conclusion that the function u k(r) is a periodic function. By
inverting equation (6.21), we immediately obtain the noteworthy result
ψ (r) = e ik·r u k(r), (6.24)
stating that any crystalline electron wavefunction is always the product between a
plane wave and a second function with the same periodicity of the lattice.
Wavefunctions of this form are referred to as Bloch functions. This result is again
known as the Bloch theorem which, therefore, can be formulated in the two
alternative forms provided in equations (6.20) and (6.24). Even in this case the
result is of general validity since it is only based on the translational invariance of the
lattice.
The importance of the Bloch theorem will be extensively exploited in the next
chapters. However, we conclude this section by a simplified argument which should
help in catching the paradigmatic relevance of the theorem. Let us consider the form
of the theorem given in equation (6.24): a special case of periodic function is

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 Solid State Physics

when u k(r) = constant. Under this condition, the Bloch functions reduce to plane
waves: crystalline electrons described by such wavefunctions grossly behave as free
particles11. Crystals where u k(r) = constant are accordingly expected to show a
metallic behaviour. On the other hand, whenever u k(r) is not a constant, the overall
behaviour of the electron system will be qualitatively different and, therefore,
addressed as non-metallic. Although very qualitative, this argument makes it clear
that the charge transport properties of condensed matter systems are solely
explained by quantum mechanical arguments, where the actual mathematical
form of the Bloch functions plays a major role.

6.4 Electrons in a periodic potential

Just as it has been possible to obtain the general form of the wavefunction of
crystalline electrons without taking into consideration any materials-specific prop-
erty, so we are about to derive the general structure of the energy spectrum for valence
electrons by only considering the periodicity of the single-particle potential
Vˆcfp(r) = Vˆcfp(r + Rl).
To this aim we discuss the simple case of a one-dimensional monoatomic crystal
under the construction usually referred as Kronig–Penney model. The situation
sketched in figure 6.1 is further simplified by approximating the crystal field
potential with a function V (x ) which consists in a periodic sequence of potential
wells spanning the core regions, separated by finite barriers occupying the interstitial
ones. This idealised situation is represented in figure 6.3. We remark that we have for
convenience set the zero of the potential at the bottom of the wells, while a and b,
respectively, indicate the width of the wells and barriers. Therefore, the lattice
periodicity is a + b or, equivalently, in this case the lattice vectors are written as
Rl = n(a + b ) with n = 0, ±1, ±2, ±3, … (see equation (2.1)). We understand that
a < b by guessing that interstitial regions are larger than core ones12.
Thanks to the crystal periodicity, it is sufficient to solve the quantum problem of a
valence electron under the action of the Kronig–Penney potential V (x ) only for a single
pair of adjacent core and interstitial regions. With reference to figure 6.3 we write
⎧ 0 0 < x < a core region
V (x ) = ⎨ (6.25)
⎩V0 − b ⩽ x ⩽ 0 interstitial region,
so that
⎧ d 2ψ (x ) 2m
⎪
⎪ + 2 e Eψ (x ) = 0 core region
⎨ 2 dx 2
ℏ
(6.26)
⎪ d ψ (x ) 2m e
⎪ + 2 (E − V0)ψ (x ) = 0 interstitial region,
⎩ dx 2 ℏ

11
In fact, the real situation is more subtle, as discussed when treating charge transport in solids. The argument
we are here developing is just qualitative.
12
This justifies the graphical representation reported in figure 6.3, but neither the mathematics developed
below nor the general results we are going to obtain actually depend on this choice.

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 Solid State Physics

Figure 6.3. Top: schematic representation of the crystal field potential Vcfp(r) acting on a valence electron in a
one-dimensional monoatomic crystal. Bottom: the corresponding Kronig–Penney model. The function V (x )
consists in a periodic sequence of potential wells of width a centred on the lattice ions, separated by finite
barriers of thickness b and height V0 spanning the interstitial regions.

where we have indicated by ψ (x ) and E the electron wavefunction and energy,

respectively. This equation is the counterpart of equation (1.22) in a simplified
notation. By introducing the shortcuts
2m e 2m e
α2 = E and − β2 = (E − V0), (6.27)
ℏ2 ℏ2
we can rewrite equations (6.26) in a more convenient way
⎧ d 2ψ (x )
⎪ dx 2 + α ψ (x ) = 0
2
⎪ core region
⎨ 2 (6.28)
⎪ d ψ (x )
⎪ − β 2 ψ (x ) = 0 interstitial region.
⎩ dx 2

Since by construction the Kronig–Penney potential is periodic, the wavefunction

ψ (x ) must be a Bloch function or, equivalently
ψ (x ) = e ikx u(x ) with u(x ) = u(x + Rl ). (6.29)

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 Solid State Physics

By inserting this form of the wavefunction into equations (6.28) we easily obtain
⎧ d 2u(x ) du(x )
⎪ dx 2 + 2ik dx + (α − k )u(x ) = 0 core region
2 2
⎪
⎨ 2 (6.30)
⎪ d u (x ) du(x )
⎪ + 2ik + (β 2 + k 2 )u(x ) = 0 interstitial region,
⎩ dx 2 dx
whose solutions are
⎧ u1(x ) = A exp[i (α − k )x ] + B exp[ −i (α + k )x ] core region
⎨ (6.31)
⎩ u2(x ) = C exp[(β − ik )x ] + D exp[ −(β + ik )x ] interstitial region.

The matching between u1(x ) and u2(x ) at the boundaries between core and interstitial
regions is obtained as customarily in quantum mechanics by imposing the continuity
of the wavefunction and of its first derivative13; these conditions lead to the
following set of four equations
du1(x ) du2(x )
u1(x = 0) = u2(x = 0) =
dx x=0 dx x=0
(6.32)
du1(x ) du (x )
u1(x = a ) = u2(x = −b) = 2 .
dx x=a dx x=−b

By inserting equations (6.31) into the boundary conditions provided by equations

(6.32), we easily obtain an algebraic system where the unknowns are the four
coefficients A, B, C and D. The system has a non-trivial solution only if the
determinant of such coefficients vanishes, that is only if
β2 − α2
sinh(βb) sin(αa ) + cosh(βb) cos(αa ) = cos[k (a + b)], (6.33)
2αβ
which represents the fundamental equation of the Kronig–Penney model: it is a
transcendental function with no closed-form solution; it can only be solved
numerically.
In order to proceed in the most straightforward way, it is useful to replace the
physical barriers with infinitesimally-thin ones, while keeping unaffected their
strength14. This is tantamount to saying that we describe the interstitial regions by
means of barriers such that
⎧V0 → +∞
⎪
⎨ with V0b = constant, (6.34)
⎪
⎩b → 0

13
We are considering barriers with finite height: accordingly, the second derivative of the wavefunction is well
defined and it is finite. Therefore, the first derivative is a continuous function.
14
It is formally proved in quantum mechanics that the penetration through a potential barrier only depends on
its strength V0b , namely on the product between the barrier height and width [7, 15, 16].

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 Solid State Physics

This is clearly a choice of mathematical convenience, which nevertheless leaves the

physical nature of the problem unchanged. The convenience lies in the fact that,
under the conditions given in equation (6.34), we have cosh(βb ) ∼ 1, sinh(βb ) ∼ βb
and V0 ≫ E so that equation (6.33) can be replaced by
sin(αa ) m eV0 ba
P + cos(αa ) = cos(ka ) with P = , (6.35)
αa ℏ2
which represents a really interesting result: since the right-hand side of this equation
is an ordinary trigonometric cosine function, then the only allowed values of the
variable (αa ) are those for which −1 ⩽ P sin(αa )/(αa ) + cos(αa ) ⩽ +1. We graphi-
cally solve equation (6.35) as shown in figure 6.4, where we have set
F (ξ ) = P sin(ξ )/ξ + cos(ξ ) and used the shortcut ξ = αa .
The main feature of this graphical procedure is that the solutions of equation
(6.35) are grouped into intervals of allowed values, separated by intervals of
forbidden ones. They will be hereafter referred to as allowed bands and forbidden
gaps, respectively. Given the definition of α reported in equation (6.27), this leads
into a remarkably important result: the Kronig–Penney model proves that the energy
spectrum of a crystalline electron consists in allowed bands separated by forbidden
gaps. Although this result has been obtained by approximating the crystal field
potential with a periodic sequence of rectangular potential wells separated by finite
barriers, it is generally valid for any type of periodic potential: therefore, the
existence of bands and gaps will be the true dominant feature of the electronic structure
of all periodically-invariant crystalline solids.
A number of interesting features are found in figure 6.4. First of all, we remark
that the higher the energy E of the electron (or, equivalently, the larger the value of
the variable ξ), the larger are the band widths and the smaller are the extensions of
the gaps. Next, we observe that as P increases, the allowed bands become less and

Figure 6.4. Graphical solution of equation (6.35) (Kronig–Penney model). The only allowed values of the
variable ξ = (αa ) are those for which −1 ⩽ F (ξ ) ⩽ +1 with F (ξ ) = P sin(ξ )/ξ + cos(ξ ). The parameters α and
a are defined in equation (6.27) and figure 6.3, respectively. The bands of allowed ξ values are shaded in light
red colour.

6-13
 Solid State Physics

less extended; in particular, in the limit P → +∞ the bands transform into a

sequence of single energy levels: the corresponding spectrum is nothing other than
that of an electron confined within an infinitely-deep potential well. Finally, as P
decreases the allowed bands become larger and larger; in the limit P → 0 all bands
merge into an interrupted continuum of allowed states, corresponding to the free
electron case.

References
[1] Martin R M 2012 Electronic Structure–Basic Theory and Practical Methods (Cambridge:
Cambridge University Press)
[2] Grosso G and Pastori Parravicini G 2014 Solid State Physics 2nd edn (Oxford: Academic)
[3] Cohen M L and Louie S G 2016 Fundamentals of Condensed Matter Physics (Cambridge:
Cambridge University Press)
[4] Bruus H and Flensberg K 2004 Many-body Quantum Theory in Condensed Matter Physics:
An Introduction (Oxford: Oxford University Press)
[5] Inkson C J 1984 Many-body Theory of Solids–An Introduction (New York: Springer)
[6] Tsvelik A M 2002 Quantum Field Theory in Condensed Matter Physics (Cambridge:
Cambridge University Press)
[7] Eisberg R and Resnick R 1985 Quantum Physics of Atoms, Molecules, Solids, Nuclei, and
Particles 2nd edn (Hoboken, NJ: Wiley)
[8] Ashcroft N W and Mermin N D 1976 Solid State Physics (London: Holt-Saunders)
[9] Colombo L 2019 Atomic and molecular physics: a primer (Bristol: IOP Publishing Ltd)
[10] Sholl D S and Steckel J A 2009 Density Functional Theory–A Practical Introduction (New
York: Wiley)
[11] Reif F 1987 Fundamentals of Statistical and Thermal Physics (New York: McGraw-Hill)
[12] Lebon G, Jou D and Casas-Vázquez J 2008 Understanding Non-equilibrium Thermodynamics
(Berlin: Springer)
[13] Glazer M and Wark J 2001 Statistical Mechanics–A Survival Guide (Oxford: Oxford
University Press)
[14] Swendsen R H 2012 An Introduction to Statistical Mechanics and Thermodynamics (Oxford:
Oxford University Press)
[15] Miller D A B 2008 Quantum Mechanics for Scientists and Engineers (New York: Cambridge
University Press)
[16] Griffiths D J and Schroeter D F 2018 Introduction to Quantum Mechanics 3rd edn
(Cambridge: Cambridge University Press)

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 IOP Publishing

Solid State Physics

A primer
Luciano Colombo

Chapter 7
Free electron theory

Syllabus—The classical and quantum theory of the free electron gas are developed to
predict its transport (both charge and heat), thermal, and optical properties. They
provide the Drude and Sommerfeld theory of metals, respectively. In the quantum case,
we also elaborate a complete description of the electron states, entirely based on the
Fermi–Dirac statistics for fermions. Merits and failures of both models are extensively
discussed by comparing their predictions to experimental evidence. Eventually we
motivate the need of a more fundamental description of crystalline quantum electron
states, which will be developed in the next chapter.

7.1 General features of the metallic state

Metals are characterised at the macroscopic level by the ability to conduct
electricity. Phenomenologically, the charge transport properties are defined by their
resistivity which typically ranges in between 10−8 and 10−6 Ωm at T = 300 K. The
presence of impurities detrimentally affects the charge transport in these materials
and, therefore, their conductivity is typically lowered by increasing the concen-
tration of defects. Finally, the resistivity is found to decrease monotonically with
decreasing temperature1.
The metallic state is very common in Nature, since more than two thirds of the
elements are in fact good conductors. They are preferentially found on the left-hand
side of the periodic table; accordingly, their atomic ground-state configuration
typically consists in a large majority of electrons hosted by core states and just a few
others found in valence states, as shown in appendix A. The number n e of valence

1
Metals are not the only solids able to conduct electricity: semiconductors do it as well. However, their charge
transport properties are remarkably different: while their native resistivity is much larger than in metals, it can
be dramatically reduced by adding impurities, a process named ‘doping’ which enables the microelectronics
technology. Furthermore, the resistivity of a semiconductor varies with temperature in a complicated way, but
it becomes very large at low temperature: strictly speaking, at T = 0 K an undoped semiconductor is in fact an
insulator. For these reasons, the free electron theory cannot be applied to semiconductors.

doi:10.1088/978-0-7503-2265-2ch7 7-1 ª IOP Publishing Ltd 2021

 Solid State Physics

electrons per cm3 is given by the product (number of atoms per mole) × (number of
moles per cm3) × (number of valence electrons per atom) or equivalently
dm
n e = NA Zv , (7.1)
A
where d m is the mass density of the metal, while the symbols NA , Zv, and A are the
Avogadro number, the number of valence electrons per atom (chemical valence), and
the atomic mass number, respectively, previously defined in sections 1.2.1 and 1.3.2.
As reported in table 7.1 this corresponds to a typical number density of the order of 1022
electrons cm−3, which is much larger than found in any ordinary atomic or molecular
gas in normal conditions of temperature and pressure2. We can also assign a volume
per electron, which corresponds to a sphere of radius re defined so that
4 1
π re3 = . (7.2)
3 ne
If we compare the calculated values of re with the typical interatomic distances in
crystals (which are of the order of few Å), we come to the conclusions that in metals
there is plenty of room available to valence electrons. Finally, we take into
consideration that they are only weakly bound to their ion core: therefore, it is
quite reasonable to assume that, upon collecting many atoms to form the crystal,
they homogeneously delocalise throughout the interstitial regions, thus giving rise to
unidirectional metal bonds, as anticipated in figure 2.22 and related discussion.
This body of phenomenological evidence supports the idea of modelling the
conduction gas of a metal as a homogeneous gas of delocalised, free, independent,
and charged particles. Although based on very drastic approximations, this picture is
nevertheless promising to describe at least the main features of metals. The key
question is: which is the most accurate level to implement this free electron gas model?

Table 7.1. Number Zv of valence electrons, electron density n e (units 1022/cm3), radius re (units Å) of the sphere
defining the volume per electron, and room-temperature resistivity ρe (units 10−8 Ω m) for some metallic
elements. The calculated Drude relaxation time τe (units 10−14 s) is also reported, as calculated by equation (7.7).

Li Na Cu Ag Au Mg Ca Fe Zn Al Sn Pb

Zv 1 1 1 1 1 2 2 2 2 3 4 4
ne 4.7 2.6 8.5 5.9 5.9 8.6 4.6 17.0 13.2 18.1 14.8 13.2
re 1.7 2.1 1.4 1.6 1.6 1.4 1.7 1.1 1.2 1.1 1.2 1.2
ρe 9.5 4.9 1.7 1.6 2.2 4.4 3.4 10.0 6.0 2.7 11.5 21
τe 0.79 2.79 2.46 3.76 2.73 0.94 2.27 0.21 0.45 0.73 0.21 0.13

2
This fact should be read as a first indication that the system of valence electrons is not at all an ordinary gas
and that it requires the development of a specific theory.

7-2
 Solid State Physics

7.2 The classical (Drude) theory of the conduction gas

A first simple approach to the physics of the free electron gas is purely classical,
mostly based on the kinetic theory of gases [1]. In the Drude theory of the metallic
state [2–4] electrons are described as point-like charged particles, confined within the
volume of a solid specimen. The very drastic approximations of free and independ-
ent particles outlined in the previous section are slightly corrected by assuming that
electrons occasionally undergo collisions with ion vibrations, with other electrons and
with lattice defects possibly hosted by the sample; the key simplifying assumption is
that we define a unique relaxation time τe (thus averaging among all possible
scattering mechanisms) defined such that 1/τe is the probability per unit time for an
electron to experience a collision of whatever kind3. This approach is usually
referred to as the relaxation time approximation. The free-like and independent-like
characteristics of the particles of the Drude gas are instead exploited by assuming
that between two collisions electrons move according to the Newtons equations of
motion, that is uniformly and in straight lines. Collisions are further considered as
instantaneous events which abruptly change the electron velocities; also, they are
assumed to be the only mechanism by which the Drude gas is able to reach the
thermal equilibrium. In other words, the velocity of any electron emerging from a
scattering event is randomly distributed in space, while its magnitude is related to the
local value of the temperature in the microscopic region of the sample close to the
scattering place (local equilibrium).

7.2.1 Electrical conductivity

The first application of the Drude theory is to predict the direct-current electrical
conductivity of a metal. Let vd be the electron drift velocity under the action of an
externally-applied uniform and constant electric field E . The overall dynamical
effect of the collisions experienced by the accelerated electrons is described as a
frictional term in their Newton equation of motion
−e E = m e vḋ + β vd , (7.3)
where β is a coefficient to be determined. Basically, the added frictional term forces
the electron distribution to relax towards the equilibrium Fermi–Dirac one when the
external electric field is removed. In a steady-state condition we have d vd/dt = 0 and
therefore
e β
− E= vd , (7.4)
me me
which naturally4 leads to defining β = me /τe . This allows us to calculate the electron
drift velocity as

3
This notion will be reconsidered in greater detail in section 7.3.3.
4
This result directly comes from the dimensional analysis of equation (7.4): β /me must be the inverse of a time
which is straightforward to identify in the relaxation time.

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 Solid State Physics

eτe
vd = − E, (7.5)
me
from which we obtain the steady-state charge current density Jq

n ee 2τe
Jq = −n ee vd = E, (7.6)
me
and the Drude expression for the direct-current conductivity σe
n ee 2τe
σe = , (7.7)
me
which links this quantity to few microscopic physical parameters associated either
with the charge carriers (e and me ) or to the specific material (n e and τe ). The
conductivity is the inverse of the electrical resistivity ρe = 1/σe , a physical property
which is easily measured: therefore, the Drude theory allows for a direct estimation
of the order of magnitude of the relaxation time related to the charge current5 which
turns out to be as small as τe ∼ 10−14 s; its predicted value is reported in table 7.1 for
some selected metallic elements. By applying the kinetic theory to the (classical)
electron gas, we can estimate the electron thermal velocity veth by means of the
equipartition theorem6 and accordingly define the electron mean free path
λ e ∼ 1 − 10Å which represents the average distance covered by an electron between
two successive collisions. It is reassuring to get a number which is comparable with
the typical interatomic distance in a crystalline solid: this supports the robustness of
the Drude model.

7.2.2 Optical properties

Another success of the classical free electron gas theory is that it correctly predicts
the optical properties of metals, which are found to strongly reflect any electro-
magnetic radiation in the visible spectrum, while at higher frequency they are able to
absorb [5], as shown in figure 7.1 in the paradigmatic case of aluminium.
In order to estimate the optical reflectivity of a free electron gas, we need to
evaluate its frequency-dependent refractive index ϵr , where ϵr is the relative
permittivity of the metal [5]. Let E(t ) = E0 exp( −iωt ) be a time-varying and uniform
electric field applied to a metallic sample, where E0 and ω are its amplitude and
frequency, respectively. Following the same path which led to equation (7.3), we
write the electron equation of motion as
m
−e E 0 exp( −iωt ) = m e vḋ (t ) + e vd (t ), (7.8)
τe

5
The reason why we are underlying that the present τe is related to the charge current will be clear in section
7.2.3 where a second relaxation time for electrons will be introduced to discuss the heat current.
6
Assuming that the electron gas is a classical gas at equilibrium, we can write me(veth )2 /2 = 3kBT /2 since its
free and independent particles have only three translational degrees of freedom [1].

7-4
 Solid State Physics

Figure 7.1. The reflectivity of aluminium. Red dots: experimental data taken from [6]. Full blue line:
reflectivity predicted by the Drude theory for ℏωp = 15.8 eV , where ωp is aluminium plasma frequency.

where we have introduced the time dependence in vd(t ) since we understand that,
under the action of an oscillating electric field, the drift velocity of a free electron
also follows a periodic variation with the same frequency. More specifically, we
write vd(t ) = vd,0 exp( −iωt ). From equation (7.8) we easily get the drift velocity7

eτe 1
vd,0 = − E 0, (7.9)
m e 1 − iωτe

and by integration we obtain the time-dependent displacement s(t ) of the electron

t
eτe 1
s(t ) = ∫0 vd (t′)dt′ =
iωm e 1 − iωτe
E 0 exp( − iωt ), (7.10)

where with no loss of generality we have set s(0) = 0 for convenience. We can now
calculate the polarisation (that is the induced electric dipole moment per unit
volume) P = −n ee s(t ) and, through the standard relation ϵ0ϵr E = ϵ0 E + P, even-
tually obtain the relative permittivity of the metal as

n ee 2 τe 1 τ 1
ϵr = 1 − =1− e ωp2 , (7.11)
ϵ0m e iω 1 − iωτe iω 1 − iωτe

7
Incidentally, through the current density j = −n ee vd , we can also obtain the Drude expression for the
alternate-current conductivity σe(ω ) = (n ee 2τe )/me(1 − iωτe ) which, as expected, reduces to equation (7.7) in
the ω → 0 limit.

7-5
 Solid State Physics

where we have introduced the plasma frequency ωp2 = n ee 2 /ϵ0me of the electron gas.
In the limit ωτe ≫ 1 (that is, for any electromagnetic wave with frequency well within
the visible spectrum or higher) this expression reduces to
ωp2
ϵr ( ω ) = 1 − , (7.12)
ω2
which indicates that whenever ω < ωp the refractive index ϵr becomes imaginary:
therefore, the electric field becomes evanescent within the material. In other words,
we have proved that no propagation of an electromagnetic wave in a metal is possible
at frequencies below the plasma frequency. The situation is summarised by stating
that for ω < ωp a metal is totally reflecting, while for ω > ωp it is able to transmit an
electromagnetic wave. The frequency-dependent reflectivity R(ω ) is defined as [5]
2
ϵr ( ω ) − 1
R(ω ) = , (7.13)
ϵr ( ω ) + 1

and eventually calculated by using equation (7.12). The result for R(ω ) is pretty well
confirmed by the experimentally observed optical reflectivity of metals, as shown in
figure 7.1 in the case of aluminium. The plasma frequency values are easily
calculated within the Drude model by simply taking the density data from table 7.1:
they are found to be of the order of ∼10−15s−1 for most metals.
The true existence of the plasma frequency has the important consequence that a
free electron gas can sustain a charge density oscillation: it corresponds to a back-
and-forth longitudinal displacement of the electron gas as a whole with respect to
the background of the positive ions (which remain clamped at their lattice
positions), as schematically shown in figure 7.2. If the electron gas is displaced
by an amount d, then a total electric field E = en ed /ϵ0 is established within the

Figure 7.2. Top: the electron gas (light blue continuum) is displaced as a whole by d, while the ions (red dots)
are kept at their lattice positions. Bottom: an electrostatic restoring electric force is established, causing plasma
oscillations.

7-6
 Solid State Physics

sample, acting as a restoring force. The resulting equation of motion of the gas as a
whole is written as
2
Ne n e
Nm e d ̈ = −Ne E = − d, (7.14)
ϵ0
where N is the total number of electrons. This is the equation of motion of a
harmonic oscillator which vibrates at the plasma frequency ω p2 = n ee 2 /ϵ0me . The
quantum counterpart of this collective oscillation is introduced as in the phonon case
(see section 3.5): a plasmon is therefore defined as a pseudo-particle (a collective
excitation, in fact) of energy ℏωp which can be created or annihilated under
suitable conditions. Because of the typical values of the plasma frequency, plasmons
have energy of the order of several eV and, therefore, they cannot be thermally
excited; this implies that, as far as its collective longitudinal oscillations are
concerned, the conduction gas of a metal lies in its ground-state at any ordinary
temperature. On the other hand, we could make a beam of fast electrons8 to pass
through a thin metallic film, collecting the energies of the emerging ones.
Experiments provide evidence that the transmitted beam shows energy losses at
discrete values, corresponding to the generation of one or more plasmons, in
excellent agreement with the prediction of the Drude theory [2, 5].

7.2.3 Thermal transport

We finally discuss the thermal conduction in metals. The basic assumption in this
case is that most of the heat current is carried by the gas of conduction electrons,
consistently with the empirical observation that metals are as good electrical
conductors as they are good thermal conductors: it is therefore natural to look at
electrons as the primary microscopic carriers for both transport phenomena. More
specifically, we understand that electrons are able to transport heat since those
coming from the hotter region of the sample carry a larger amount of thermal energy
than the electrons moving from the colder region.
The approximation to consider only electrons as heat carriers implies that in the
Drude theory we formally assume κtot = κ e + κ l ∼ κ e , where κ e and κ l are the
separate contributions due to the electronic and ionic degrees of freedom, respec-
tively. In order to calculate the actual form of κ e , we can proceed by analogy9 with
the lattice case discussed in section 4.3 by setting
1
κ eDrude = τe 〈(veth )2 〉cVe(T ), (7.15)
3
where cVe(T ) is the constant-volume specific heat of the electron gas. It is important
to remark that in this equation we used the same symbol τe as before, but with a

8
In this kind of experiment incident electrons typically have a kinetic energy in the range 1–10 keV.
9
The analogy is supported by the fact that, in order to calculate the lattice thermal conductivity as in equation
(4.40), we used the corpuscular language of phonons. So, we can simply replace any phonon-like quantity by
its electron-like counterpart.

7-7
 Solid State Physics

subtly different meaning: in equation (7.15) it is understood as the relaxation time

occurring in the heat current phenomenon, while in equation (7.7) it was associated
with the charge current one. This abuse of notation suggests that within the present
free electron theory we are assuming the charge current and heat current relaxation
times to be just the same: indeed a useful approximation, which however will be
revised in section 7.3.4. By describing the conduction gas classically, we calculate10
〈(veth )2〉 = 3kBT /me and cVe(T ) = 3n ekB /2 and, by means of equation (7.7), we predict
the ratio between its thermal and electrical conductivity to be

κ eDrude 3 ⎛ k ⎞2
= ⎜ B ⎟ T, (7.16)
σe 2⎝ e ⎠

which is in remarkable good agreement with the experimental Wiedemann-Franz law

stating that in most metals the ratio between the thermal and electric conductivities
of the conduction gas is proportional to T at sufficiently high temperatures (room
temperature belongs to this range). To a very good extent, the proportionality
constant is found to be about the same for all metals. Equation (7.16) not only
accurately predicts the high temperature trend of the κ e /σe ratio, but it also provides
the estimation of the constant

κ eDrude 3 ⎛ k ⎞2
= ⎜ B ⎟ = 1.11 · 10−8 W Ω K−2, (7.17)
σeT 2⎝ e ⎠

which is known as the Lorenz number. Finally, we remark that this result critically
depends on the assumption that the relaxation times limiting charge and heat
currents are the same.

7.2.4 Failures of the Drude theory

From the discussion developed so far, one could draw the conclusion that Drude
theory is basically sufficient to explain the main physical features of electron gas in a
metal, given its apparent success in all topics where we have applied it.
Unfortunately, this is not true.
First of all, we observe that the predicted value for the Lorenz number is (roughly)
only half of the measured one [2, 3]. Second, the ability to explain the phenomeno-
logical Wiedemann–Franz law is just casual since it is based on two large numerical
errors pointing in opposite directions: (i) the actual specific heat of an electron gas at
the typical metallic density is much smaller than predicted classically and (ii) the
typical electron velocities are much larger than calculated by equipartition. By
chance these two rather inaccurate estimations almost perfectly compensate for each
other, thus giving an illusory impression of robustness to the Drude theory. In
addition, the Lorenz number is not a constant at low temperatures [7], mainly because
it is found that κe is a function of temperature. As a matter of fact, only a full

10
As before, these results directly derive from the application of the equipartition theorem to the electron gas.

7-8
 Solid State Physics

quantum theory is able to correct such discrepancies so as to match the experimental

evidences.
In addition to the above failures, the Drude theory is also unsuccessful in
describing the electron gas under the action of both an electric and magnetic field,
as found when investigating the Hall effect [2, 3]. Let us consider the situation shown
in figure 7.3 where a constant and uniform magnetic field B = (0, 0, B ) is applied
normal to the current density j generated by a constant and uniform electric field
E = (Ex, Ey, 0). As before, the equation of motion for the electrons of the conducing
gas are Newton-like
m
−e E − e vd × B = m e vḋ + e vd , (7.18)
τe
where the left-hand side is now given by the Lorentz force −e(E + vd × B). If we
consider a steady-state regime, we have vḋ = 0 and the equation of motion leads to
eτe eτ eτe eτ
vd, x = − Ex − e B vd, y and vd, y = − Ey + e B vd, x , (7.19)
me me me me
By imposing the condition that no transverse current flows in the y direction (an
open-circuit configuration corresponding to the condition vy = 0) we get
Ey eτe
=− B, (7.20)
Ex me
which allows us to define the Hall coefficient RH as
Ey Ey 1
RH = = =− , (7.21)
jx B σeExB n ee
where jx is the current density along the x direction and equation (7.7) has been used
for the conductivity σe . Therefore, Drude theory predicts that the Hall coefficient is
independent of the applied magnetic field and, in any case, negative. Unfortunately,
both conclusions are wrong!
In order to enlighten these failures, let us plot as shown figure 7.4 the quantity
−1/n eeR H* versus the applied magnetic field, where R H* is the result of a direct
experimental measurement of the Hall coefficient. Plotting this quantity, we directly
measure the deviation of the Drude theory from the real physical situation. This is

Figure 7.3. Left: schematic representation of the Hall effect. The + and − signs respectively indicate the
depletion/accumulation of electrons on the opposite sides of the metallic sample (shown in grey) caused by the
magnetic field B = (0, 0, B ) (red arrow). The applied electric field E = (Ex, Ey, 0) is shown by blue arrows.
When the circuit is open in the y direction, only the current density jx = −en evd,x is observed (black arrow).

7-9
 Solid State Physics

Figure 7.4. The experimentally observed dependence of the Hall coefficient R H* (blue dots) on the applied
magnetic field in aluminium (nominal chemical valence Zv = 3). The two physical parameters are represented
by the dimensionless quantity −1/n eeR H
* and ω τ , respectively. Data are taken from [8].
c e

usually done by reporting on the horizontal axis the dimensionless quantity ωcτe ,
where ωc = eB /me is the cyclotron frequency. In figure 7.4 we illustrate the
paradigmatic case of aluminium, showing that −1/n eeR H* is not at all a constant
and, therefore, that R H* is experimentally found to depend on the magnetic field,
contrary to the prediction of the Drude theory; furthermore, at high magnetic field it
assumes positive values. This latter feature is quite intriguing: first of all, we observe
that −1/n eeR H* = n e* /n e , where n e* is the experimental density of the electron gas
which turns out to differ from the one calculated by equation (7.1) within the free
electron theory (in the case of aluminium shown in figure 7.4 the high field value of
the Hall coefficient suggests that there is just one carrier per atom, instead of three);
next, the change in sign suggests that the carriers have positive charge. These results,
similarly found in other metals like for instance zinc and cadmium, do not find an
explanation within the classical free electron theory.

7.3 The quantum (Sommerfeld) theory of the conduction gas

The main source of failure of the Drude theory lies in fact that the electrons have
been treated classically. The next obvious step in improving our theory, is, therefore,
to apply quantum mechanics to the conduction gas.

7.3.1 The ground-state

Let us consider a metal specimen at zero temperature. Since its bulk properties do
not depend on the shape, for mathematical convenience we will consider a cubic
sample with side L and faces normal to the x, y, and z Cartesian axes. The single-
particle wavefunction for any (free and independent) electron of the conduction gas
is obtained by solving the Schrödinger equation

7-10
 Solid State Physics

ℏ2 2
− ∇ ψ (r) = Eψ (r), (7.22)
2m e
where E is the electron energy. By imposing the Born–von Karman condition stated
in equation (1.4), we easily get the normalised wavefunction
1 1
ψk(r) = 3/2
exp(i k · r) = exp(i k · r), (7.23)
L V
where V = L3 is the system volume and the electron wavevector k has the following
Cartesian components11
2π 2π 2π
kx = ξx kx = ξy kx = ξz , (7.24)
L L L
with ξx, ξy, ξz = 0, ±1, ±2, ±3, …. We stress that (i) the free electron wavefunction
given in equation (7.23) has been labelled by k which plays the role of a quantum
number for the crystalline states12, (ii) the wavefunction given in equation (7.23) does
obey the Bloch theorem discussed in section 6.3: in this specific case we simply have
u k(r) = 1. The electron energy is
ℏ2k 2 ℏ2
E=
2m e
=
2m e
(kx2 + k y2 + k z2 ), (7.25)

a result which makes quite evident the function of quantum numbers associated with
kx, ky, and kz. By using the quantum mechanical operator pˆ = −i ℏ∇ , we easily
obtain the electron momentum
p = ℏk, (7.26)
and the corresponding electron velocity v = ℏk/me .
Because of equation (7.24), the allowed wavevectors are spaced by 2π /L along
any direction and, therefore, their number density in the reciprocal space is
L3 /(2π )3 = V /(2π )3. Furthermore, since the electron energies only depend on the
magnitude of k (and not on its direction in space), the number of allowed
wavevectors corresponding to quantum levels with energy in between E and E +
dE is given by the product between their number density and the infinitesimal
volume 4πk 2dk of the reciprocal space13. While the discussion has so far proceeded
in analogy with the vibrational case discussed in section 3.7, now we must duly take
into account that we can place two electrons (with opposite spin) on each allowed
quantum level: as a matter of fact, each k actually corresponds to two different
electron states with the same energy14. Therefore, by calculating

11
We remark that in this chapter the Latin indices i and j will be used to label particles, while the Cartesian
components will be indicated explicitly by x, y, and z.
12
The wavevector k is the counterpart of the principal, angular, and magnetic quantum numbers found in
atomic physics [9] and outlined in section 1.2.
13
This volume corresponds to the spherical shell contained in between the two surfaces E and E + dE.
14
By assumption, there is no external magnetic field acting on the system.

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 Solid State Physics

V
G (E )dE = 2 4πk 2dk (7.27)
(2π )3

V
= (2m e )3/2 E1/2 dE , (7.28)
2π 2ℏ3
we either define the electronic density of states (eDOS) G (E ) and provide a direct way
to calculate the number of electron states with energy in the interval [E , E + dE ] given
by G (E )dE .
The ground-state configuration of the conduction gas in a metal is found at
temperature T = 0 K by placing the N free electrons on the various quantum levels
provided by equation (7.25), starting from the lowest one. This is a subtle task: we
know that each k vector corresponds to two different states and that electrons obey
the Pauli principle; accordingly, we can place just one electron on each spin-resolved
state (which, in practice, corresponds to populate pairs of electron states degenerate
in energy). This way of proceeding is perfectly consistent with the statistical
arguments developed in section 6.2, provided that we take the results there obtained
in the limit T → 0. Since the total number N of electrons is finite (although very
large, as discussed in section 7.1), this necessarily implies that all quantum states are
totally filled up to a maximum energy EF, while for any E > EF we find empty states.
Hence
EF
N = ∫0 G (E ) dE (7.29)

V 3/2
= ( 2m e ) EF3/2 (7.30)
3π 2ℏ3

2
= G (E F ) E F , (7.31)
3
from which we calculate the Fermi energy of the free electron gas
ℏ2
EF = (3π 2 )2/3 n e2/3, (7.32)
2me
where n e = N /V is the electron density: EF represents the energy of the highest
occupied state at zero temperature. The corresponding ground-state energy EGS of the
electron gas can be calculated either using the eDOS or summing over the single-
particles energies
EF
V
E GS = ∫0 E G (E ) dE =
5π 2ℏ3
(2m e )3/2 EF5/2 (7.33)

ℏ2k 2
=2 ∑ , (7.34)
k < kF
2m e

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 Solid State Physics

where kF = 2meEF /ℏ2 is the Fermi wavevector

kF = (3π 2n e )1/3 , (7.35)
which corresponds to the radius of the sphere in the reciprocal space containing the
N /2 states able to host N electrons in pairs with opposite spin15. Its surface marks
the boundary between filled and empty states at T = 0 K: it is commonly referred to
as the Fermi surface of the metal. Typical values of the Fermi energy and wavevector
are reported in table 7.2. It is interesting to observe that kF is of the order of ∼1 Å−1
and, therefore, the de Broglie wavelength of the matter waves [9] associated with the
most energetic electrons is comparable to the typical interatomic distances. This
suggests that the free electron picture is conceptually wrong: conduction electrons
are not free because their interactions with ions are negligibly small but, rather,
because matter waves freely propagate within a periodic lattice of ions.
The Fermi velocity vF
ℏkF
vF = , (7.36)
me
represents the quantum expectation value for the maximum electron speed in the
conduction gas. As reported in table 7.2, its values are typically quite large, contrary
to what was predicted by the classical kinetic theory according to which they should
be zero at T = 0 K; we also observe that they even exceed the room-temperature
thermal velocity of a classical particle with mass me . This is indeed an interesting
feature, providing evidence that, consistently with the conclusion of the previous
section, the conduction gas can hardly be treated as a classical one. This conclusion
is naturally explained by introducing the Fermi temperature TF = EF /kB, defining the
inherent temperature scale of the conduction gas in a metal: such a gas can be
treated classically only at temperatures comparable to or larger than TF, but given
the typical values reported in table 7.2, this situation is never reached because TF

Table 7.2. Fermi energy EF (units eV), Fermi wavevector kF (units 108 cm−1), Fermi velocity vF (units 108 cm s−1),
and Fermi temperature TF (units 104 K) for some metallic elements. Values are approximated to the first digit.

Li Na Cu Ag Au Mg Ca Fe Zn Al Sn Pb

EF 4.7 3.2 7.0 5.4 5.5 7.1 4.7 11.1 9.5 11.7 10.2 9.5
kF 1.1 0.9 1.4 1.2 1.2 1.4 1.1 1.7 1.6 1.7 1.6 1.6
vF 1.3 1.1 1.6 1.4 1.4 1.6 1.3 2.0 1.8 2.0 1.9 1.8
TF 5.5 3.8 8.2 6.4 6.4 8.2 5.4 13.0 11.0 13.6 11.8 11.0

15
This calculation is simple: the product between the volume of the sphere and the number density of allowed
wavevectors (4πkF3/3)[V /(2π )3] = N /2 immediately drives to equation (7.35).

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 Solid State Physics

exceeds the melting temperature of the metal. We draw the conclusion that the
electron conduction gas in a metal is genuinely quantum at any temperature at which
the solid state exists.

7.3.2 Finite temperature properties

Let us now consider a metal in equilibrium at temperature T > 0 K. In this case the
eDOS is written as
G (E , T ) = G (E ) nFD(E , T ) (7.37)

V 1
= (2m e )3/2 E1/2, (7.38)
2π ℏ
2 3
1 + exp[(E − μc )]/ kBT

where we have combined the expression given in equation (7.28), which is a mere
counting of states, with the finite-temperature probability nFD(E , T ) that the
quantum level E is occupied, a correction entering our theory through equation
(6.7). The G (E , T ) function is plotted in figure 7.5 (thick blue line), together with its
zero-temperature counterpart (thin black line). We remark that in plotting this
figure we have neglected the temperature dependence of the chemical potential and,
accordingly, we have set μc = EF at any T ⩾ 0. We will very soon critically
re-address this assumption, proving that it is valid to a very good extent.
The number N of electrons is obviously unaffected by temperature and we can
therefore cast the normalisation condition (previously expressed as in equation
(7.29)) in a new form

Figure 7.5. The electronic density of states G (E , T ) (thick blue line) of a metallic free electron gas in
equilibrium at temperature T > 0 K. The thin black line represents the plot of its zero-temperature counterpart
(see equation (7.28)). The energy range ∼kBT marked by a red double arrow corresponds to the interval over
which the Fermi–Dirac distribution is non-constant: it is centred at the zero-temperature value of the chemical
potential, namely at the Fermi energy EF .

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 Solid State Physics

+∞ +∞
N= ∫0 G (E , T ) dE = ∫0 G (E ) nFD(E , T ) dE , (7.39)

which allows us to interpret the shaded area of figure 7.5 as the conserved number of
electrons. Since this notion is valid for any selected range of energy, we can develop
a new interesting concept.
At first we remark that the area under the G (E , T ) function corresponding to the
energy interval EF ⩽ E ⩽ +∞ represents the number of those few electrons that,
upon increasing temperature from zero to T, have been promoted to energies above
the Fermi energy. Obviously, this promotion has emptied an equal number of states
just below EF: this number corresponds to the area in between the function G (E , T )
and its zero-temperature counterpart and calculated for 0 ⩽ E ⩽ EF . This phenom-
enon is usually referred to as thermal excitation of electrons and it is summarised by
stating that at any finite temperature a number of electrons of the conduction gas are
promoted from quantum states just below the Fermi energy to states just above it. The
thermal excitation phenomenon mostly involves electrons with energy in the interval
∼kBT centred at EF.
The time has come to calculate once and for all the actual μc (T ) dependence, thus
checking the accuracy of the assumption which has replaced it with EF at any
temperature. To this aim we are going to follow a two-step procedure: first, we
calculate N as a function of μc and T; next, we invert the expression so obtained to
eventually get μc (T ). The integral appearing in equation (7.39) can only be calculated
by the asymptotic Sommerfeld expansion [3, 10], as outlined in appendix F. Given the
key role of such an expansion, the resulting quantum theory is commonly referred to as
the Sommerfeld theory of the metallic state. By using equation (F.10) we can write
μc
π 2 (1) 7π 4 (3)
N= ∫0 G (E ) dE +
6
G (μc ) (kBT )2 +
360
G (μc ) (kBT )4 + ⋯ , (7.40)

where G (m)(μc ) indicates the mth order derivative of the zero-temperature eDOS
calculated at energy μc . Even if we are now acknowledging that the chemical
potential is a function of temperature, we can nevertheless say that, as shown in
figure 6.2, μc ∼ EF whenever T ≪ TF; this allows us to approximate the integral
appearing in the above equation as
μc EF
∫0 G (E ) dE ∼ ∫0 G (E ) dE − (EF − μc ) G (EF ) =
(7.41)
= N − (EF − μc ) G (EF ),

where we used equation (7.29). The graphical rendering of this calculations is

reported in figure 7.6. By inserting this result into equation (7.40) and retaining just
the first leading term of the Sommerfeld expansion we easily get
π2 G (1)(μc )
EF − μc = (kBT )2 , (7.42)
6 G (EF )

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 Solid State Physics

Figure 7.6. Graphical rendering of the calculation of the integral given in equation (7.41). The chemical
potential and Fermi energy are indicated by μc and EF , respectively. The function G (E ) is the zero-temperature
eDOS given in equation (7.28).

which represents the formal justification of the adopted approximation μc ∼ EF: the
difference between the Fermi energy and the chemical potential is proportional to T 2
and, therefore, to a large extent it is negligible in the limit of small temperatures16.
Another consequence of this result is that
⎡ π2 ⎛ T ⎞ ⎤
2
μc (T ) = EF ⎢1 − ⎜ ⎟ ⎥, (7.43)
⎢⎣ 12 ⎝ TF ⎠ ⎥⎦

which further confirms that for any T ≪ TF (see table 7.2) we can safely set
μc (T ) = EF. While this is a common practice, the reader is warned that identifying
the Fermi energy with the chemical potential is just a (motivated and convenient)
approximation, but the two concepts should not be confused.
The method of the Sommerfeld expansion is also useful to calculate the internal
energy of the free electron gas which is obtained by generalising equation (7.33) as
+∞ +∞
U= ∫0 E G (E , T ) dE = ∫0 E G (E ) nFD(E , T ) dE , (7.44)

By defining the zero-temperature internal energy as

EF
U0 = ∫0 E G (E ) dE , (7.45)

and making use of equation (7.42) we proceed with the Sommerfeld expansion of
equation (7.44) as follows (see appendix F)

16
This result suggests that figure 6.2 has been obtained for very high T2,3,4 temperatures, that is, much higher
than typical temperatures of physical interest. As a matter of fact, the calculations performed to obtain
figure 6.2 were executed for T2,3,4 of the order of ∼10 4 K.

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 Solid State Physics

π2
U = U 0 + (μc − EF ) EF G (EF ) + [G (EF ) + EF G (1)(EF )] (kBT )2 + ⋯
6
π2 π2
= U0 − (kBT )2 EF G (1)(μc ) + [G (EF ) + EF G (1)(EF )] (kBT )2 + ⋯ (7.46)
6 6
π2
= U0 + G (EF ) (kBT )2 + ⋯ ,
6
which eventually leads to a full quantum expression of the specific heat cVe(T ) of the
free electron gas
1 ∂U 1 π2
cVe(T ) = = G (EF ) k B2 T + ⋯
V ∂T V V 3
π2 T
= n ekB +⋯ (7.47)
2 TF
3 ⎛1 T ⎞
= n ekB ⎜ π 2 ⎟ + ⋯.
2 ⎝ 3 TF ⎠

By comparing this expression with the classical result cVe(T ) = 3n ekB /2 provided by
the Drude theory developed in section 7.2, we understand that quantum mechanics
predicts a lower specific heat than classical physics, in full agreement with
experimental observations. Furthermore, the quantum correction term put in
parenthesis in the above equation introduces a linear dependence on T which,
once again, corrects a major failure of the Drude theory (according to which cVe(T )
should be constant). Anyway, we stress that the electron contribution to the total
specific heat of a metallic crystal is proportional to T /TF which, at temperatures of
physical interest, makes it really marginal as compared to the lattice contribution cVl
discussed in sections 4.1.2 and 4.1.3.
It is instructive to consider in more detail what happens at low-temperature: to
this aim, we will make use of equations (7.47) and (4.11), respectively17. Basically,
they dictate that
cVe(T ) = ξ1 T and cVDebye(T ) = ξ2 T 3, (7.48)

where ξ1 = (π 2n ekB /2TF ) and ξ2 = 12Rπ 4 /5VTD. In figure 7.7 the variation of the
quantity
1 tot 1
cV = (cVe + cVl ) = ξ1 + ξ2 T 2, (7.49)
T T
upon the square temperature is shown in the case of copper (red dots), where we
have expressed the lattice contribution to the specific heat by the Debye result setting
cVl = cVDebye . The observed linear trend (blue dashed line) in the reported low

17
We know that the Debye model does not provide the most fundamental description of the lattice specific
heat cVl , but for the purpose of the present discussion either it is accurate enough and it predicts the needed
analytic trend of cVl .

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 Solid State Physics

Figure 7.7. The quantity cVtot /T for copper (red dots) measured at low-temperature (data taken from [11]). The
blue dashed line is the trend predicted by equation (7.49).

temperature range provides evidence that the electronic contribution is here

dominant.
This way of representing the laboratory data has become a widely adopted
practice, since the zero-temperature value of the quantity cVtot /T directly provides an
experimental determination of ξ1, that is, of the Fermi temperature18. However, a
non negligible discrepancy is often found between the experimental and the
theoretical19 values; for instance, this discrepancy is as large as 48% in the case of
aluminium. In order to reconcile the free electron theory with experimental evidence,
it is customary to replace the bare electron mass me by an effective mass m e* which
basically corrects the fact that the electrons of the conducting gas are not really free,
but rather diffuse within a crystalline environment and, therefore, their inertia to
motion is different than in empty space. The Fermi temperature is corrected by a
factor m e* /me and, therefore, by setting a suitable value of effective mass it is possible
to fit the theoretical prediction onto experimental data20.
Finally, at higher temperatures the ionic degrees of freedom dominate the specific
heat or, equivalently, cVtot /T ∼ cVl /T . This is qualitatively shown in figure 7.8: in metals
the electronic and lattice contributions to the specific heat are comparable only at very
low temperatures, that is, up to just few degrees Kelvin, for most metallic systems.
The Sommerfeld theory of the free electron gas also provides a more correct
estimation of the electron contribution to the thermal conductivity κ e than the Drude
theory. While we maintain the kinetic theory as our reference theoretical framework,
we now replace in equation (7.15) the correct expression for the specific heat
provided in equation (7.47) and estimate the electron square velocity by means of
equation (7.36) so that
1 3 ⎛1 T ⎞
κ eSommerfeld = τe vF2 n ekB⎜ π 2 ⎟ , (7.50)
3 2 ⎝ 3 TF ⎠

18
We remark that a similar way of plotting the specific heat of an insulating material would provide a straight
line with the zero-temperature intercept found right at the origin of the axes: in these materials the electronic
contribution is absent. This is consistent with the experimental evidence that electrons do not provide any
measurable contribution to the specific heat of a non metallic system at room temperature.
19
We recall that we have defined TF = EF /kB , where EF is given in equation (7.32).
20
The effective mass concept is widely used in solid state physics. Therefore, one should always pay attention
to the genuine meaning of effective mass which is used in the various contexts.

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 Solid State Physics

Figure 7.8. A qualitative comparison between the electron (red line) and lattice (black line) contributions to
the total specific heat (blue line) of a metallic system in the range of very small temperatures (just a few degrees
Kelvin above zero temperature).

which leads to
κ eSommerfeld π 2 ⎛ kB ⎞2
= ⎜ ⎟ = 2.44 × 10−8 W Ω K−2 (7.51)
σeT 3 ⎝e ⎠
a prediction in fairly good agreement with the experimental values of the Lorenz
number which, for the metal elements listed in table 7.2, has an average value of
2.41 × 10−8 W Ω K−2 . This remarkable result derives from the rectification of the
double error contained in the classical theory, namely: the Sommerfeld expression
for cVe(T ) now includes a corrective term which decreases the Drude estimation by a
factor proportional kBT /EF , while the Fermi velocity is larger than the classical
average thermal velocity by an inverse factor EF /kBT . The two errors cancel out in
the Drude theory, thus providing the correct order of magnitude for the Lorenz
number, but still remain in its separate evaluation of the specific heat and electron
speed.
Finally, the Sommerfeld theory also overcomes other failures of the classical
theory, remarkably in describing thermoelectric effects. For instance, Drude theory
wrongly predicts the ratio Q = ∣E∣/∣∇T ∣ where ∇T is the gradient of temperature
applied across a metal sample and E is the observed electric field established within
it. Q is a material-specific property named thermopower: its determination does
require the use of the Fermi–Dirac statistics for the conduction electrons [3, 10].

7.3.3 More on relaxation times

In our discussion on transport coefficients σe and κ e we have twice introduced the
notion of relaxation time which, although conceptually different in the two cases,
was considered the same for charge and heat currents. It is now necessary to
reconsider this aspect in greater detail.
Let us start by readdressing the direct-current conductivity. Electrons, during
their drift motion under the action of an external electric field E , undergo scattering

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 Solid State Physics

with lattice defects and ionic oscillations21. The former provide a constant
contribution τd to the electron relaxation time, while the effect of the ionic oscillation
can be described as electron–phonon scattering events: their contribution τph(T ) is
inherently dependent on temperature since the phonon population of each mode is
so. If we assume that the two mechanisms are independent (that is, if the number of
defects is small enough to leave unaffected the vibrational spectrum of the system),
then we can apply the same Matthiessen rule already introduced in section 4.3 to
understand thermal transport and write
1 1 1
= + . (7.52)
τe τd τph(T )

By now inserting this expression for the electron relaxation time into equation (7.7),
we immediately obtain the resistivity ρe of a metal in the form
me 1 me 1 me 1
ρe = 2
= 2
+ 2
= ρd + ρph (T ), (7.53)
n ee τe n ee τd n ee τph(T )

where the two contributions are referred to as the residual resistivity and ideal
resistivity, respectively, since ρd is the only one active even at zero temperature, while
ρph (T ) is the only one found even in a totally defect-free system. The electron–phonon
scattering largely affects the relaxation time, which is typically decreased from 10−11 s
at T = 0 K down to 10−14 s at room temperatures. By multiplying the Fermi velocity
by τe we can easily estimate the order of magnitude of the electron mean free path λ e to
be as large as dozens of nm at room temperature or dozens of μm at zero temperature.
This is indeed a much more accurate estimation of λ e than provided by the Drude
theory and, more importantly, it better proves that the average distance covered
between two successive collisions is much larger than the lattice interatomic spacing:
as far as charge current phenomena are concerned, the electrons in a metal can be
really considered as free, that is not colliding with lattice ions.
The discussion so far has not yet considered the collisions between electrons
which represent another source of scattering affecting the relaxation time: however,
neglecting electron-electron scattering is largely justified by the occurrence of (i)
screening effects and (ii) exclusion principle. Screening is something we can under-
stand only by going beyond the independent electron approximation and, therefore,
it falls beyond the level of treatment we are developing: in fact, screening is an
advanced topic of solid state theory which is presented elsewhere [3, 10, 12]. Here we
limit ourselves to observing that the motion of the electrons in the conducing gas is
highly correlated and, therefore, they can adjust in such a way as to largely cancel
the single-particle long-range Coulomb field. The resulting screened potential turns
out to have the form of a Yukawa potential: a Coulomb term multiplied by an
exponentially decaying term. The interaction strength is accordingly reduced to a
negligibly small value above a characteristic distance named screening length,

21
The effect of the regular ions of the ideal lattice can be included in the electron effective mass, as discussed in
the previous section.

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 Solid State Physics

typically calculated of the order of ∼1Å. Such a dramatic reduction of the

interaction range makes the mean free path between two electron–electron scattering
events much larger than in the case of impurity or phonon scattering which,
therefore, can be considered as the by far dominant ones at room temperature. As
for the Pauli principle, the scattering between two electrons can only occur if there
are vacant states in which to accommodate the electrons after the collision. In
practice, the exclusion principle reduces the frequency of electron–electron scattering
events because it makes fewer scattering channels available to electrons [2].
Now that we have better characterised the different scattering mechanisms affecting
electrons during their drift motion, let us consider how an externally applied electric
field affects their quantum states, that is their wavevector. In the absence of field, the
Fermi surface is a sphere centred at k = 0 (see equation (7.35) and related comments)
with a boundary which is slightly blurred by the thermal excitation mechanism. Under
the action of the field E the sphere is shifted by an amount that corresponds to a
wavevector variation Δk = −(eτe /ℏ)E for each electron22. The situation is pictorially
represented in figure 7.9. The relaxation time entering this equation is the same
discussed above, but it is now naturally interpreted as the time needed to the Fermi
sphere to relax back to its unshifted position, after that the electric field is turned off. The
key point is that the microscopic events driving this displacement are precisely the
collisions suffered by the electrons. More specifically, the relaxation is ruled over by
those scattering processes removing one electron from the right side of the shifted
Fermi sphere and adding it to its left side. As shown in figure 7.9 we estimate a
wavevector change as large as ∼2kF for these processes.
It is now time to address the thermal current case and its corresponding relaxation
time. A similar analysis of the scattering events as above can be elaborated when a

Figure 7.9. Left: the Fermi sphere centred at k = 0 in absence of electric field (kF is the Fermi wavevector, that
is, the radius of the Fermi sphere whose blurred boundary is not shown explicitly for sake of clarity). Middle:
an electric field E is applied so that a constant force F = −e E acts on each electron causing a Δk shift of the
centre of the Fermi sphere. Right: by turning off the applied electric field, the Fermi sphere relaxes back to its
equilibrium position because of electron scattering events pictorially shown by purple arrows (the filled and
empty dots represent the initial and final electron state before and after the scattering event, respectively).

22
This result is obtained by integrating the electron equation of motion under the action of the field E .

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 Solid State Physics

metal is under the action of a thermal gradient ∇T . In this case, the blurring of the
Fermi sphere is different on its opposite sides taken along the direction of ∇T
because of the different local temperature. We can equivalently state that there are
more thermal excitations on the hotter side of the Fermi sphere than on the colder side.
Following the same reasoning as in the charge current case, the relaxation time
entering equation (7.50) is now interpreted as the time needed to the Fermi sphere to
even its left and right blurring, after that the temperature gradient is turned off. Once
again, this evolution is regulated by microscopic events that mainly consist in
electron–phonon scattering processes. In this case, however, they occur by means of
either large or small momentum changes, as pictorially indicated in figure 7.10.
In summary, just one leading scattering process has been identified in the case of a
charge current (associated with a large ∼2kF wavevector change), while two unalike
processes (characterised by a large and small wavevector change, respectively) are at work
in the case of a heat current. Since in this latter case more scattering occurs, we expect the
relaxation time associated with a heat current is shorter than the one associated with a
charge current. This qualitatively explains why, contrary to what is predicted by the
Drude and Sommerfeld free electron theories (see equations (7.17) and (7.51)), the Lorenz
number does depend on temperature23, as found experimentally at low temperatures.
In concluding this section we observe that for both a charge and a heat current we
have understood that the overall relaxation time actually depends on temperature,
since phonon populations enter into their evaluation. The separate calculation of τe
for both cases leads to the accurate prediction of the temperature dependence of the
electrical resistivity and thermal conductivity of a metal as reported elsewhere [3, 4].

Figure 7.10. The Fermi sphere of a metal under the effect of a temperature gradient. The cold (cyan) and hot
(red) boundaries are blurred (dashed line) to a different extent since they feel a different local temperature. For
illustration purposes the blurring effect is much magnified. Yellow lines pictorially represent electron–phonon
scattering events which can imply a small as well as a large momentum change. They drive the Fermi sphere
back to its equilibrium state as soon as the temperature gradient is removed. This occurs by displacing
electrons from thermally-excited states (filled dots) to empty ones (empty dots).

23
More precisely, we have eventually understood that it is incorrect to assume that the relaxation time for a
charge and a heat current is the same. In other words, the simplification of the τe factor operated in deriving
equations (7.17) and (7.51) is only valid as a first approximation.

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 Solid State Physics

7.3.4 Failures of the Sommerfeld theory

The Sommerfeld theory outclasses the Drude one by more accurately predicting
many physical properties of metals; it also enlightens some important concepts like
the difference between the chemical potential and the Fermi energy or the real need
to treat the electron conduction gas as a fermion gas obeying the Fermi–Dirac
statistics. However, it cannot yet be regarded as the most complete and predictive
quantum theory of electron states in a crystal, since its predictions are still not in
good agreement with experiments in some important cases.
First of all, we remark that the Sommerfeld theory for the charge current is
basically the same as the Drude one and, therefore, it suffers the same limitation, in
particular as regards the wrong predictions about the Hall coefficient24. This is
mainly due to the fact that in deriving such a coefficient the fermionic nature of the
charge carriers is not explicitly taken into account25. Even the alternate-current
conductivity provided by the two free electron models is only grossly adequate in
describing metal reflectivity, but it falls short with other optical properties of metals
like, notably, their colour. Finally, the Fermi surface of real metals is not a simple
sphere with radius kF [3, 4].
A part for these phenomenological failures, the Sommerfeld theory is unable to
explain a very fundamental fact: why in Nature do insulators exist? Our basic
assumption was to treat the system of valence electrons as a gas of delocalised charge
carriers. Why is this a reasonably good approximation in some materials (metals)
and not in many others (insulators)? The Sommerfeld theory does not provide an
answer to this question. We need a more refined approach to the electronic structure
of a crystalline solid.

References
[1] Reif F 1987 Fundamentals of Statistical and Thermal Physics (New York: McGraw-Hill)
[2] Kittel C 1996 Introduction to Solid State Physics 7th edn (Hoboken, NJ: Wiley)
[3] Ashcroft N W and Mermin N D 1976 Solid State Physics (London: Holt-Saunders)
[4] Hook J R and Hall H E 2010 Solid State Physics (Hoboken, NJ: Wiley)
[5] Fox M 2001 Optical Properties of Solids (Oxford: Oxford University Press)
[6] Ehrenreich H, Phillipp H R and Segall B 1962 Phys. Rev. 132 1918
[7] Ziman J M 1960 Electrons and Phonons (Oxford: Oxford University Press)
[8] Lück R 1966 Phys. Status Solidi 18 49
[9] Colombo L 2019 Atomic and Molecular Physics: A Primer (Bristol: IOP Publishing)
[10] Grosso G and Pastori Parravicini G 2014 Solid State Physics 2nd edn (Oxford: Academic)
[11] Corak W S, Garfunkel M S, Satterthwaite C B and Wexler A 1955 Phys. Rev. 98 1699
[12] Jones W and March N H 1973 Theoretical Solid State Physics vol 1 (New York: Dover)

24
This also reflects in a wrong prediction about the magnetoresistance: the free electron theory, in both
formulations, estimates that the resistance of a metal sample measured normally to an externally applied
magnetic field does not depend on the field. This is not true, as experimentally found in many real metals [3, 4].
25
See section 7.2.1 where σe was calculated without any need to use statistical laws.

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 IOP Publishing

Solid State Physics

A primer
Luciano Colombo

Chapter 8
The band theory

Syllabus—The band theory for the crystalline electron states is developed through a
hierarchy of increasingly accurate models. At first we discuss the nearly free electron
approach, where the crystal field potential is treated just as a perturbation on the free
electron states. Despite its qualitative character, this model is nevertheless able to
support the key concept that energy states are grouped into allowed bands, separated
by forbidden gaps. The corresponding band filling will allow us to distinguish among
metals, insulators, and semiconductors. Next we will elaborate the tight-binding theory
which leads to the accurate prediction of the dispersion relations for the crystalline
electrons. A number of general features about the band structure will be eventually
discussed, with focus on electron dynamics. In particular, we will discuss the concepts
of hole and effective mass, indeed two key ingredients of the microscopic theory of the
charge transport in solids.

8.1 The general picture

8.1.1 Bands and gaps
As explained in section 7.3.4, the Sommerfeld theory is unable to explain why some
materials are metals and others not, although the corresponding atomic elements
have a similar ground state electronic configuration. Ultimately, the limitations of
this theory are related to the fact that valence electrons are treated as free and
independent particles. In order to improve our description of electron states in
crystalline systems, we should therefore re-address these assumptions and accord-
ingly develop a more refined theory.
Actually, it is not necessary to question the entire theoretical apparatus set up in
chapter 1 since we already decided once and for all to treat electrons within the
single-particle approximation: according to what is explained in section 1.4.1, each
electron is assumed to independently move under the action of a local crystal field
potential Vcfp(r). The electron quantum many-body problem is therefore disen-
tangled into a set of similar single-particle Schrödinger equations (1.22), which

doi:10.1088/978-0-7503-2265-2ch8 8-1 ª IOP Publishing Ltd 2021

 Solid State Physics

provide the energy and the wavefunction for each electron. The implication of this
approximation is twofold: while electrons will be similarly treated as independent
particles as in the Sommerfeld theory of metals, nevertheless they are subject to a
potential Vcfp(r) = Vcfp(r + Rl) which has the same periodicity of the crystal. This
immediately calls into question two general quantum mechanical results, namely: (i)
the energies of the crystalline electron states are grouped into bands, separated by
forbidden gaps (see section 6.4) and (ii) the wavefunction must have the Bloch form
(see section 6.3).
Let us consider in more detail the structure of allowed bands and forbidden gaps
obtained within the Kronig–Penney model and reported in figure 6.4. We remark
that, although this result has been obtained for a simplified form of Vcfp(r), it has a
general validity: the energy spectrum of an electron subject to any periodic potential
consists in bands and gaps. The parameter ξ = αa appearing in equation (6.35) is
nothing other than the normalised single-electron energy E and, therefore, we can
draw a conceptual diagram representing the band structure of a crystal such as the one
shown figure 8.1: here we report the case of both a free electron (left) and an electron
subject to a periodic potential (middle); for sake of comparison, the energy spectrum
of an electron confined with an infinite potential well is also reported (right): ideally
it represents the extreme case of an electron so tightly bound to its nucleus that it is
not affected by the crystalline environment.
We can easily read the physical situation as follows. If we consider the case of a
one-dimensional infinite chain, we understand that each potential well corresponding
to a core region is just equivalent to any other and, therefore, the electron quantum
states there confined have the same discrete energies as in any other similar well.

Figure 8.1. Comparison of the energy levels allowed to a free electron (left), to an electron subject to periodic
crystalline potential (middle) and to an electron confined within an infinite potential well. This diagram
represents the graphical conceptualisation of what is obtained within the Kronig–Penney model and reported
in figure 6.4.

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 Solid State Physics

Since, however, the wells are not actually infinitely deep, there is a non-zero tunnelling
probability for one electron to diffuse from a given core region to a neighbouring one.
This, in turn, makes the set of discrete confined levels (each of which would be
degenerate in the idealised situation of infinite wells) to open into a band of allowed
crystalline states. It is convenient to label bands by a band index n which we must duly
append to any symbol identifying electron energies En and wavefunction ψn(r). On the
other hand, so as not to complicate formalism too much, we will hereafter omit the
apex (R) identifying the specific ionic configuration for which the electron problem is
solved (see equation (1.22)). It will be understood that, according to the adiabatic
approximation, if ions are displaced then the electron problem will have to be
rewritten and solved again.
Let us now consider the implications of the Bloch theorem. We know that the
wavefunction of an electron moving in a periodic potential has the form of a product
between a plane wave exp(ik · r) and a periodic function u k(r) = u k(r + Rl) (see
equation (6.24)). This implies that the wavevector k is the real quantum number
identifying the crystalline state and, therefore, we agree to index the wave function as
ψnk(r) and the corresponding energy as Enk . However, a new problem now arises:
how does the wavevector k enter a conceptual diagram like the one shown in
figure 8.1? The answer is: it doesn’t! In other words, this way to represent the allowed
states is unable to provide any information about the relationship between the
wavevector and the corresponding energy. This is enough to understand that we
cannot settle to represent the band structure of a crystal only this way: we do need
dispersion relations Enk = En(k).

8.1.2 The weak potential approximation

A first qualitative attempt to calculate the dispersion relations consists in considering
the periodic potential as a small perturbation with respect to the free electron
situation. While this approach, usually referred to as nearly free electron theory
[1–3], is unable to provide quantitative Enk = En(k) relations for real crystals, it
nevertheless provides very useful qualitative information about their band structure.
Following a pedagogical approach already adopted several times in this textbook,
we will develop this model just in one dimension, so as to keep the mathematics
simple and focus on the physical implications.
Let us consider a monoatomic linear chain with lattice spacing a; assuming that a
valence electron is subject to a weak periodic potential, we indicate by
Vˆpert(x ) = Vˆpert(x + sa ) (with s any positive or negative integer) the operator
describing the perturbation with respect to the free electron situation. Because of its
periodicity, we can represent the perturbation as a Fourier series
+∞
⎛ 2πm ⎞
Vˆpert(x ) = − ∑ Vm cos ⎜ x⎟ , (8.1)
⎝ a ⎠
m=1

where, in order to fulfil the convention adopted in figure 6.3(top), we assume that
Vm > 0. To the first order in the perturbation, the correction ΔE pert to the free
electron energy is calculated as

8-3
 Solid State Physics

∫ ϕ*(x)Vˆpert(x)ϕ(x) dx
ΔEpert = , (8.2)
∫ ϕ*(x)ϕ(x) dx
where ϕ(x ) is the unperturbed wavefunction: it is the one-dimensional counterpart of
the general solution given in equation (7.23) and corresponds to a plane wave1. This
result of standard quantum mechanical perturbation theory must be handled with some
care: the unperturbed energy levels E (0) = ℏ2k 2 /2me are twofold degenerate since the
progressive ϕp(x ) = exp(ikx ) and regressive ϕr(x ) = exp( −ikx ) waves correspond to
the same electron energy. Therefore, the more rigorous way to calculate the integrals
appearing in equation (8.2) is to replace the unperturbed wavefunctions with two
orthogonal linear combinations of ϕp(x ) and ϕr(x )—which we will indicate by ϕ1(x )
and ϕ (x )—and to impose the constraint ∫ ϕ *(x )Vˆpert(x )ϕ (x ) dx = 0 [4, 5]. It is
2 1 2
easy to prove that this constraint is fulfilled for any value of k only by the real
functions ϕ1(x ) = cos(kx ) and ϕ2(x ) = sin(kx ). Let us proceed by inserting ϕ1(x )
into equation (8.2)
⎡ +∞ ⎤
∫ ϕ1*(x )⎢ −
⎢⎣
∑ Vm cos ( 2πam x)⎥⎥ϕ1(x) dx
⎦
m=1
ΔE pert =
∫ ϕ1*(x )ϕ1(x ) dx
+∞
∑ Vm ∫ cos2(kx ) cos ( 2 πm
a
x ) dx
m=1 (8.3)
=−
∫ 2
cos (kx )dx
+∞
∑ Vm ∫ [1 + cos(2kx )] cos ( 2 πm
a
x ) dx
m=1
=− .
∫ [1 + cos(2kx )] dx

The numerator is nonzero only if the periodicity of cos(2kx ) and cos(2πmx /a ) are
the same or, equivalently, only if k = mπ /a . For these values of the wavevector we
straightforwardly calculate ΔE pert = Vm /2. On the other hand, by inserting ϕ2(x ) into
equation (8.2) we obtain the same constraint on k and similarly calculate
ΔE pert = −Vm /2. The conclusion is remarkable: the perturbation operated by the
weak potential on the free electron energies affects only those pairs of states lying at
the Brillouin zone boundaries (which correspond to m = ±1, ±2, ±3, …) by shifting
the two degenerate states by a similar amount Vm /2 but in opposite directions.

1
We remark that in this calculation we are taking an infinite system and, therefore, unperturbed wavefunctions
are not normalised as in equation (7.23). Accordingly, in equation (8.2) the expectation value of the
perturbation operator is properly normalised [4, 5].

8-4
 Solid State Physics

Figure 8.2. Right: full lines (blue and red) represent the dispersion relations E = E (k ) of a crystal treated
within the nearly free electron (NFE) model; the dashed line (black) corresponds to the dispersion relation
E = ℏ2k 2 /2me of a free electron. The weak NFE potential opens gaps at the zone boundaries; the first gap has
width V1. Right: the corresponding band structure diagram. The reduced zone scheme consists in folding the
NFE dispersions within the first Brillouin zone (1BZ).

In figure 8.2 it is shown the dispersion relation of the one-dimensional crystal

treated in the weak potential approximation and, for comparison, the dispersion
predicted by the free electron model. The first important feature to observe is that the
true existence of a periodic potential acting on the valence electrons opens gaps in the
energy spectrum, in full agreement with the Kronig–Penney model. Furthermore, by
only considering the vertical ordering of the allowed energies (that is, by neglecting
the role played by the wavevector) we recover the band diagram shown in figure 8.1.
These two ways of representing the energies of crystalline electrons are fully
equivalent and equally useful, but only through the dispersion relations can we
graphically render the role of the wavevector. Next, we rigorously observe that the
perturbative calculation only indicates that gaps are opened at k = mπ /a; on the
other hand, in figure 8.2 the dispersion relation in proximity of the Brillouin zone
boundaries is shown to deviate from the purely parabolic trend predicted by the free
electron model. In order to calculate the exact En(k ) values near such boundaries a
more refined calculation is indeed required. Let us for instance consider the
boundary of the first Brillouin zone at m = 1. The arguments developed after
the derivation of equation (8.3) indicate that here the only non vanishing term of
the perturbation potential is V1 cos(2πx /a ) so that the Schrödinger equation can be
written in the form
⎡ ℏ2 ⎛ 2πx ⎞⎤
⎢− ∇2 + V1 cos ⎜ ⎟⎥ψ (x ) = Eψ (x ). (8.4)
⎣ 2m e ⎝ a ⎠⎦

8-5
 Solid State Physics

EF

EF

Figure 8.3. Band filling at T = 0 K. The allowed energy levels occupied by electrons are shown in blue or red,
while empty ones are shown in grey. For metals (left) the Fermi energy EF falls within the upmost partly
occupied band. For non-metals (right) EF coincides with the highest level of the last fully occupied band.

Since we understand that the potential term appearing in this quantum problem is
weak, in the spirit of the perturbation theory we attempt a trial solution of the form
ψ (x ) = ξ1 exp(ikx ) + ξ2 exp[i(k − 2π / a )x ], (8.5)
where the first term corresponds to the free electron case, while the second term
describes the wavefunction distortion generated by the perturbation. The constants
ξ1 and ξ2 are determined by a boring, but straightforward calculation2 which leads to
ℏ2k 2 ℏ2π ⎡
E (k ) = + k′ ± k′2 + (m eaV1/2π ℏ2)2 ⎤⎦ , (8.6)
2m e m ea ⎣
with k′ = π /a − k . This result reduces to the free electron one for any value of k well
within the Brillouin zone, while it provides the bending of the dispersion relation
nearby its boundaries, as shown in figure 8.2. The same calculation is performed for
any m > 1, leading to similar results.

8.1.3 Band filling: metals, insulators, semiconductors

Before developing a theory more rigorous than the nearly free electron one, we
address another important qualitative question which is inherently linked to the
actual existence of a band structure, namely: how are the allowed states populated by
electrons? To answer, it is useful to adopt the kind of band representation shown in
figure 8.3.

2
The trial solution must be at first inserted into equation (8.4); the resulting equation is then multiplied by
exp(ikx ) and eventually integrated over the space. These two last steps are repeated a second time but now
multiplying by exp[i (k − 2π /a )x . The resulting two equations form an algebraic system where ξ1 and ξ2 are the
unknown variables. By imposing the secular determinant to be zero in order to have nontrivial solutions, we
get a second order equation in E (k ) whose solutions are given by equation (8.6).

8-6
 Solid State Physics

Let us suppose that the crystal is in thermal equilibrium at temperature T = 0 K

and that a total number Nval of valence electrons are available: we want to elaborate
a procedure for filling the allowed bands. In order to do so, we must operate under
the Pauli principle: the maximum number of electrons that we can accommodate on
each level is equal to double the degree of degeneracy of that level3, since only up to
two electrons with opposite spin can correspond to the same quantum state labelled
by the given pair of band index n and wavevector k . Accordingly, we start by placing
the maximum possible number of electrons on the lowest level of the lowest allowed
band. Having filled the first level, we pass to the second one of that same band,
arranging the electrons with similar criteria as before. Then we proceed in the same
way for all the other levels of the first band until it is completely filled. At this point,
in order to place the remaining electrons, we have to pass directly to the second
band, since no states are available in the forbidden gap separating the first band
from the second one. By repeating the same procedure as in the previous case, the
second band will also be filled; next, the third one and so on … until all the Nval
valence electrons are eventually used. As in the case of the free electron gas, the
energy EF of the highest occupied level in this T = 0 K condition will be referred to as
Fermi level.
The intriguing point is that, if we consider the position of the Fermi level, only two
possible cases are given: either (i) EF falls within the last (that is: highest in energy)
band or (ii) EF coincides with the upmost level of the last band. The situation is
pictorially summarised in figure 8.3. The two cases are inherently different since the
last band is either partially or completely occupied and, therefore, according to the
position of the Fermi level we distinguish between metals (first case) and non-metallic
materials (second case). While this distinction could appear as mostly conventional,
we can motivate it by a simple qualitative argument addressing the charge transport
properties in the two opposite cases. In metallic systems an applied electric field
shifts the electron distribution similarly to the case of a free electron gas (see
figure 7.9) simply because there are vacant states just above EF: this generates a
crystalline quantum state able to carry electric current. On the other hand, in the
case of a totally filled last band an applied electric field is unable to promote
electrons above EF since the energy it provides is not large enough to bypass the
energy gap: in this condition the material is an insulator. There are, however, solids
whose energy gap between the last fully occupied band and the first empty one is
sufficiently small that at any finite temperature some electrons are thermally excited
to the upper band. The graphical rendering of the situation is reported in figure 8.4.
These materials are, therefore, insulators at zero temperature and (poor) conductors
at T > 0 K: they are named semiconductors.
A more fundamental explanation of the charge transport characteristic will be
developed in section 8.3.3, but as of now we can already understand why in Nature
do materials exist with metallic or insulating or semiconducting behaviour

3
The qualitative discussion so far developed has not yet highlighted the fact that states belonging to different
bands, but with the same wavevector, can be degenerate in energy. This will be fully exploited in the next
section. Here it is sufficient to admit that, in general, energy levels can be degenerate.

8-7
 Solid State Physics

T =0K T >0K

EF

Figure 8.4. Graphical rendering of the thermal excitation (red arrows) of electrons (black dots) in a small-gap
insulator. White dots: emptied energy levels. The topmost occupied level is shifted from the Fermi energy EF
position to an intermediate position of the upper band, thus reproducing a situation qualitatively similar to
that of a metal (see figure 8.3).

T =0K

Egap
EF

Figure 8.5. Standard nomenclature of the band theory (only the topmost occupied band and the first empty
one are shown).

(something that the free electron theory was unable even to predict): ultimately, the
ability of a material to carry an electric current is determined by its band structure and
by the relative filling with valence electrons. We also understand that this is a truly
quantum mechanical property since it is derived from general properties of electrons
subject to a periodic potential4.
We complete the nomenclature commonly used within the band theory by
introducing some further conventions. First of all, we agree to refer to the highest
energy level of a band as the ‘band top’ and, likewise, to the lowest energy level as the
‘band bottom’. Furthermore, in non-metallic systems the last occupied band is called
the valence band, while the first empty band is called the conduction band. The two
bands will be hereafter indicated by the acronyms VB and CB, respectively. The
energy difference between the VB top and the CB bottom will be indicated with the
explicit name of forbidden gap: it represents the interval of forbidden energies that
separates the last totally full band from the first completely empty one. We will
always indicate with the symbol Egap the energy amplitude of such a forbidden gap.
These conventions are summarised in figure 8.5.

4
Historically, this achievement was welcome as one of the most convincing successes in support of quantum
mechanics [6].

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 Solid State Physics

8.2 The tight-binding method

The time has come to proceed with a rigorous calculation of the band structure of a
crystalline solid. As in the case of lattice dynamics, we will first consider a model
one-dimensional crystal where the formal development can be exploited in full
analytical detail and the most relevant physical features emerge in a simple way.
Next, we will outline the calculation of the band structure of real solids, using the
formal results reported in appendix G.

8.2.1 Bands in a one-dimensional crystal

Let us consider a monoatomic linear chain of atoms with lattice spacing a so that ion
positions are given by xs = sa with s = 0, ±1, ±2, ±3, …. Let us also suppose that
there is just one valence electron for each atom in the chain. The Born–von Karman
boundary condition given in equation (1.4) is applied to a crystal portion containing
a suitably large number N of atoms (and, therefore an equal number Nval = N of
valence electrons).
The preliminary step in our approach is to consider the case of a single isolated
atom of the same chemical species present in the chain. Let V̂a be the quantum
operator describing the potential Va felt by the valence electron and let us suppose
that the corresponding Schrödinger problem
⎡ ℏ2 ⎤
⎢− ∇2 + Vˆa⎥ϕa = Eaϕa , (8.7)
⎣ 2m ⎦

has been solved by means of the standard methods of atomic physics [7–9]. In our
formalism ϕa e Ea are the atomic wavefunction and energy of the atomic states,
respectively.
Once the atom is placed in some lattice position along the chain, we can assume to
a very good approximation that its valence electron is now subject to a potential
Vc(x ) written as5
Vc(x ) = Va + ΔV (x ), (8.8)
where ΔV (x ) describes the difference between the crystalline environment and the
isolated atom situation. Our physical intuition suggests that ΔV (x ) is vanishingly
small in the core regions, while it significantly differs from zero in the interstitial
ones, as qualitatively reported in figure 6.3. Obviously ΔV (x ) = ΔV (x + sa ). The
crystalline Schrödinger problem is therefore written as
⎡ ℏ2 d 2 ⎤
⎢− + ˆc(x )⎥ψc(x ) = Ecψc(x ),
V (8.9)
⎣ 2m dx 2 ⎦

where V̂c(x ) is the quantum operator corresponding to the potential given in

equation (8.8), while ψc(x ) and Ec are the wavefunction and energy of the crystalline

5
We choose to align the monoatomic chain along the x direction.

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 Solid State Physics

states, respectively. For further convenience, we recast this equation in a more

compact form

ℏ2 d 2
Hˆ ψc(x ) = Ecψc(x ) where Hˆ = − + Vˆc(x ). (8.10)
2m dx 2
By multiplying on the left by ψc*(x ) and by taking the integral over the crystal
volume containing the selected N atoms we get

ψc*(x )Hˆ ψc(x ) = Ecψc*(x )ψc(x ) → ∫ ψc*(x)Hˆ ψc(x)dx = Ec∫ ψc*(x)ψ (x)dx. (8.11)

Since it is understood that ψc(x ) is (or can straightforwardly be) normalised we

have

∫ ψc*(x)Hˆ ψc(x)dx = Ec. (8.12)

The calculation of the energies of the valence electrons is therefore traced back to
evaluate integrals like the one appearing on the left-hand side of this equation. To
this aim we should know ψc(x ).
The key idea is to represent ψc(x ) as a linear combination of atomic wave-
functions (LCAO approximation), each centred on a given lattice site. Since we
are now addressing VB states, our physical intuition leads us to use a basis set
made of atomic wavefunctions ϕa(x ) corresponding to valence states which are
filled when the atom is in its ground configuration. This is tantamount to
writing
N
1
ψc(x ) = ∑ exp(iksa) ϕa(x − sa), (8.13)
N s=0

where k is the wavefunction of the electron matter wave. Thanks to the prefactor
1/ N such a wavefunction is properly normalised6. Since the acting potential Vc(x ) is
periodic, we must verify that this guessed form of ψc(x ) has the due mathematical
property defining a Bloch function. To this aim we proceed with the following
calculation
N
1
ψ (x ) = ∑exp(iksa) ϕa(x − sa )
N s=0
N (8.14)
1
= exp(ikx ) ∑exp[ − ik(x − sa)] ϕa(x − sa )
N s=0
= exp(ikx ) u(x ),

6
This holds true under the plain assumption that the atomic wavefunctions are also normalised.

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 Solid State Physics

where it is easy to realise that u(x ) = u(x + a ) as requested (see section 6.3). The
wavefunctions given by equation (8.14) are known as Bloch sums.
Let us then proceed with the calculation of the integral appearing in equation
(8.12): by there inserting the Bloch sum given by equation (8.13) we get
N N
1 1
Ec = ∑ ∑exp( − iks′a )exp(iksa )∫ ϕa*(x − s′a )Hˆ ϕa(x − sa )dx
N N s ′= 0 s = 0
N N
1
= ∑ ∑exp[ik (s − s′)a ]∫ ϕa*(x − s′a )Hˆ ϕa(x − sa )dx (8.15)
N s ′= 0 s=0
N N
1
= ∑ ∑exp[ik (s − s′)a ]∫ ϕa*(x′)Hˆ ϕa(x′ − (s − s′)a )dx′,
N s ′= 0 s=0

where in the last step we performed the change of variable x′ = x − s′a . This is
mostly convenient since, for any value of s selected by the corresponding sum, the
variable s″ = s − s′ assumes the values 0, 1, 2, 3, … upon summing over s′.
Therefore, we can write
N N
1
Ec =
N
∑ ∑ exp[ik (s − s′)a ] ∫ ϕa*(x′)Hˆ ϕa(x′ − (s − s′)a )dx′
s ′= 0 s = 0
N N
1
=
N
∑∑ exp(iks″a ) ∫
ϕa*(x′)Hˆ ϕa(x′ − s″a )dx′
(8.16)
s=0 
s ″=
0 
these values are all equal for any s
N
1
=
N
N ∑ exp(iks″a ) ∫ ϕa*(x′)Hˆ ϕa(x′ − s″a )dx′ .
s ″= 0

It is now useful to split the sum over s″ by isolating the s″ = 0 term

N
Ec = ∫ ϕa*(x′)Hˆ ϕa(x′)dx′ + ∑ exp(iks″a ) ∫ ϕa*(x′)Hˆ ϕa(x′ − s″a)dx′, (8.17)
s ″= 1

so as to separately treat the two terms that appear on the right-hand side of this
equation.
In the first term both the wavefunction ϕa and its complex conjugate are
calculated in the same position x′. Since they are atomic wavefunctions, they are
pretty much localised in the neighbourhood of the ion core they belong to.
Accordingly, if x′ lies in an interstitial region, we can to a good approximation
assume ϕa(x′) = 0 = ϕa*(x′). On the other hand, if x′ is instead close to a core region,
then we can locally assume Vc(x′) = Va , according to equation (8.8). This reasoning
leads to

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 Solid State Physics

⎡ ℏ2 ⎤
∫ ϕa*(x′)Hˆ ϕa(x′)dx′ ≃ ∫ ϕa*(x′)⎢ −
⎣ 2m
∇2 + Vˆa⎥ϕa(x′)dx′
⎦
(8.18)
= Ea ∫ ϕa*(x′)ϕa(x′)dx′
= E a,
because of the already mentioned normalised character of the atomic wavefunction.
In the second term on the right-hand side of equation (8.17) the ϕa wavefunction
and its complex conjugate are calculated in positions (x′ − s″a ) and x′, respectively.
Even in this case we can take profit from their localised character: if the distance s″a
between such positions is large enough, then one out of the two wavefunctions turns
out to be negligibly small. Or, equivalently: the localised character of the basis set
functions used in the linear combination allows us to assume that the integral contained
in this term is nonzero only provided that x′ and (x′ − s″a ) fall near to two first next-
neighbouring lattice positions, that is, only when s″ = ±1. Under this assumption, we
are left to calculate just two integrals for (x′ − s″a ) = x′ ± a , something that can be
done straightforwardly once the atomic wavefunctions are known. We remark that
two such integrals are equal by symmetry and that even now we can set Vc(x′) = Va ,
thus handling them as in the previous case. More explicitly we have
⎡ ⎤
∫ ϕa*(x′)Hˆ ϕa(x′ ± a)dx′ ≃ ∫ ϕa*(x′)⎢⎣− 2ℏm ∇2 + Vˆa⎥⎦ϕa(x′ ± a)dx′ = γ,
2
(8.19)

where γ is in principle calculated once and for all, since all ingredients are known7.
This remarkable results leads to the following dispersion relation EVB(k ) for the
valence band
EVB(k ) = Ea + γ exp(ika ) + γ exp( −ika ) = Ea + 2γ cos(ka ). (8.20)
Typically it turns out that γ > 0.
A similar calculation can be executed for the conduction band, provided that a
properly different set of atomic wavefunctions is used; more specifically, it is
intuitive to assume that we must represent crystalline CB states (that are empty at
T = 0 K) by atomic wavefunctions φa(x ) describing excited states. The correspond-
ing CB integrals of the same kind as in equation (8.19) are typically negative and,
therefore, we conveniently write the dispersion relation ECB(k ) for the conduction
band as
E CB(k ) = Ea′ − 2δ cos(ka ), (8.21)

7
In practice, it is possible to adopt two different strategies: the most fundamental approach consists in the
numerical calculation of the γ integral by using some explicit (exact or approximated) representation of the
atomic wavefunctions [7–9]; alternatively, integrals like this one are fitted on a suitable set of available
experimental data [1, 10]. The first approach is more fundamental and superior, but it could result in a heavy
numerical effort; the second approach is much more computationally light, but it is limited in accuracy and
transferability. In modern solid state physics the two approaches are, respectively, referred to as an ab initio or
a semi-empirical theory.

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 Solid State Physics

ECB (k)

EVB (k)

−π/a +π/a

Figure 8.6. Valence (VB, blue) and conduction (CB, red) band of a monoatomic linear chain with lattice
spacing a. Left: dispersion relations calculated by the tight-binding method and represented in the reduced
zone scheme. Right: corresponding band diagram.

where Ea′ ≠ Ea is the energy of the atomic excited states described by φa(x ) and it is
understood that δ is the absolute value of the integral ∫ φa*(x′)Hˆ φa(x′ ± a )dx′. It is
interesting to observe that both VB and CB dispersion relations can be obtained
directly from the exact result provided by the Kronig–Penney model through
equation (6.33) by exploiting the LCAO approximation [11].
The theoretical procedure we have developed takes the name of tight binding
theory (or method), where the locution ‘tight-binding’ underlines the fact that the
crystalline wave function is built out of atomic wave functions. In this respect, the
tight binding theory is dual to the nearly free electron one: the actual potential felt by
electrons is considered pretty much atom-like (at least in the core regions). As in the
case of phonons (see section 3.2), it is customary to adopt the reduced zone scheme
for plotting the electron dispersion relations; in other words, the EVB(k ) and ECB(k )
curves are plotted for wavevector values within the first Brillouin zone, that is, for
k ∈ [−π /a, +π /a ] as reported in figure 8.6.

8.2.2 Bands in real solids

The tight-binding theory can also be applied to three-dimensional solids [1, 10, 12, 13]
in any possible crystal structure or chemical composition, as well as containing an
arbitrary number of valence electrons. Although the theory is developed in the same
way as described in the previous section, the resulting mathematics is definitely
more complicated, as shown in full detail in appendix G: here we simply outline
the procedure from a conceptual point of view and discuss a few paradigmatic
applications.
The starting point is to write the crystalline wavefunction in a LCAO form by
using a set of suitable localised orbitals {φα lb(r) = φα(r − Rl − Rb)} centred on the

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 Solid State Physics

different ion positions8; the label α stands for the full set of quantum numbers
defining the corresponding state. In principle, such orbitals can be true atomic
wavefunctions which, however, form a non-orthogonal basis set since orbitals
centred on different lattice positions are not so; alternatively, an orthogonalisation
procedure can be operated, as detailed in appendix G, still preserving the s-, p-, d-, …
character of the atomic orbitals.
In order to set up a formalism naturally obeying the Bloch theorem, the following
Bloch sums are defined
1
φαBloch
bk (r) = ∑ ei k·R φα(r − Rl − Rb),
l
(8.22)
N l

where N is the number of unit cells contained in the crystal portion subject to the
periodic Born–von Karman boundary condition. The electron wavefunction for the
nth band is accordingly cast in the following LCAO form
1
ψnk(r) = ∑ B˜nαbφαBloch
bk (r)
Nb αb
1
= ∑ ei k·R B˜nαbφα(r − Rl − Rb)
l
(8.23)
NNb α lb
1
= ∑ Bnαlb(k)φα(r − Rl − Rb),
NNb α lb

where Nb is the number of atoms in the lattice basis, B˜nα b are the LCAO expansion
coefficients, and for brevity we have set Bnα lb(k) = exp(i k · Rl)B˜nα b .
By inserting the trial wavefunction given in equation (8.23) into the eigenvalue
equation9
⎛ ℏ2 ⎞
⎜− ∇2 + Vˆcfp(r)⎟ψnk(r) = En(k)ψnk(r), (8.24)
⎝ 2m e ⎠

we are reduced to solving the secular problem

∑[Hα′ l′ b′,αlb − En(k)Sα′ l′ b′,αlb ]Bnαlb(k) = 0, (8.25)

α lb

where
ℏ2 2
Hα ′ l ′ b ′ , αlb = 〈φα ′(r − Rl ′ − Rb ′)∣ − ∇ + Vˆcfp(r) ∣ φα(r − Rl − Rb)〉, (8.26)
2me

8
We recall that, consistently with the notation used elsewhere in this Primer, the vectors Rl and Rb ,
respectively, indicate the unit cell position and the ion position within the lattice basis.
9
We are working in the single-particle approximation developed in section 1.4.1 and, therefore, each electron is
considered under the action of a crystal field potential Vcfp(r).

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 Solid State Physics

are the elements of the tight-binding matrix while

Sα′ l ′ b′,αlb = 〈φα′(r − Rl ′ − Rb′) ∣ φα(r − Rl − Rb)〉 (8.27)
are the elements of the overlap matrix: in the case where the φα orbitals are
orthogonal, it reduces to the unit matrix.
When implementing this theory in practice, we must cope with a diagonalisa-
tion problem (see equation (8.25)) which must be repeated as many times as the
number of k wavevectors we like to use for plotting the dispersion relations. Since
a diagonalisation task corresponds to an overall computational effort showing a
cubic scaling with the matrix rank, we need strategies to reduce as much as
possible the number of tight-binding and overlap matrix elements to be evaluated
(which is itself a very demanding task). To this aim, we first of all observe that the
number of integrals entering the theory through equations (8.26) and (8.27) can be
really huge, depending on the completeness of the basis set which has been chosen,
as well as on the space extension of the orbitals there included10. Many different
approaches can be followed and approximations can be adopted in order to reduce
this number as much as possible, still keeping a meaningful description of the true
physics. For instance we can choose a minimal basis set consisting of just a few
orbitals for each ion, and/or we can restrict to close-neighbours interactions only,
and/or we can agree to determine the surviving matrix entries through an empirical
fitting procedure, instead of calculating them explicitly by some first principles
method. A detailed discussion of these issues is reported in appendix G. All in all, the
tight-binding method—in whatever flavour—has been extensively applied in solid
state physics and crystalline bands have been accordingly calculated for most
materials [10, 13, 14].
In order to illustrate a practical application of the tight-binding method, we will
illustrate the case of silicon and gallium arsenide, which represent two prototypical
materials of paramount importance for applications in the information technol-
ogy. Silicon is a group-IV elemental semiconductor with diamond structure, while
gallium arsenide is a III–V compound semiconductor with zincblend structure (see
section 2.3.2). In both cases we have eight valence electrons per unit cell (four for
each silicon atom and three or five for the gallium or arsenic atom, respectively).
The dispersion relations will be calculated with a minimal tight-binding basis set
consisting in just a few atomic orbitals whose energy is close to the energy of the
highest VB states and the lowest CB ones: accordingly, we will focus our attention
just on the crystalline states which fall near the forbidden gap. This is justified by
the fact that only electrons in this energy window are affected by thermal
excitation and, therefore, mainly contribute to transport and optical properties.
Nevertheless, it is in principle conceivable the use of a larger tight-binding basis set
which would enable the calculation of a wider spectrum of bands [14]. We will
follow the standard convention to represent the En(k) dispersions by choosing k to

10
It is clear from equations (8.26) and (8.27) that the larger the delocalisation of the φα orbitals, the larger is the
number of integrals to be evaluated.

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 Solid State Physics

vary along the high-symmetry directions of the irreducible part of the 1BZ (see
figure 2.16).
In figure 8.7 the band structure of silicon is reported, where the energy scale has
been chosen by setting its zero at the VB top: the four valence or conduction bands
correspond to negative or positive energies, respectively. This is obviously purely
conventional: the physics does not change if the band structure is rigidly shifted
upwards or downwards by an arbitrary amount; for example, the optical properties
of a crystal solely depend on energy differences between different electronic states
which, of course, are not altered by a rigid displacement of the entire band
structure. At T = 0 K each valence band is fully occupied11 while all conduction
bands are empty. This band structure suggests a number of meaningful physical
features.
First of all, we observe that along some direction bands can be degenerate in
energy. Let us for instance consider the valence bands along Λ and Δ directions: it
seems we have three bands. On the other hand, the same valence bands appear in
the number of four if we look at the Σ direction. Obviously, the number of bands
cannot change by varying the direction of the wavevector in the reciprocal space,
as confirmed by the careful analysis of what happens at the X point as far as the
two valence bands higher in energy are concerned: coming from the Σ direction
they end up overlapping, merging into two different but degenerate sequences of
quantum states along the Δ direction. There are also cases in which degenerate states
are found only at a specific k -point, as at Γ (three degenerate valence bands) or X (two
different pairs of degenerate bands). This adds a very practical motivation to use the
combination of the band index and the wavevector to label the electron crystalline
states.
Next, we can estimate Egap Si
= 1.1 eV . This allows us to elaborate a more
quantitative distinction between insulators and semiconductors than provided in

Figure 8.7. Band structure of silicon, as obtained by a tight-binding calculation (see appendix G for technical
details). The energy gap is in yellow colour. Tight-binding parameters are taken from [15].

11
As well as all other lower-energy bands not represented in figure 8.7. We remark once again that we are not
representing the full band structure, but only its part close to the energy gap. In other words, figure 8.7 is the
three-dimensional counterpart of figure 8.6.

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 Solid State Physics

section 8.1.3. More precisely, it is customary to define semiconductor any non-metallic

crystalline system (i.e. any solid whose last occupied band at T = 0 K is fully filled)
with an energy gap of the order of ∼1 eV. According to this criterion, we classify
silicon as a semiconductor12.
Another interesting feature is that VB top and the CB bottom correspond to
different wavevectors, that is, they are not vertically aligned. Therefore, silicon is
said to be an indirect gap semiconductor. This intrinsic characteristic determines
much of its application potential in technology; for example, this is the reason why
crystalline silicon is not a good material for optoelectronics, as more extensively
discussed in section 9.4.
Let us now turn to consider figure 8.8 where the band structure of gallium arsenide is
reported, following the same conventions and notation as in the previous case. Even
this material can be classified as a semiconductor since we have Egap GaAs
= 1.43 eV . The
most important difference with silicon is that gallium arsenide is a direct gap
semiconductor: its BV top and CB bottom are both found at the Γ point of the
Brillouin zone. On the other hand, the general topological features and degenerate
character of the bands are quite similar to the case of silicon, to report the fact that the
two materials have identical crystalline structure and very similar electronic structure.
We conclude this overview with an observation, full of consequences, about the
two dispersions at the VB top and CB bottom. It is better to focus on the case of
GaAs (neglecting, for simplicity, the degeneracy of its valence states), because it is
more immediate to read, and to consider the band topology just near the Γ point.
The similarity with the bands calculated through the simple one-dimensional model
of section 8.2.1 (see figure 8.6) is really remarkable: this motivates us to continue the
study of the bands in one dimension, confident that their main physical properties
will be (at least qualitatively) similarly valid even in the case of real three-dimen-
sional crystals.

Figure 8.8. Band structure of gallium arsenide, as obtained by a tight-binding calculation (see appendix G for
technical details). The energy gap is in yellow colour. Tight-binding parameters are taken from [15].

12
To give a more direct feeling about the difference between a true insulator and a semiconductor, we observe
SiO2
that the native oxide of silicon, that is SiO2, has an energy gap as large as Egap = 8.9 eV , which corresponds
to an experimentally observed excellent insulating behaviour at any temperature of physical interest.

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 Solid State Physics

In concluding this discussion on the band structure of real solids we remark that
the width of the energy gap is an intrinsic property of an insulating material which,
however, is affected by temperature and by any applied stress, as ultimately due to
the variation of the interatomic distances with respect to the zero temperature
equilibrium configuration. Some details on this issue are reported in appendix H.

8.3 General features of the band structure

Interestingly enough, the general features of the band structure discussed so far are
similarly found both in the nearly free electron and tight-binding models, although
they are based on somewhat opposite approaches to the problem. Therefore, it is
quite reasonable to expect that most of the physical implications deduced from the
band structure predicted by them apply to real solids, where an intermediate situation
is likely expected to occur.
In order to explore the most relevant of such implications, we will investigate a
one-dimensional model solid whose electrons are accommodated on band states
described by dispersion relations E = En(k ). The corresponding matter waves will be
described by ψnk (x ) wavefunctions. This choice of convenience will make the
formalism simple and the underlying physics transparent, but is should be noted
that the band theory can be developed at a more mathematically rigorous level, as
found in [1, 2, 12].

8.3.1 Parabolic bands approximation

If we concentrate on considering only the band portions close to the VB top and the
CB bottom, we recognise that in these regions the bands have a trend that, in very
good approximation, can be described as parabolic (see figures 8.7 and 8.8 at the Γ
point). Therefore, whenever we are interested in studying the physics of electrons in
the proximity of the forbidden gap, we can meaningfully use the so called parabolic
bands approximation.
This approximation is straightforwardly applied in the one-dimensional case by
taking the limit of k → 0 of equations (8.20) and (8.21)
⎡ (ka )2 ⎤
lim E VB(k ) = Ea + 2γ ⎢1 − ⎥
k→0 ⎣ 2 ⎦
(8.28)
⎡ (ka )2 ⎤
lim ECB(k ) = Ea′ − 2δ⎢1 − ⎥.
k→0 ⎣ 2 ⎦

Taking this limit is justified by the fact that at k = 0 we found both the VB top and
the CB bottom, as shown in figure 8.6. A general remarkable feature is drawn: the
thermal excitation of electrons basically occurs within a parabolic band scheme.

8.3.2 Electron dynamics

It is very convenient to address electron dynamics within a semi-classical scheme
according to which: (i) electron energy states are described quantum mechanically,

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 Solid State Physics

but (ii) their equations of motion are classical. This approximated scheme is
trustworthy when one wants to study the motion of the electrons over a length
scale much larger than the interatomic distances. This is, for instance, the relevant
case of motion under the action of an externally applied and slowly varying electric
field, that is an electric field which is practically constant over the length scale of
interatomic lattice distances. On the other hand, the results of this approximation
can hardly be extended to the case of nanostructures [16, 17], that is to solid state
systems whose structural features display on the 10−9 m scale: here a full quantum
theory of electron transport is needed, as detailed elsewhere [12, 17–19].
Let us at first consider the dynamics of an electron in the absence of an external
electric field. If it is accommodated on the nth band, then its velocity vn(k ) is given by
the group velocity of the corresponding matter wave
dωn(k ) 1 dEn(k )
vn(k ) = = , (8.29)
dk ℏ dk
where of course we used the En(k ) = ℏωn(k ) matter–wave duality relation. This
result is as much mathematically straightforward as it is dense in physical
implications: the velocity depends on electron wavevector and the kind of band on
which the electron is accommodated. In other words, electrons with identical
wavevector, but belonging to different bands, have different velocity; equation
(8.29) indicates that their velocity is calculated as the angular coefficient of the band
tangent at the selected k-point. Similarly, an electron accommodated on a given
band will have different velocity depending on its wavevector. Incidentally, this is
one of the most important reasons why it is so necessary to know in detail the band
structure of a solid: our ability to correctly describe the electron dynamics ultimately
depends on it13. In any case, this result indeed represents a major conceptual
difference with respect to a pure free electron theory, where just the same (average)
drift or thermal velocity was attached to each electron. Another intriguing feature is
that a VB electron and a CB electron with the same wavevector move in opposite
directions, as clearly understood by inspection of figure 8.6: indeed an unexpected
result which, as soon to be explained, has many implications in the theory of charge
transport mediated by electrons subject to a periodic crystal field potential.
In the one-dimensional case the calculation of the electron velocity is easy: simply,
we must combine equation (8.29) with equations (8.20) and (8.21) to obtain the
results summarised in figure 8.9. Interestingly enough, this simple case illustrates
quite an important new feature, namely: once a band is selected, it is possible that an
electron placed there has zero velocity (this corresponds to a zone-centre or zone-
boundary wavevector) and it also moves in both possible directions, depending on its
wavevector (in a one-dimensional system, it can move both to the right and to the
left).

13
Soon we will meet a second very important reason of importance of bands, related to the concept of electron
effective mass.

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 Solid State Physics

Figure 8.9. Left: the band structure of a one-dimensional crystal. Left: the velocity v VB(k ) and vCB(k ) for the
valence and conduction band, respectively.

These results are naturally extended to the three-dimensional case by setting

1
vn(k) = ∇k En(k), (8.30)
ℏ
where we understand that ∇k = (∂/∂kx )iˆ + (∂/∂ky )j
ˆ + (∂/∂kz )kˆ . Once again, the
important result is that the velocity of a crystalline electron has magnitude,
orientation and direction depending on its actual quantum state, which is in turn
characterised by the pair of band index and wavevector.

8.3.3 Electric field effects

Let us now apply a constant and uniform electric field E along a one-dimensional
crystal. Within the semi-classical scheme a driving force
dk
F = −e∣E∣ = ℏ , (8.31)
dt
is calculated, governing the drift motion of the electron. The solution of this
equation of motion is
e∣E∣
k (t ) = k 0 − t, (8.32)
ℏ
where k0 is the electron wavevector at time t = 0, that is, when the electric field is
turned on. By making use of equation (8.29) this result reflects in a time-dependent
electron velocity
1 dEn
vn(k , t ) = , (8.33)
ℏ dk (t )

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 Solid State Physics

suggesting the practical rule that under the action of an electric field, the electron
velocity at time t is calculated by evaluating the slope of band tangent at the point k (t )
given in equation (8.32). This result has a quite interesting implication, as we easily
understand by considering the case of an electron in the valence band: under the
action of the electric field, which we consider oriented to the left with no loss of
generality, the wavevector varies linearly with time, assuming gradually increasing
values and, therefore, it will sooner or later end up reaching the right edge of the
1BZ. However, given the crystalline periodicity, the k = +π /a value defines a
quantum state equivalent to the one described by k′ = k + G with G = −2π /a a
reciprocal lattice vector. This is tantamount to saying that the electron, once it
reaches the right edge, is flipped back to a state corresponding the left one. Next, as
time goes by, the electron will again assume increasing wavevector values, as before
eventually reaching the right edge of 1BZ: here its wavevector will be flipped back
once more. And so on … This periodic back-and-forth variation of k (t ) in the
Brillouin zone will continue as long as the electric field is present. This phenomenon
is described by saying that under the action of an electric field a band electron is
subjected to Bloch oscillations: their graphical rendering is reported in figure 8.10.
We remark that this result has been obtained by guessing the equation of motion
(8.31) where no scattering phenomena appear, contrarily to what we discussed in
section 7.1. This is of course a very crude approximation: in practice, it is very
difficult to experimentally observe Bloch oscillations in real materials just because
ionic motions and defects disturb the electron motion. Such oscillations are only
detected at low temperature and in chemically pure systems, since the occurrence of
such circumstances makes the periodic variation of k (t ) only marginally affected by
electron–phonon and electron-defect scattering events or, equivalently, the friction
term appearing in equation (7.3) to play a marginal role.

t1
EVB (k) t2
t3

G = −2π/a
k
−π/a k0 k1 k2 k3 +π/a

Figure 8.10. Graphical rendering of a Bloch oscillation. A valence band electron is initially located on the state
with wavevector k0. By applying a constant and uniform electric field E oriented to the left, the wavevector
changes through the sequence k1 < k2 < k3 < ⋯ which correspond to the values at time t1 < t2 < t3 < ⋯
predicted by equation (8.32). Eventually the electron will occupy a state described by a wavevector falling just
at the right edge of the first Brillouin zone, which is represented by the shaded area. Because of the crystalline
periodicity, this k = +π /a value defines a quantum state equivalent to the one described by k′ = k + G with
G = −2π /a and, therefore, we can state that the electron is effectively flipped back to the left edge of the zone.
This oscillation lasts until the electric field is turned off.

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 Solid State Physics

Let us now derive a new result of major conceptual importance by considering a

completely filled band: in this case we have

∑ vn(k ) = 0, (8.34)
k ∈ 1BZ

because for any electron with velocity vn(k ) we have another one with velocity
−vn(k ), as easily understood by applying equation (8.29) to the band structure
reported in figure 8.9. This implies that the current density vector14 is zero, since it is
proportional to the sum of the velocities of all available charge carriers (in the
present case: all the electrons accommodated on the band). The striking feature of
the full-filling situation is that this conclusion applies even if the crystal is under the
action of an external electric field: although electrons are accelerated by the field as
predicted by equation (8.33), their new velocities nevertheless correspond to values
already found for that band, as summarised in figure 8.10, and therefore corre-
sponding to occupied quantum states. In other words, equation (8.34) still holds
under the action of an electric field. The conclusion is remarkable: a completely filled
band cannot conduct electricity, since the current density vector associated with its
electrons is in any case zero.
This outcome is really very important since it quantitatively justifies the
classification presented in section 8.1.3 where we discussed the possible band filling
situations and correspondingly associated the metallic or insulating behaviour. On
the one hand: metals can transport electricity simply because they have the topmost
band only partially occupied and, therefore, equation (8.34) does not hold or,
equivalently, they have an inherently non-zero current density vector. On the other
hand, the same argument applies to semiconductors to explain their (poor)
conducting behaviour at finite temperature: because of the thermal excitation of
electrons from the VB to the CB (see figure 8.4), a semiconductor material results as
having two partially filled bands which provide a non zero density current vector.
However, since Jq is proportional to the carrier density, we easily understand that
such a material can only be a poor conductor: the Fermi–Dirac distribution function
is significantly blurred by temperature only in the range ∼kBT around the Fermi
level and, therefore, just very few electrons are in fact promoted to the CB, while
only an equally small number of empty states are created in the VB15. Finally, an
insulator has such a large energy gap that the number of electrons actually promoted
to the CB by thermal excitation is so small that their contribution to the current
density vector is totally negligible.

14
The charge current density vector Jq appearing in the Ohm law Jq = σ E is defined as Jq = (∑i qi vi)/V , where
the sum is over all the carriers with charge qi and moving with velocity vi , while V is the system volume. In the
case where all carriers have the same charge q and move with the same drift velocity vd , we recover the
elementary definition Jq = nq vd .
15
This argument rigorously applies to intrinsic semiconductors only, that is, to undoped ones. Doping
significantly alters this picture, as discussed in section 9.1.1.

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 Solid State Physics

8.3.4 Electrons and holes

The case of semiconductors, characterised by two partially filled topmost bands, is
especially intriguing and deserves more attention.
Let us consider a semiconductor with Nval valence electrons and suppose that NCB
ones have been promoted to the conduction band by thermal excitation, while
NVB = Nval − NCB electrons are left in the valence band. If an electric field is applied,
the NCB electrons in CB provide a contribution Jq,CB to the total charge current
density as large as

Jq,CB ∼ ∑ vCB(kocc ), (8.35)

kocc ∈ CB

where the sum runs over the wavevectors corresponding to the occupied CB states.
Similarly, the VB electrons provide a contribution

Jq,VB ∼ ∑ vVB(kocc ), (8.36)

kocc ∈ VB

where in this case the sum runs over the wavevectors of the VB states still occupied
by the unexcited electrons. It is useful to rewrite this current density contribution in a
different way

Jq,VB ∼ ∑ v VB(kocc ) + ∑ v VB(kempty ) − ∑ v VB(kempty )

kocc ∈ VB kempty ∈ VB kempty ∈ VB
(8.37)
∼ ∑ v VB (k ) − ∑ v VB(kempty ),
∀ k ∈ VB kempty ∈ VB

where the ∑kempty∈VBv VB(kempty ) term is obtained by summing over the wavevectors
corresponding to the emptied states in the valence band. By applying the result 8.34,
we understand that the first term on the right-hand side of the above equation is zero
since it contains a sum over all the states of a totally filled band. This leads to the
remarkable result

Jq,VB ∼ − ∑ v VB(kempty ), (8.38)

k empty ∈ VB

which in physical terms is interpreted as if the Jq,VB contribution is given by a new type
of charge carriers which occupy the VB states left empty by excited electrons. These
new carriers have opposite charge with respect to electrons, as proved by the minus
sign appearing in equation (8.38), and they equal in number the thermally promoted
electrons. The new carriers are referred to as holes and, because of their positive
charge, their acceleration has the same direction as the applied electric field. The
picture emerging from this analysis is really compelling: the thermal excitation
phenomena deeply affect the zero temperature insulating behaviour of semiconductors
in that a double population of carriers is generated, respectively, electrons in CB and

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 Solid State Physics

holes in VB. Their number is just the same since temperature cannot of course
generate or annihilate charge carriers.
The advantage of building a theory of electric transport based on two types of
carriers is mainly practical and can be explained in a basic way by using a simple
hydrodynamic analogue. Let us consider the case of a tube only partially filled with a
fluid: by placing the tube in vertical position, we will observe an air bubble at the
top. If we now overturn the tube, we observe a flow of fluid matter downwards that
we could in principle describe by calculating the equations of motion of all the
running molecules. An alternative, but much simpler, way is to describe the observed
situation by calculating the trajectory of just the air bubble; in doing so we introduce
a fictitious ‘air bubble’ particle (which is a ‘hole’ of molecules) with an interesting
characteristic: subject to the action of the gravitational force, it always moves
upwards as if it has a negative mass. The parallel with the case of electrons and holes
in a semiconductor is really transparent: the valence band conduction is more easily
described in terms of the motion of just a few holes, instead of considering all the
unexcited electrons still accommodated there.
Thermally generated electrons (in CB) and holes (in VB) are named intrinsic
carriers since their number is only determined by the combination of intrinsic
materials properties (mainly, the energy gap) and the environment temperature.
Their separate contributions to the total current density vector add up: as a
matter of fact, electrons and holes do carry opposite charges but they also have
opposite velocity at equal wavevector (see figure 8.9) and the current density
is given by products (charge) × (velocity). We nevertheless remark that the
magnitudes of Jq,VB and Jq,CB are different since electrons and holes have unalike
absolute velocity at equal wavevector (the slope of the VB and CD bands are
inherently different).

8.3.5 Effective mass

We now readdress the electron dynamics under the action of an external electric
field E , preliminarily treated in section 8.3.3 where, however, we only considered
the effect of the external force −e∣E∣. Even if we still assume for the moment an
idealised picture where no defects are present and ions are clamped at their ideal
positions, we know that an electron travelling within a crystal experiences as well the
interactions with the lattice and with all the remaining electrons. In short, we are
faced with the problem of inserting the fundamental notion that we are dealing with
band electrons, rather than free electrons.
By considering an infinitesimal time interval dt, the work done by external force
−e∣E∣ changes the electron energy by the amount
dEn(k ) = − e∣E∣ vn(k ) dt
−e∣E∣ dEn(k ) (8.39)
= dt ,
ℏ dk

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 Solid State Physics

leading to
dk −e∣E∣
= . (8.40)
dt ℏ
The rate of change of the electron velocity is
dvn dv dk 1 d 2En(k ) −e∣E∣
= n = , (8.41)
dt dk dt ℏ dk 2 ℏ
so that its equation of motion is conveniently written in the form
⎛ 1 d 2En(k ) ⎞−1 dvn
−e∣E∣ = ⎜ 2 ⎟ , (8.42)
⎝ℏ dk 2 ⎠ dt
which indeed represents a very suggestive way of putting things: the quantity
⎛ 1 d 2En(k ) ⎞−1
m e* = ⎜ 2 ⎟ , (8.43)
⎝ℏ dk 2 ⎠

plays the role of an electron effective mass, fully incorporating any band feature
through the second derivative of the En(k ) dispersion.
The conclusion is striking: whenever we aim at investigating the dynamics of a band
electron under the action of an external force, we can treat it as a free particle,
provided that its rest mass is replaced by the effective mass m e*. It is hard to
underestimate the importance of this unexpected result: we have traced the over-
whelmingly complicated dynamical problem of an electron simultaneously subject
to an external applied field and to the crystalline potential to that of a free particle
just subject to the external force, at the only price of replacing me with m e*. This is yet
another issue supporting the fundamental role played by the band structure in the
theory of the crystalline solid state.
Since the value of m e* depends on the dispersion En(k ) we understand that the
effective mass of an electron is not unique, but it depends on the specific quantum
state on which it is accommodated. In other words, while all electrons have the same
rest mass, they have different m e* values, depending (i) on the actual band they
occupy and (ii) on the wavevector describing their crystalline state, as reported in
figure 8.11 in the case of a one-dimensional crystal. Therefore, within the same
system we will observe that electrons on different quantum states will be differently
accelerated by the applied electric field. Accurate calculations of the bands in real
semiconductors [20–23] show that the values of m e* in typical semiconductors are
smaller than the rest mass value by a factor in between 10 and 100. The effective
mass can also assume negative values, whenever the band assumes a downward
curvature: this implies that an electron is accelerated along the direction of the
electric field (contrary to what happens for a free electron). There are also states for
which the effective mass is infinitely large: they correspond to the flex points of the
E = En(k ) dispersion curve. When an electron is accommodated on such states, an
electric field—regardless of its strength—cannot accelerate it. All features so far
discussed also apply to holes and, therefore we introduce the more refined formalism

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 Solid State Physics

Figure 8.11. Left: dispersion relation (top) and effective mass (bottom) for a conduction band electron (filled
dot). Right: dispersion relation (top) and effective mass (bottom) for a valence band hole (empty circle).

⎛ 1 d 2E CB(k ) ⎞−1 ⎛ 1 d 2EVB(k ) ⎞−1

m e* = ⎜ 2 ⎟ m * = ⎜
h ⎟ , (8.44)
⎝ℏ dk 2 ⎠ ⎝ ℏ2 dk 2 ⎠

for the effective mass of a CB electron and a VB hole, respectively.

The extension to the case of three-dimensional crystals is straightforward, but it
still requires some attention to mathematical details because of the term containing
the second derivative of the band energy. More precisely, in real materials we
calculate a tensorial effective mass for both electrons and holes as
⎛ 1 d 2E (k ) ⎞−1 ⎛ 1 d 2E (k ) ⎞−1
CB VB
m e,* ij =⎜ 2 ⎟ m h, ij = ⎜ 2
* ⎟ , (8.45)
⎝ ℏ dkidkj ⎠ ⎝ ℏ dkidkj ⎠

where i and j are the Cartesian indices. The tensorial character has an important
physical consequence: by applying an electric field E = (Ex, Ey, Ez ) the equation of
motion of, say, an electron must be written in the following matrix form
⎛ ⎞
⎛ Ex ⎞ ⎜ m e,* xx m e,* xy m e,* xz ⎟ ⎛ ax ⎞
⎜ ⎟ ⎜ ay ⎟ ,
−e⎜ Ey ⎟ = ⎜ m e,* yx m e,* yy m e,* yz ⎟ ⎜ ⎟ (8.46)
⎜ ⎟ ⎜ ⎟ ⎝ az ⎠
⎝ Ez ⎠ ⎜ m e,* zx m e,* zy m e,* zz ⎟
⎝ ⎠

where a = (ax , ay, az ) is of course its acceleration. This implies that the net force
along the ith direction is
−eEi = m e,ixax + m e,iyay + m e,iz az , (8.47)
or, equivalently, the acceleration is not necessarily along the direction of the external
field, as the case of a free electron. It is quite intuitive to understand that this

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 Solid State Physics

property, closely related to the underlying band structure, largely affects the charge
transport properties of a real crystal. The very same conclusions are drawn for holes.
The practical calculation of m e* and m h* is made easy by the parabolic band
approximation introduced in section 8.3.1. To be concrete, let us consider the two
paradigmatic cases of silicon and gallium arsenide: figures 8.7 and 8.8 suggest that
within the energy window limited to just above the CB bottom and just below VB
top their bands can be parabolically approximated. Once again: these are the only
crystalline states which are interested by the thermal excitation phenomena and,
therefore, this is the actual energy interval where electrons and holes are, respec-
tively, promoted to and generated in conducting states. Two important details need
nevertheless to be duly taken into account: (i) we must include in our model the
existence of two almost degenerate valence bands; and (ii) the silicon CB is actually
not symmetric around its minimum. Including these features defines the minimum
level of sophistication to elaborate a meaningful picture. The true occurrence of two
valence bands in this parabolic model implies that we should expect the existence of
two different kinds of holes, accommodated on bands with different curvature or,
equivalently, with different effective mass: the higher (smaller) curvature corresponds
to the smaller (higher) value of m h*. It is therefore customary to distinguish between
light holes and heavy holes, respectively.
The model is implemented in practice as shown in figure 8.12 where the bands are
represented as

bottom
ECB
top
EVB

Γ Γ

Figure 8.12. Parabolic band approximation applied to the case of gallium arsenide (left), a direct gap
semiconductor, and silicon (right), an indirect gap semiconductor. The Γ point lies at the centre of the 1BZ.

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 Solid State Physics

Table 8.1. The energy gap Egap (eV units) and the electron m e* and hole m h* effective mass of some
semiconductors. In a few cases distinction is made between the longitudinal m e,* and transverse m e,* ⊥ effective
mass of electrons, as well as between the light hole m lh
* and heavy hole m * effective mass. All effective masses
hh
are in units of the electron rest mass me .

Si Ge AlP AlAs GaP GaAs InP InAs CdTe CdSe GaN

Egap 1.11 0.66 2.43 2.16 2.26 1.43 1.35 0.36 1.50 1.73 3.40
m e* 0.13 0.50 0.13 0.067 0.07 0.028 0.11 0.13 0.20
m e,* 0.98 1.58
m e,* ⊥ 0.19 0.08
m h* 0.67 0.40 0.33 0.35 0.40 0.80
m lh* 0.16 0.04 0.49 0.12
*
m hh 0.50 0.30 1.06 0.50

bottom ℏ2k 2
E CB(k ) = E CB + + f (k )
2m e*
lh top ℏ2k 2
E VB (k ) = E VB − (8.48)
2m lh*
hh top ℏ2k 2
E VB (k ) = E VB − ,
*
2m hh

where f (k ) is a suitable function correcting for the possibly non-perfect parabolic

curvature of the conduction band (in gallium arsenide this term is missing), while m lh*
and m hh* are the light hole and heavy hole effective mass, respectively.

The non perfect symmetry of the conduction band at its minimum (typical of
semiconductors with indirect gap) is properly taken into account by calculating two
different values of the electron effective mass, according to the actual reciprocal space
direction chosen to measure the band curvature. Let us consider the silicon case
reported in figure 8.7: the minimum of the CB is found along the Δ direction, linking
the Γ and X points of the 1BZ. The standard notation is to name ‘longitudinal’ the
effective mass value obtained by considering the band curvature along Δ and to use
the symbol m e,* to indicate it; on the other hand, the value calculated through the
curvature found along a direction normal to Δ is in turn named the ‘transverse’
effective mass with symbol m e,* ⊥. In table 8.1 we report the effective mass values for
electrons and holes as calculated for some semiconductor materials.

8.4 Experimental determination of the band structure

Many experimental techniques used to measure the band structures in solids use
magnetic fields to enable the Landau quantisation of electronic orbits. Other
methodologies are instead optical: they measure either photon absorption or
reflection phenomena occurring during the interaction between a solid specimen

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 Solid State Physics

and some electromagnetic probe. Since the detailed description of such magnetic or
optical techniques can hardly be exploited by means of such an elementary theory of
the solid state as presented in this Primer, we limit ourselves to outlining just the
photoelectron spectroscopy (PES) technique, which is (at least conceptually) very
simple, while directing the interested reader to other textbooks [1, 2, 22, 24] for a
more thorough presentation.
PES is the solid state counterpart of the photoelectric effect [7, 8] in that an
electron initially located on an occupied crystalline band state with energy En is
promoted, upon absorption of a photon with energy ℏω, to an empty state with
energy Eempty above the vacuum level E vacuum of the investigated material16. The
excited electron eventually escapes from the solid, moving as a free particle with
kinetic energy Eout simply given by the balance
Eout = Eempty − E vacuum
 (8.49)
= (En + ℏω) − E vacuum.
While the photon energy ℏω is controlled by the experimental setup, a direct
measurement of Eout allows us to determine the energy En of the crystalline band
state (the vacuum level is known). The intensity of the PES signal is proportional to
the number of electrons occupying the initial state or, equivalently, to the density of
electronic states at energy En.
From this simplified explanation it is deduced that PES just provides the band
diagram reported in figure 8.2 left, but not the dispersion relations E = En(k) which
indeed require the knowledge of both the wavevectors and the corresponding
energies. This limitation is overcome by a more advanced version of this exper-
imental technique, known as angle-resolved photoelectron spectroscopy (ARPES).
We observe that, in general, the photoemission process must conserve both energy,
as reported in equation (8.49), and momentum. In particular, it is proved [1] that
parallel to the crystal surface the electron momentum obeys the following con-
servation law
ℏk (surf)
n = ℏk (surf)
empty + ℏG
(surf)
, (8.50)

where G(surf) is a reciprocal lattice vector of the surface17, while ℏk (surf)

n and ℏk (surf)
empty
are the surface projections of the initial and final electron momenta. Therefore, if we
measure the angle θ at which the electron is emitted18, then we can write [1]
ℏ∣k (surf)
empty∣ = (2m eEout )
1/2
sin θ , (8.51)

16
The vacuum level is the energy of a free electron outside a solid material. The minimum work that must be
spent on promoting a crystalline electron to the vacuum level is commonly referred to as the work function of
that material.
17
Crystalline surfaces have their own two-dimensional crystallography which, similarly to the bulk one, is
described in terms of direct and reciprocal lattice vectors [25–27].
18
More precisely, θ is the angle formed by the crystal surface normal and the emission direction of the electron.

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 Solid State Physics

which in turn provides ℏkn through equation (8.50). Excluding the details of the
experimental apparatus and protocol of measurement, this is enough to understand that
equations (8.49) and (8.51) are enough to reconstruct the En versus k relationship.

8.5 Other methods to calculate the band structure

The accurate determination of the band structure of crystalline solids is an art
extensively developed in the second half of the XXth century, in parallel with the
development of increasingly powerful digital computers: advances in theoretical
methods and numerical techniques have been tightly interlaced and mutually
beneficial. A number of different methods have been set up, the tight-binding
approach—here privileged for pedagogical reasons—being just one among many
others. The mathematics of such methods represents a very technical issue of the
solid state theory, which is fully exploited elsewhere [2, 12, 13]. Here we limit
ourselves to outlining some general features.
The conceptual framework is that defined by the adiabatic, frozen core, non-
magnetic, and single-particle approximations. The Schrödinger problem to be solved is
provided by equation (1.22), where the local potential Vcfp(r) acting on the electron is
typically determined by a self-consistent procedure. Also, the single-particle wave-
function must have the form of a Bloch wavefunction. Finally, core and valence
wavefunctions have a remarkably different space dependence: they both display strong
atomic-like oscillations near each ion, while in the interstitial regions core wave-
functions are vanishingly small and valence ones are instead slowly-varying plane-wave
like. This ultimately dictates that core and valence wavefunctions are orthogonal.
A first class of band structure methods is based on the idea of representing the
crystalline states as Bloch wavefunctions independent of the energy of the valence
state of interest. This is the case of the tight-binding method where atomic orbitals
are used to create a Bloch state; the same concept is also adopted by using
orthogonalised plane waves (OPWs), where the orthogonality between core and
valence states is enforced by constructing the valence Bloch state by means of plane
waves suitably orthogonalised to core states.
A different choice is operated in the so called cellular methods, where just a single
Wigner–Seitz cell is considered, where the single-electron potential is approximated
within a sphere centred on each lattice site so as to describe an isolated ion (and,
therefore, is spherically symmetric), while outside the sphere is taken to be zero. The
radial Schrödinger equation for such a muffin tin potential is then solved and these
solutions are used as a basis set for the crystalline wavefunctions. More specifically, a
set of augmented plane waves (APWs) are generated, consisting in a combination of a
spherical wave (describing the electron state within the Wigner–Seitz cell) and a plane
wave (describing the electron state in the interstitial regions). Suitable boundary
conditions are applied at the borders of the Wigner–Seitz cell.
Finally, a completely different strategy is to eliminate the core states from our
electronic structure problem. This can be done by replacing the real crystalline
potential with a suitable pseudo-potential providing the accurate determination of
both valence and conduction band states. The key idea is to replace the true strong

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 Solid State Physics

potential with a much softer one within the core regions. This replacement is
operated under the constraint that the corresponding pseudo-wavefunctions do
represent the genuine crystalline wavefunctions outside the core region, while no
care is played within it. Also, the pseudo-wavefunctions must be norm-conserving.
The pseudo-potential calculations are typically performed by OPWs expansions.

References
[1] Singleton J 2001 Band Theory and Electronic Properties of Solids (Oxford: Oxford University
Press)
[2] Ashcroft N W and Mermin N D 1976 Solid State Physics (London: Holt-Saunders)
[3] Kittel C 1996 Introduction to Solid State Physics 7th edn (Hoboken, NJ: Wiley)
[4] Miller D A B 2008 Quantum Mechanics for Scientists and Engineers (New York: Cambridge
University Press)
[5] Griffiths D J and Schroeter D F 2018 Introduction to Quantum Mechanics 3rd edn
(Cambridge: Cambridge University Press)
[6] Eisberg R and Resnick R 1985 Quantum Physics of Atoms, Molecules, Solids, Nuclei, and
Particles 2nd edn (Hoboken, NJ: Wiley)
[7] Colombo L 2019 Atomic and Molecular Physics: A Primer (Bristol: IOP Publishing)
[8] Demtröder W 2010 Atoms, Molecules and Photons (Berlin: Springer)
[9] Bransden B H and Joachain C J 1983 Physics of Atoms and Molecules (Harlow: Addison-
Wesley)
[10] Harrison W A 1980 Electronic Structure and the Properties of Solids (New York: Dover)
[11] Morrison M A, Estle T L and Lane N F 1976 Quantum States of Atoms, Molecules, and
Solids (Upper Saddle River, NJ: Prentice-Hall)
[12] Grosso G and Pastori Parravicini G 2014 Solid State Physics 2nd edn (Oxford: Academic)
[13] Bassani F and Pastori-Parravicini G 1975 Electronic States and Optical Transitions in Solids
(Oxford: Pergamon)
[14] Papacostantopoulos D A 2015 Handbook of the Band Structure of Elemental Solids 2nd edn
(New York: Springer)
[15] Vogl P, Hjilmarsson H H and Dow J D J 1983 Phys. Chem. Solids 44 365
[16] Davies J H 1998 The Physics of Low-dimensional Semiconductors (Cambridge: Cambridge
University Press)
[17] Di Ventra M 2008 Electrical Transport in Nanoscale Systems (Cambridge: Cambridge
University Press)
[18] Ridley B K 1988 Quantum Processes in Semiconductors (Oxford: Oxford University Press)
[19] Heikkilä T T 2013 The Physics of Nanoelectronics (Oxford: Oxford University Press)
[20] Grundmann M 2010 The Physics of Semiconductors (Heidelberg: Springer)
[21] Cardona M and Yu P Y 2010 Fundamentals of Semiconductors (Heidelberg: Springer)
[22] Seeger K 1989 Semiconductor Physics (Heidelberg: Springer)
[23] Balkanski M and Wallis R F 1989 Semiconductor Physics and Applications (Oxford: Oxford
University Press)
[24] Fox M 2001 Optical Properties of Solids (Oxford: Oxford University Press)
[25] Prutton M 1983 Surface Physics (Oxford: Oxford University Press)
[26] Zangwill A 1988 Physics at Surfaces (Cambridge: Cambridge University Press)
[27] Bechstedt F 2003 Principles of Surface Physics (Berlin: Springer)

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 IOP Publishing

Solid State Physics

A primer
Luciano Colombo

Chapter 9
Semiconductors

Syllabus—The physics of charge carriers in a semiconductor and related transport

phenomena is developed considering both the case of a drift current generated by
electric fields and the case of a diffusion current generated by concentration gradients.
In this context we introduce the concept of mobility, relaxation time and conductivity,
which have paramount importance in semiconductor physics and engineering. We then
move on to study the carrier statistics in a semiconductor at thermal equilibrium,
eventually leading to the general laws that predict the concentration of charge carriers
in intrinsic and extrinsic semiconductors. Next, we outline the non-equilibrium
processes involving the generation and recombination of carriers and we obtain the
most general form for the continuity equation for the charge current. This chapter ends
with a phenomenological description of the optical properties of semiconductors,
mainly developed through the concept of inter-band transitions.

9.1 Some preliminary concepts

In our presentation of elementary solid state physics, semiconductors play a twofold
pedagogical and applicative role: on the one hand, they represent a paradigmatic
playground for developing a microscopic theory of charge transport in non-metallic
systems (that is in systems where the free electron model cannot be used as effectively
as in metals); on the other hand, they have a fundamental role in modern
information and communication (nano)technologies. For these reasons they deserve
the careful treatment here developed where both issues are addressed, together with
a statistical treatment of charge carrier populations either in equilibrium condition
or out of equilibrium.
Throughout this chapter we assume that the band structure of a semiconductor is
known, as obtained by applying anyone of the methods developed in chapter 8.
Therefore, concepts like the valence (VB) and conduction (CB) band, the energy gap
and the effective mass of charge carriers will be extensively used.

doi:10.1088/978-0-7503-2265-2ch9 9-1 ª IOP Publishing Ltd 2021

 Solid State Physics

9.1.1 Doping
We will distinguish between intrinsic and extrinsic semiconductors, provided that a
pristine sample or a doped one (see section 2.5.1) is, respectively, considered.
Basically, they differ in that only thermally excited carriers from the VB to the CB
are available in pristine materials (hereafter referred to as intrinsic carriers), while
additional electrons in CB and holes in VB are added in doped semiconductors
(hereafter referred to as extrinsic carriers). This is of course the result of doping, that
is, the alteration of the chemistry of the pristine material by insertion of donor or
acceptor impurities: in the first case, the impurity atoms carry an excess of electrons
with respect to the pristine material, while in the second case they lack of electrons
or, equivalently, they carry an excess of holes with respect to the intrinsic
population. In engineering applications doped semiconductors are mainly used.
Given the fundamental role played in semiconductor physics by doping, a
preliminary question should be addressed, namely: what is the effect of doping on
the underlying band structure of the material? The answer to this question is not at all
trivial, since it requires the extensive use of the quantum mechanical perturbation
theory: the presence of a dopant defect is described as a perturbation to the ideal case
of perfect crystal and the way such a perturbation affects the band structure is
accordingly calculated1. The graphical rendering of this concept is reported in
figure 9.1, which will soon be explained in full detail.
In principle, by using perturbative methods it is possible to rigorously calculate
the effects of dopant species both on the electron energy levels (and, therefore, the

Figure 9.1. Graphical rendering of the effect of n-doping on the crystal field potential (Kronig–Penney model)
in silicon.

1
The tacit assumption is that the density of dopants is not too high, so we can still think in terms of a pristine
material—with its own band structure—and a perturbation due to the dirty chemistry. There are cases of
extremely high doping levels for which this assumption is actually not valid and, therefore, a different
theoretical approach should be followed. This topic is treated in specialised textbooks [1–4].

9-2
 Solid State Physics

new band structure) and on the wavefunctions describing the corresponding

quantum states (and, therefore, their localised or extended character). This is a
heavy indeed approach and we rather prefer a semi-quantitative argument based on
the Kronig–Penney model (see section 6.4) which, despite its simplified character,
nevertheless leads to the correct framing of the problem.
Let us consider the case of an ideal one-dimensional semiconductor: the crystal
field potential felt by an electron in conduction band is represented in figure 9.1, top
left. To make things concrete, we suppose that the semiconductor is silicon. Let us
now suppose that, following a doping process, a single lattice site has been occupied
by an arsenic atom: this situation is usually referred to as n-doping, where n stands
for ‘negative’ as justified by the observation that arsenic injects an excess electron
(which is a negative charge) into the pristine material. To a very good approx-
imation, we can describe the variation of potential due to the Si → As substitution as
a variation of the depth of the potential well at the lattice site where this substitution
has occurred. Since the atomic number of arsenic is greater than that of silicon [5],
we can qualitatively predict that the variation consists in a depth increase of the well:
the As nucleus more strongly attracts an electron than a silicon nucleus does. This is
graphically rendered in figure 9.1, top right.
After doping, the resulting state of affairs can be described as the sum of an ‘ideal
situation’ and a ‘perturbation’: the ‘ideal situation’ corresponds to a potential profile
typical of a non-doped crystal, while the ‘perturbation’ only consists in a depth
variation of the potential well at the lattice site interested by the Si → As substitution.
Graphically, the situation is described in figure 9.1, bottom right. The energy levels of
the doped crystal are ultimately obtained by adding to the normal band structure of
the ideal crystal the discrete levels confined in the additional potential well generated
by doping. In practice, it is assumed that the additional well is so shallow that only one
state can be there confined. Moreover, the explicit perturbative calculation provides
evidence that the new energy level falls within the forbidden gap of the intrinsic
semiconductor. In particular, in the case of n-doping this level always falls near the CB
bottom. In summary, this simple model allows us to state that the effect of a single Si
→ As replacement on the silicon band structure is equivalent to the addition of a single
energy level in the forbidden gap, just below the conduction band. As befits electron
states of a potential well, also the new state generated by the doping is confined at the
site interested by the Si → As replacement: for this reason, it is called impurity level.
Equivalently, we can state that the matter wave describing an electron accommodated
on the impurity level is mostly confined nearby the dopant site.
To complete the picture, we must clarify an important issue: so far it was
supposed to insert just one dopant atom into the host lattice. In fact our model is
still valid even if many dopant impurities are placed at as many lattice sites, as long
as their density is not too high. If the dopants are sufficiently diluted, they are on
average at a fairly large distance from each other. Because of their confined
character, the new impurity levels do not interact2: each will be confined in its

2
Equivalently: the tails of the matter waves describing the impurity states have a negligibly small overlap.

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 Solid State Physics

own well. Since all the wells are just the same (all dopants are of the same chemical
species), each of them contributes to the band structure with the same impurity level
as all the others. In brief: if they have been inserted ND doping elements and a low-
dilution situation is in fact verified, then the band structure will have only one
ND-fold degenerate impurity level.
The model we have developed can be straightforwardly extended to the case of a
dopant lacking a valence electron with respect to the replaced silicon atom: a
situation that is reported as p-doping. Let us consider, a Si → B substitution: since
boron has a smaller atomic number than silicon [5], the corresponding potential well
is shallower. The situation is summarised in figure 9.2. There should be noted an
important difference with respect to the previous case: the term describing the
‘perturbation’ seems to represent a potential barrier, rather than a well. However, we
remark that in this case we are discussing the motion of the carriers in the valence
band: therefore, we are allowed to look at the barrier as a potential well for the holes
accommodated in the VB. All the previous results also apply to p-doping: the
impurity levels are degenerate in energy (in the low dilution limit) and they lie in the
forbidden gap, but this time just above the top of the VB (as proved by an explicit
perturbative calculation).
Finally, in figure 9.3 we sketch the band structure of a semiconductor in the
intrinsic case as well as in the two different cases of n- and p-doping. The impurity
levels for n-doping are called donor levels because at T = 0 K they are fully occupied
by electrons that, upon heating, can be easily excited to the CB. In other words, since

Figure 9.2. Graphical rendering of the effect of p-doping on the crystal field potential (Kronig–Penney model)
in silicon.

Figure 9.3. The band structure of an n-doped (left), intrinsic (middle), and p-doped (right) semiconductor.
Impurity levels are marked by a thick black line: they lie just below the CB minimum and just above the VB
maximum in the case of n-doping and p-doping, respectively.

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 Solid State Physics

the donor levels lie just below the CB minimum, the thermal excitation mechanism is
much more efficient for them, than for the intrinsic electrons in the VB. The levels
introduced by p-doping are, instead, called acceptor levels because at T = 0 K they
are empty and become populated by thermal excitation of electrons (eventually
trapped there) from the VB.
Nowadays, the exact position of donor and acceptor levels in the gap of any
semiconductor is precisely known [3] and, therefore, it is possible to engineer the
band structure according the specific application of interest. In practice, by selecting
the kind and density of dopant impurities it is possible to have a (doped) semi-
conductor with any density of holes in VB and/or electrons in CB as required by the
target application.

9.1.2 Density of states for the conduction and valence bands

For further convenience we elaborate an explicit expression of the eDOS (previously
introduced in section 7.3.1 only for metals) for a specific semiconductor band n.
To this aim, we will combine the parabolic bands approximation outlined in section
8.3.1 with the concept of effective mass introduced in section 8.3.5.
Let us consider a generic band En(k) and indicate by Gn(E )dE the corresponding
number of electron states with energy in the interval [E , E + dE ]. If we know the
single-electron energies, then we can calculate the eDOS Gn(E ) as
V
Gn(E ) = ∑ δ(E − En(k)) =
(2π )3
∫k∈1BZ δ(E − En(k)) d k, (9.1)
k ∈ 1BZ

where the discrete sum takes into account that only a set of discrete k -points are
rigorously allowed in the 1BZ, while the integral expression is a good approximation
valid in the limit of a very large crystal when we can treat the wavevector as a
continuous variable3. It is convenient to normalise the eDOS with respect to the
crystal volume, so as to obtain an expression independent of geometrical factors. To
this aim, we introduce the density of states per unit volume gn(E ) for the nth band as
Gn(E ) 1
gn(E ) =
V
=
(2π )3
∫k∈1BZ δ(E − En(k)) d k, (9.2)

which allows us to directly calculate the total density of states (per unit volume) as
gtot(E ) = ∑n gn(E ).
While equation (9.2) represents the most accurate way to calculate the eDOS for
each band, in the following we can take profit from the twofold fact that: (i) most of
the semiconductor physics is ruled over by carriers in the proximity of the forbidden
gap; and (ii) the valence and the conduction bands are to a very good approximation

3
We have similarly found this twofold way of writing a density of states in section 4.1.3, where the phonon
density of states was used in both forms to evaluate the heat capacity. In general, each allowed k wavevector
has a volume Δk = (2π )3/V in the reciprocal space, so that for any function f (k) it holds
∑k f (k) = [V /(2π )3]∑k f (k)Δk . In the limit Δk → 0 (that is in the limit of a very large crystal) we can
therefore replace any sum ∑k (⋯) with the integral expression [V /(2π )3] ∫ (⋯) d k .

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 Solid State Physics

parabolic near such a gap. This twofold observation greatly simplifies the mathe-
matics since we can straightforwardly extend to semiconductors the result given in
equation (7.28), which was derived under a similar parabolic k-dependence of the
energy in the free electron gas. Therefore, we define the CB density of states per unit
volume ḡCB(E ) in parabolic bands approximation as

4π 3/2
g¯CB(E ) = (2m e*) (E − Ec )1/2 , (9.3)
h3
where Ec is the energy at the bottom of the CB. Similarly, we define the VB density of
states per unit volume ḡVB(E ) in parabolic bands approximation as

4π 3/2
g¯VB(E ) = (2m h*) (E v − E )1/2 , (9.4)
h3
where E v is the energy at the top of the VB.
Equations (9.3) and (9.4) are deceptively simple at first sight, but they hide a
subtlety: the effective mass is used for both electrons and holes, but it is left
ambiguous which actual value should be used. More specifically, we must be more
accurate in defining whether we should use the m lh* or the m hh
* mass for holes, or the

m e, or the m e, ⊥ mass for electrons, as explained in section 8.3.5. In order to work

* *

out a general theory, it is customary to define effective masses suitably averaged for
the density of states calculations: they are usually referred to as density of states
effective masses.
Let us consider first the case of three-dimensional non-perfectly parabolic
conduction band. In this case the electron tensorial mass defined in equation
(8.45) is replaced by an average scalar quantity defined as
1/3
m e* = (m e,*xx m e,*yy m e,* zz ) . (9.5)

If we further consider the case of a non-symmetric CB shown in figure 8.12, typically

found in indirect gap semiconductors with diamond structure, we conveniently
define a density of states effective mass which properly takes into account the CB
anisotropicity by setting

⎡ 2 ⎤1/3
m e* = N 2/3 ⎣m e,* (m e,* ⊥ ) ⎦ , (9.6)

where the numerical pre-factor N appears since the CB has N equivalent minima (for
silicon N = 6, while for germanium N = 4).
Finally, if we consider the case of degenerate valence bands (see figure 8.12), we
define a density of states effective mass which properly takes into account the
existence of light and heavy holes by setting
* )3/2 ]2/3 ,
m h* = [(m lh* )3/2 + (m hh (9.7)

which holds for semiconductors with both diamond and zincblend structure.

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 Solid State Physics

We conclude this part by pointing out that, in order not to overcharge the
formalism, we have used the same symbol for the effective mass and the density of
states effective mass. This will not generate any ambiguity in the following.

9.2 Microscopic theory of charge transport

Charge carriers can be accelerated, so as to give rise to a conduction regime, either:
(i) by an external electric field (which generates a drift current); or (ii) by the
existence of a spatial gradient of their concentration (which generates a diffusion
current). In both cases we have to explicitly consider the contributions to the current
density Jq of both electrons and holes.

9.2.1 Drift current in a weak field regime

Let us consider the case in which an external electric field E is applied to a
semiconductor. For simplicity, we assume a constant, spatially uniform and not too
intense field. This last condition is normally referred to as the weak field regime and
it corresponds to a wide class of situations really encountered in engineering
applications. From a physical point of view, we can consider ‘weak’ an electric
field provided that it is able to accelerate electrons only up to a velocity not larger
than their classical thermal velocity veth = 3kBT /m e* .
The semi-classical calculation that solves the CB electron dynamics proceeds
similarly to section 7.2 and the key result provided by equation (7.6) only needs to be
corrected by replacing the electron rest mass by its effective counterpart, which
introduces full information on the underlying band structure, thus obtaining4
n ee 2τe
Jq,CB = −n ee vd = E, (9.8)
m e*
from which we understand that the CB electron transport is ruled over by three
intrinsic material parameters, namely: (i) the electron effective mass m e*; (ii) the
relaxation time τe for the scattering events; and (iii) the density n e of the electrons in
the CB. While we have already developed robust knowledge on m e*, we still need to
deepen the analysis of τe and n e . This will be done soon, respectively, in sections 9.2.2
and 9.2.3, but we can anticipate that the important issue for technological
applications is that all these quantities are not assigned once and for all, but instead
can be engineered according to specific needs. In fact: the number of electrons in the
CB can be controlled by the doping level; the effective mass of the carriers is
determined by the band structure, that is by the selected semiconductor; the
relaxation time is governed by the kind and by the amount of imperfections present
in the lattice (both features affect the frequency of electron-defect scattering events),
as well as by the temperature (which determines the phonon population and,
therefore, the occurrence of electron–phonon scattering events). Given the

4
We are using the same notation introduced in section 8.3.4.

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 Solid State Physics

fundamental role played in semiconductor transport theory by the combination of

these three parameters, it is quite useful to define the electron mobility
eτ
μe = e , (9.9)
m e*
which is typically expressed in units [cm2 V−1 s−1]. This quantity provides a direct
measure of the ease with which electrons can be accelerated, since the mobility
increases with increasing relaxation time (i.e. with decreasing frequency of scattering
events) and decreasing effective mass (which measures inertia to motion). The direct-
current electrical conductivity is written as
e 2τe
σe = n e = n e e μe , (9.10)
m e*
which is the counterpart of equation (7.7) valid for metals.
The above analysis is not complete since at T > 0 K in any semiconductor
(regardless if doped or intrinsic) both electrons and holes are in principle found,
respectively, in CB and VB. In other words, we must duly take into account the
existence of two kinds of charge carriers. The hole transport in VB is basically
described by the same equations (9.8)–(9.10) as above, provided that suitable values
for the effective mass m h*, relaxation time τh and carrier density n d are used. In brief,
the total current density Jq,tot in a semiconductor is written as
Jq,tot = Jq,CB + Jq,VB = (σe + σh ) E = σtot E, (9.11)
where we have introduced the total direct-current conductivity
σtot = e(n eμe + nhμh ), (9.12)
as the sum of the electron and hole ones. This equation makes it clear that the
electron and hole contributions to the total current are always additive (that is, they
have the same sign) since electrons and holes carry an opposite charge, but also
move with opposite drift velocity under the action of the external electric field E . The
electron and hole mobility values in some semiconductors are reported in table 9.1.

9.2.2 Scattering
As anticipated, we now address the relaxation time topic. In order to make the
physical picture clear, we are going to discuss it in the specific case of electrons.

Table 9.1. Room temperature electron μe and hole μh mobility of some semiconductors in units cm2 V−1 s−1.

Si Ge AlP AlAs GaP GaAs InP InAs CdTe CdSe GaN

μe 1350 3900 80 1000 300 8500 4000 22600 1050 650 300
μh 480 1900 180 150 400 600 200 100

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 Solid State Physics

The approach to the scattering problem will be phenomenological, a more advanced

and rigorous discussion is reported in [6].
In a real semiconductor there is a great variety of electron scattering mechanisms,
basically due to two different classes of phenomena: (i) scattering by lattice
vibrations; and (ii) scattering by defects (native, contaminant or dopant), either
charged or electrically neutral. They differently affect electron mobility and we
assume to be valid, as in the case of metals discussed in section 7.3.3, the Matthiessen
rule, according to which each scattering mechanism acts individually. Therefore, the
resulting total relaxation time τe,tot is given by
1 1
= ∑ , (9.13)
τe,tot i
τe,i

where the sum runs over all the different scattering mechanisms labelled by the i
index. This assumption is useful to develop a very detailed analysis by raising the
question: what would be the electron mobility μe,i if only the ith scattering mechanism
was present? It is possible to answer to this question since the direct consequence of
equation (9.13) is that we can write
1 1
μe,tot
= ∑μ , (9.14)
i e,i

where μe,tot is the real electron mobility5. Since any mechanism has a very specific
temperature dependence, the Matthiessen rule allows for studying them separately,
thus determining which one dominates in different temperature regimes.
In order to address the scattering by lattice vibrations it is very convenient to
adopt the phonon picture, which allows for a very effective corpuscular language.
The scattering phenomena are therefore described as impacts between particles
(electrons/holes and phonons, respectively) which, in a weak field regime, are treated
as elastic: phonon scattering events modify electron trajectories, leaving their
energies unaffected.
Long wavelength acoustic phonons give rise to a periodic compression/dilatation
of the lattice: it is as if a spatially modulated pressure was applied. Pressure, in turn,
alters the band structure6 and it is precisely in this phenomenon that the effect of
scattering from acoustic phonons resides: due to the band structure modification, the
effective mass of electrons is affected, and so is their mobility. Since the amplitude of
acoustic oscillations depends on temperature, electron mobility will also depend on
temperature. It is experimentally found that for this mechanism μe ∼ 1/T 3/2 , that is:
the scattering by acoustic phonons dominates (or, equivalently, it is maximally
effective in decreasing the electron mobility) at high temperatures. In the case of
elemental semiconductors this is the leading mechanism due to lattice vibrations.
In compound (that is, polar) semiconductors long wavelength optical phonons
generate an internal electric field that, obviously, interacts with the charge carriers

5
Real in this framework means the experimentally measured value.
6
In appendix H we outline the variation of the energy gap, as a simple example of pressure effects.

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 Solid State Physics

altering their dynamics: also in this case the scattering by optical phonons results in a
decreasing mobility as the temperature increases. This dependence is very complex,
but if the temperature is not too high it is experimentally found that for this
mechanism μe ∼ 1/T .
Finally, piezoelectric scattering is observed in those semiconductors that show
polarisation under deformation. This is for instance the notable case of GaAs and
II–VI semiconductors: acoustic phonons induce polarisation and the resulting
electric field affects electron mobility. For this mechanism, experimental data
provide a μe ∼ 1/T1/2 dependence.
We now move to consider the scattering by defects and, to keep things simple, we
limit our discussion to point defects only. Let us start by considering the case of a
charged defect, which is commonly associated with the presence of a contaminating
impurity or an ionised dopant: an electron approaching the defect site will be
affected by Coulomb attraction or repulsion which, of course, will affect its
dynamics and ultimately its mobility. Also for the Coulomb scattering there is
observed a dependence on temperature that, this time, is found in the form
μe ∼ T 3/2 . The result is qualitatively explained by observing that upon increasing
temperature, the carrier kinematics becomes faster and, therefore, electrons on
average remain near charged defects for a shorter time: ultimately, the scattering
mechanism becomes less effective. Limited to this case, we can conclude that the
mobility increases with increasing temperature.
Finally, there is also observed the scattering by neutral (not electrically charged)
defects. These are non-ionised dopants or contaminants, as well as isovalent
substitutional defects (for example: a Ge atom replacing a Si atom in a silicon
crystal). In this case the scattering is purely mechanical, basically due to the local
variation of the crystal potential in the proximity of the defect. This scattering
phenomenon modifies the dynamics of the carriers, but it is practically independent
of temperature; in any case it is measured to be comparatively much less important
than the other mechanisms.
In figure 9.4 we report the temperature dependence of the electron mobility in an
n-doped GaAs sample: the different μe,i values (corresponding to an ideal mobility
only limited by just the single ith scattering mechanism) are shown by coloured lines,
while the experimental (total) mobility is shown by dots. In order to qualitatively
justify the observed T-dependence, we can focus on the two main scattering
mechanisms, namely: the optical phonon and Coulomb ones. Accordingly, we
approximate equation (9.14) as follows

1 1 1
∼ + , (9.15)
μe,tot μe,opt.phonon μe,Coulomb

and elaborate the following rationale: at low temperatures the Coulomb scattering
mechanism dominates in limiting the electron mobility and, therefore, we can set
μe,tot ∼ μe,Coulomb ; in the opposite regime of high temperatures, the optical phonon
scattering mechanism dominates and therefore we approximate μe,tot ∼ μe,opt.phonon ;

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 Solid State Physics

Figure 9.4. Black dots: electron mobility as function of temperature (bi-log scale) in an n-doped GaAs sample
(data taken from [7]). Colour lines: contribution of each single scattering mechanism.

at intermediate temperatures, experimental data show the transition between the two
limiting situations.
We remark that (i) all the arguments developed here apply both in the case of
electrons and in the case of holes and (ii) all the conclusions drawn in this section are
actually generally valid for any semiconductor. In particular, this holds for the
Coulomb scattering which, in doped semiconductors, is governed by the actual
concentration of donors or acceptors. This gives rise to a new question: how does the
carrier mobility depend on the dopant concentration? The experimental data show
that at low doping levels the carrier mobility is basically unrelated to the actual
number of impurities per unit volume, while in heavily doped samples the mobility
monotonically decreases with increasing defect concentration. This is shown in
figure 9.5 in the case of silicon, for both electrons and holes. The experimental trends
are explained by these simple arguments: at low dopant concentrations, the carrier
scattering is dominated by phonon-like mechanisms, while the Coulomb mechanism
plays a marginal role; therefore, the mobility in this regime is basically constant (we
are keeping constant the temperature of the sample). In the opposite case of heavy
doping, the Coulomb scattering dominates in increasing reason with the number of
impurities present: more dopants → increased scattering efficiency → smaller carrier
mobility.
The result shown in figure 9.5 deserves a further comment. Depending on the
specific technological application, we may be interested in maximising either the
density of carriers or their absolute mobility. In this respect, figure 9.5 clearly shows
that the two conditions of ‘high doping’ and ‘high mobility’ are in contrast as
regards their overall effect on the ability to conduct electric current. The mutual
influence of temperature and doping level is revealed in a complex dependence of

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 Solid State Physics

Figure 9.5. Electron and hole mobility in silicon at room temperature as function of the dopant concentration
(semi-log scale). Data taken from [8].

mobility on these two factors, as shown in figure 9.6 in the specific case of electrons
in an n-doped silicon sample7. So, as far as the ability to support high intensity
electric currents is concerned, the ‘best’ semiconductor is not necessarily the one with
maximum carrier mobility (a condition that necessarily corresponds to a low
concentration of carriers, given the fact that their number is obviously determined
by the number of dopant atoms), nor the one with very high concentration of
carriers (a condition that necessarily implies a low mobility). In other words, in
practical applications it will always be necessary to determine the optimal balance
between dopant concentration and carrier mobility: this is an illuminating example
of ‘materials engineering’.

9.2.3 Carriers concentration

The last parameter to be discussed affecting charge transport is the concentrations of
carriers, that is, the number of electrons and holes per unit volume in CB and VB,
respectively. Like mobility, these concentrations vary with temperature. We discuss
the paradigmatic case of an n-doped semiconductor, whose band structure is
sketched in figure 9.3.
At zero temperature all impurity levels are occupied by electrons confined in the
potential well associated with the dopant atom, while the CB is empty. Upon
increasing temperature, a number of such electrons are thermally excited to the

7
We remark that in the high doping regime (that is, for concentrations larger than 1017 dopants cm−3) the
T-dependence of electron mobility is similar to the trend reported in figure 9.4: this provides full consistency
with the discussion developed in section 9.2.2.

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 Solid State Physics

Figure 9.6. Electron mobility in a n-doped silicon sample as function of temperature and dopant concentration
(bi-log scale). Data taken from [9].

conduction band: this process is possible even at moderately low temperatures due to
the proximity of the impurity level to the CB bottom. By progressively increasing the
temperature, the number of excited electrons increases correspondingly, until all the
impurity levels have lost their electrons8: therefore, a further temperature increase
does not cause any increase of the electron population in CB. This regime lasts until
such a high temperature is eventually reached to make possible the direct VB → CB
excitation. Starting from this condition, any further temperature rise will accord-
ingly increase the carrier concentration in CB, directly taking more and more
electrons from the VB. This promotion becomes exponentially efficient with
increasing temperature.
In figure 9.7 the variation of the electron concentration in the conduction band is
reported as a function of the inverse temperature for the case under discussion. We
can identify three regions with different physical features:
• in the high temperature range the concentration of carriers is very high: in
fact, not only have all impurities been ionised, but also the thermal excitation
of electrons from the valence band to the conduction band is particularly
efficient. In this temperature range the conductivity is dominated by the
electrons coming from the VB, that is, by the intrinsic carriers9: accordingly,
this regime is called ‘intrinsic region’;
• in the intermediate temperature range the concentration of carriers remains
basically constant because (i) all impurities have been ionised, but (ii) the
temperature is no longer so high as to generate a sizeable number of electron

8
It is equivalently said that all dopant atoms have been ionised.
9
This is the only mechanism in fact at work in the absence of any doping.

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 Solid State Physics

Figure 9.7. Qualitative trend of the electron concentration in conduction band (log scale) in an n-doped
semiconductor as function of inverse temperature.

excitations from the VB to the CB. We accordingly refer to this intermediate

situation as the ‘extrinsic region’, since the conductivity is dominated by
electrons mainly provided by an extrinsic cause, such as doping;
• in the low temperature range no VB carrier is directly excited to CB, while the
number of those made available by doping decreases as the temperature
decreases. This temperature regime is referred to as the ‘freeze-out region’.

It is important to point out that for all semiconductors used in modern

information technologies, the room temperature falls within the intrinsic region:
this implies that at normal operating temperatures the number of carriers in a
semiconductor device is typically regulated by the doping level. Finally, we remark
that the above qualitative discussion and conclusions are of general validity, that is,
they also apply to holes in p-doped materials.

9.2.4 Conductivity
Equation (9.12) makes it clear that the total direct-current conductivity depends on
the concentration of both carrier types, as well as on their mobilities. Since we
learned that such quantities depend on temperature, we understand that
σtot = σtot(T ). By combining the results discussed in sections 9.2.2 and 9.2.3, we
obtain the qualitative T-dependence of σtot shown in figure 9.8 for an n-doped
semiconductor. This picture is interpreted as follows10:
• in the intrinsic region the higher the temperature, the larger is the concen-
tration of carriers. A remarkable feature of this region is that the carriers
number increases more rapidly than the mobility decreases because of
phonon scattering (which, as we know, is the leading scattering mechanism

10
It is of course understood that the n-doping level is kept constant.

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 Solid State Physics

Figure 9.8. Qualitative trend of the conductivity, electron mobility, and electron concentration in conduction
band (log scale) in an n-doped semiconductor as function of inverse temperature.

in this temperature range). All in all, conductivity rapidly increases with

temperature;
• in the extrinsic region the conductivity is mainly affected by the variation of
mobility with temperature, instead basically with the concentration of carriers
being constant. The conductivity variation with temperature is overall similar
to the mobility one;
• in the freeze-out region both the concentration of carriers (fewer and fewer
electrons are thermally excited) and the mobility (which, in this region, is
severely limited by Coulomb scattering) decrease with decreasing temper-
ature. The conductivity follows the same behaviour.

Once again, this case-study highlights general features that are similarly valid in
p-doped semiconductors.

9.2.5 Drift current in a strong field regime

A tacit assumption in the discussion so far developed is that both τe and τh scattering
times are independent of the applied electric field. This condition is physically
verified only if the drift velocity of carriers is smaller than their thermal velocity
e ∣ (a condition that represents the typical weak field regime situation). If, in
∣vd∣ ⩽ ∣v th
contrast, we suppose application of a very high electric field, we enter a different
conduction regime: carriers can be accelerated up to largely exceed their thermal
velocity. We will then talk about conduction in a strong field regime.
We know from basic thermodynamics [10–12] that the velocities of the particles
forming a molecular gas are associated with its temperature; by extending this

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 Solid State Physics

concept to the gas of charge carriers, we guess that the strong field regime
corresponds to a situation where the temperature of such a gas is very high, even
higher than the temperature of the crystal lattice (determined by the ionic
vibrations). This makes it possible for carriers to transfer energy to the lattice11. In
the case we are discussing, the transfer of energy from charge carriers to the lattice is
obviously mediated by the electron–phonon scattering events which, unlike what
happens in a weak field, must be now treated as inelastic collisions12. On the other
hand, the very fact that the lattice temperature increases is equivalent to saying that
in a strong field regime the scattering events generate phonons: the energy needed to
generate them is subtracted from the gas of charge carriers by a mechanism that
becomes more and more efficient as the accelerating external field increases. Because
of this, the drift velocity increases as a function of the applied field, until it eventually
reaches a saturation value vsat which cannot be exceeded because so many phonons
are generated to prevent further carrier acceleration.
It is easy to elaborate an order-of-magnitude estimate of the saturation velocity
by simply evaluating the energy balance for electrons in a saturation condition13: the
energy gained per unit time by a single electron because of the acceleration
impressed by the external field E must be equal to the energy lost by emission of
a phonon in the same time span. This balance is translated in the equation
ℏωphonon
−e vsat · E = , (9.16)
τe
where the term on the right-hand side is given by the ratio between the energy
ℏωphonon of the emitted phonon and the average time τe separating two consecutive
scattering events. Since the saturation velocity is the upmost value of the drift
velocity, we can insert into equation (9.16) the definition given in equation (7.5) and
thus obtain
ℏωphonon
∣vsat ∣ = , (9.17)
m e*

providing the saturation velocity in terms of the energy of the generated phonon. In
figure 9.9 we report the variation of the electron and hole drift velocity as a function
of the applied electric field in the paradigmatic cases of silicon and germanium. The
saturation velocity is easily extracted from these plots.

9.2.6 Diffusion current

We now consider a situation where no electric field is applied to the semiconductor,
but the charge carriers are non uniformly distributed along, say, the z direction.

11
This occurs in analogy to what happens when two ordinary molecular gases at different temperatures come
into contact: the warmer one gives energy to the colder one, causing a rise in temperature.
12
Interestingly enough, this is definitely a non-adiabatic effect.
13
The exact value of the saturation velocity can only be obtained by a careful calculation of the electron–
phonon scattering rate [4].

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 Solid State Physics

Figure 9.9. Room temperature electron (full lines) and hole (dotted lines) drift velocity in silicon (red) and
germanium (blue) as a function of the strong electric field. Data for silicon are taken from [8]. Data for
germanium are taken from [13].

To be more specific, we assume that n e = n e(z ) is the concentration of electrons in CB.

By considering an arbitrary section of the sample, we understand that it is crossed by
electrons along both z-directions because of their thermal motion. However, the
numbers of carriers crossing the section from −z or from +z directions are unequal,
since the concentration is not uniform. Accordingly, the net flux of electrons Φe (that
is, the total number of electrons crossing the section per unit time) is just the balance
between the these two currents. Elementary diffusion theory [14] proves that
dn e(z )
Φe = −De , (9.18)
dz
which is referred to as the Fick diffusion law: the net flux is proportional to the
concentration gradient. The proportionality constant De is known as the electron
diffusivity; it is easily calculated as
λ e2 2
De = λ eveth = = (veth ) τe, (9.19)
τe
where λ e is the electron mean fee path, that is, the average distance covered between
two consecutive scattering events (occurring at 1/τe rate). By inserting in equation
(9.19) the result14 veth = kBT /m e* , we obtain the Einstein relation for the electron
diffusivity

14
We stress that in this case the electrons are assumed to move uni-directionally only along z and, therefore,
equipartition imposes that m e*(veth )2 /2 = kBT /2 .

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 Solid State Physics

kBT kT
De = τe = B μe , (9.20)
me* e
where we made used of the definition given in equation (9.9). This is quite an
important result, predicting that if electrons have high mobility, then they also have
high diffusivity; furthermore, diffusivity is found to grow with temperature. Both
features are nicely confirmed by experiments. The diffusion of electrons along their
concentration gradient is obviously translated into a diffusion current whose density
is easily calculated as ( −e )Φe .
A gradient of hole concentration nh(z ) in VB similarly gives rise to a diffusion
current density ( +e )Φ h of positive charges, which is governed by the very same
equations as above. More specifically
dnh(z ) kBT
Φ h = −Dh where Dh = μ , (9.21)
dz e h
where the hole diffusivity D h = (kBT /e )μ h has been introduced.

9.2.7 Total current

The most general case that can be given is that of charge transport in a three-
dimensional semiconductor with non uniform distribution of carriers and subject to
the action of an electric field E which, for simplicity, we will consider weak. This
situation is found in many devices used in information and communication
technologies. In these conditions the total current density Jq,tot is calculated as the
sum of the drift and diffusion ones, each provided by both electrons and holes.
By combining the results of section 9.2.1 and section 9.2.6 we can write
⎧ JCB = eμe n e E + eDe∇n e
Jq,tot = JCB + JVB with ⎨ (9.22)
⎩ JVB = eμh nh E − eD h∇nh ,

where the three-dimensional formulation of the concentration gradients is reported

for sake of generality.

9.3 Charge carriers statistics

In order to complete the microscopic theory of charge transport we must face the
problem of how to determine in the most general way the carrier concentration in
both the conduction and valence band. To this aim, we must separately consider the
equilibrium and out-of-equilibrium circumstances. In the first case we will refer to a
situation where the concentration of free carriers (whether electrons in CB or holes
in VB) is uniquely determined by temperature. The system will be, therefore, in
conditions of constant temperature and will not experience any perturbative external
action. In contrast, in the second case we will refer to a physical situation where the
concentration of free carriers is influenced by external factors (like, for instance, the
interaction with an electromagnetic field or the injection of extra carriers from some
external source) in addition to thermal effects.

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 Solid State Physics

9.3.1 Semiconductors in equilibrium

In a semiconductor in equilibrium at temperature T the density of electrons in
conduction band is given by
+∞
n e(T ) = ∫E c
g¯CB(E ) nFD(E , T ) dE , (9.23)

where, according to the discussion developed in section 9.1.2, we have introduced

the notation ḡCB(E ) for the CB eDOS. Similarly, the density of holes in valence band
is given by
Ev
n h (T ) = ∫−∞ g¯VB(E ) [1 − nFD(E , T )] dE , (9.24)

where the term in square brackets indicates the probability that a VB state with
energy E is not occupied by an electron or, equivalently, it indicates the probability
to find there a hole. Obviously, ḡVB(E ) is the VB eDOS. In both cases we make use of
the parabolic bands approximation, which is just fine for the following
developments.
The explicit expressions for the eDOS provided by equations (9.3) and (9.4) must
be, respectively, inserted into equations (9.23) and (9.24). In particular, in equation
(9.23) all energies belong to the CB so that E − μc ≫ kBT ;15 therefore, the Fermi–
Dirac distribution can be conveniently approximated as
⎛ E − μc ⎞
nFD(E , T ) ∼ exp⎜ − ⎟, (9.25)
⎝ kBT ⎠

which leads to
⎧ ⎛ Ec − μc ⎞
⎪ n e(T ) = Nc exp ⎜ − ⎟
⎨ ⎝ kBT ⎠ (9.26)
⎪ 3/2
⎩ Nc = 2(2πm e*h−2kBT ) ,
where Nc is known as the effective density of states in CB. It is a very important
parameter for applications, varying over a wide range of values; for instance, in the
case of Si, Ge, and GaAs we have: NcSi = 2.7 × 1019 cm−3, NcGe = 1.0 × 1019 cm−3,
and NcGaAs = 4.4 × 1017 cm−3, respectively.
As for holes, we must consider that in equation (9.24) all energies belong to the
VB so that ∣E − μc ∣ ≫ kBT and, therefore, in this case the Fermi–Dirac distribution
can be conveniently approximated as
⎛ μ − E⎞
nFD(E , T ) ∼ 1 − exp⎜ − c ⎟, (9.27)
⎝ kBT ⎠

15
This is indeed a subtle step: we are implicitly assuming that the Fermi level is at the top of the valence band
and that, similarly to metals, the chemical potential and the Fermi energy can be to a good extent identified.
These assumptions will soon be readdressed and justified.

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 Solid State Physics

which leads to
⎧ ⎛ μ − Ev ⎞
⎪ nh(T ) = Nv exp ⎜ − c ⎟
⎨ ⎝ kBT ⎠ (9.28)
⎪ 3/2
⎩ Nv = 2(2πm h*h−2kBT ) ,
where Nv is known as the effective density of states in VB. There results:
NvSi = 1.1 × 1019 cm−3, NvGe = 3.9 × 1018 cm−3, and NvGaAs = 9.7 × 1018 cm−3.
Equations (9.26) and (9.28) provide evidence that the concentration of free carriers
does depend on the position of the chemical potential μc , a remarkable result indeed,
which forces us to consider in more detail the problem of where exactly the chemical
potential is located16 and, more importantly, how its position depends on temper-
ature and doping condition.

9.3.2 Chemical potential in intrinsic semiconductors

If an intrinsic semiconductor is in equilibrium at temperature T the numbers of
electrons in CB and holes in VB are just the same, since for any electron thermally
excited a corresponding hole is generated. This means that n e(T ) = nh(T ) or
equivalently
⎛ Ec − μc ⎞ ⎛ μ − Ev ⎞
Nc exp⎜ − ⎟ = Nv exp⎜ − c ⎟, (9.29)
⎝ kBT ⎠ ⎝ kBT ⎠
from which we get
Ec + E v 1 N
μc = − kBT ln c , (9.30)
2 2 Nv
defining the position of the chemical potential as a function of temperature. From
this result we immediately obtain that EF = limT →0 μc = (Ec + E v )/2 = Egap /2: the
Fermi level of an intrinsic semiconductor is placed right at the centre of the forbidden
gap. In many contexts the EF = Egap /2 energy is referred to as the intrinsic Fermi
level.
The μc = μc (T ) dependence predicted by equation (9.30) is in fact very weak since
we have that ln Nc /Nv ∼ ln m e* /m h* ∼ 0: the logarithm of the mass ratio is actually
very small since in most semiconductors the electron and hole effective masses are
not that much different (see table 9.1).

9.3.3 Chemical potential in doped semiconductors

In doped semiconductors the carrier concentration is determined by the combination
of thermal excitation and ionisation phenomena of dopant atoms. In this case it is

16
This information will also teach us where the Fermi level of a semiconductor must be placed. This is not a
trivial problem: if we simply adopt the definition of Fermi energy as ‘the energy below which any allowed level
is fully occupied at T = 0 K’, we are left with a large ambiguity, since any energy in between the forbidden gap
does fulfil this definition.

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 Solid State Physics

really needed to make a systematic distinction between thermally generated intrinsic

carriers and doping related extrinsic carriers (see discussion in section 9.1.1). We will
separately address the n-doping and p-doping case.
Let us consider first an n-doped semiconductor where ND identical donors per
unit volume have been added, each generating an impurity level with energy ED just
below the CB minimum (see figure 9.3). In equilibrium at temperature T we will find
n e(T ) = n e(i)(T ) + n D(T ), (9.31)

electrons per unit volume in CB, where n e(i)(T ) is the concentration of intrinsic
electrons (that is, electrons promoted from the valence band) and nD(T ) ⩽ ND is the
concentration of extrinsic electrons (that is, electrons promoted from the impurity
level). We can distinguish three temperature ranges:
• at low temperature (freeze-out region) n e(i)(T ) is negligibly small, while the
number of extrinsic electrons is just equal to the number of ionised donors
which, in turn, corresponds to the number of empty impurity levels.
Combining these arguments with equation (9.28), we can write
⎛ Ec − μc ⎞ ⎛ ED − μc ⎞
Nc exp⎜ − ⎟ = ND[1 − nFD(ED, T )] ∼ ND exp ⎜ ⎟, (9.32)
⎝ kBT ⎠ ⎝ kBT ⎠

where we have approximated the Fermi–Dirac distribution function as we did

in section 9.3.1. This result immediately leads to
Ec + ED kT N
μc = + B ln D , (9.33)
2 2 Nc
which indicates that at T = 0 K the Fermi level of an n-doped semiconductor is
placed right in between the donor level and the bottom of the conduction band;
• at intermediate temperature (extrinsic region) we still have n e(i)(T ) ∼ 0, but all
donors have been ionised. This is tantamount to writing n e(T ) = ND so that

ND
μc = Ec + kBT ln , (9.34)
Nc
• at high temperature (intrinsic region) the population of intrinsic electrons
largely exceeds the extrinsic one, although all donors are ionised: therefore,
the semiconductor actually behaves as if it were intrinsic. According to the
conclusion reported in section 9.3.2, we can consider μc basically independent
of temperature and set it at the Egap /2 midgap position.

This analysis is fully consistent with experimental evidence, as shown in figure 9.10
in the case of n-doped silicon. The same figure also reports the opposite case of
p-doping where similar arguments as before can be developed. More specifically, for
a p-doped semiconductor we have
nh(T ) = n h(i)(T ) + nA(T ), (9.35)

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 Solid State Physics

Figure 9.10. Position of the chemical potential μc in n-doped (dotted lines) and p-doped (full lines) silicon as
function of temperature and doping level. The μc position is reported relative to the intrinsic Fermi level
Egap /2 = 0.56 eV .

where n h(i)(T ) and nA(T ) ⩽ NA are the concentration of intrinsic and extrinsic holes,
respectively. It is assumed that NA acceptors per unit volume have been placed in the
material, each generating an impurity level EA just above the VB maximum (see
figure 9.3). By increasing temperature we span the usual three regions:
• at low temperature (freeze-out region) the number of intrinsic holes is
negligibly small and therefore
E v + EA kT N
μc = − B ln A , (9.36)
2 2 Nv
which indicates that at T = 0 K the Fermi level of a p-doped semiconductor is
placed right in between the acceptor level and the top of the valence band;
• at intermediate temperature (extrinsic region) nA(T ) = NA , while n h(i)(T ) is still
vanishingly small. This leads to
NA
μc = E v − kBT ln , (9.37)
Nv
• at high temperature (intrinsic region) the p-doped semiconductor actually
behaves as intrinsic.

9.3.4 Law of mass action

By combining the two general expressions (9.26) and (9.28) we obtain the note-
worthy result

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 Solid State Physics

⎛ E − Ev ⎞ ⎛ Egap ⎞
n e(T )nh(T ) = NcNv exp⎜ − c ⎟ = NcNv exp⎜ − ⎟, (9.38)
⎝ kBT ⎠ ⎝ kBT ⎠

known as law of mass action17. It is valid for any semiconductor in whatever intrinsic
or extrinsic configuration and states that the product between the electron and hole
concentrations is independent of the chemical potential or, equivalently, of the doping
level; rather, it only depends on the energy gap.
In the specific case of an intrinsic semiconductor, we always have n e(T ) = nh(T )
and, therefore, the law of mass action takes the intriguing form
⎛ Egap ⎞ ⎛ Egap ⎞
n i(T ) = NcNv exp⎜ − ⎟ ∼ T 3/2 exp⎜ − ⎟, (9.39)
⎝ kBT ⎠ ⎝ 2kBT ⎠

which provides the intrinsic carrier population n i(T ) in its explicit dependence on
temperature. This equation makes it clear why at high temperature the intrinsic
carriers largely exceed in number the extrinsic ones.
We finally remark that in a doped semiconductor the number of electrons is not
equal to the number of holes: which of the two carrier population is larger depends
on the actual doping level. It is therefore useful to distinguish between majority
carriers and minority carriers: in n-doped (p-doped) semiconductors the majority
carriers are the electrons (holes), while the minority carriers are the holes (electrons).

9.3.5 Semiconductors out of equilibrium

The external actions affecting the carrier populations, in addition to ordinary
thermal excitation effects, are basically the injection or extraction of carriers and the
photo-generation or recombination of electron–hole pairs due to photon absorption or
emission.
The injection of carriers is a process that generates an excess of electrons or holes
(as compared to the concentration that these carriers would have in equilibrium
conditions) through a suitable system of electrical contacts coupling the semi-
conductor to some external circuit. The same setup is used in the process of carrier
extraction. On the other hand, the photo-generation corresponds to the promotion
of electrons at higher energy levels (with a corresponding generation of holes in the
initial energy level) by absorption of photons with appropriate energy. Several
absorption mechanisms are possible [16], among which we recall (i) the inter-band
mechanism, where an electron is excited directly from the VB to the CB (this process
involves photons of energy ℏωphoton ⩾ Egap ) and (ii) the transition from a band state
to an impurity level or viceversa 18. Recombinations occur whenever an electron–
hole pair is annihilated because of inverse transitions. This can happen through
either a radiative or a non-radiative process. In the first case the recombination

17
This wording is taken from the physical chemistry chapter dealing with chemical reactions [15].
18
More specifically: an electron initially placed in VB could be promoted to an acceptor or to a donor level, as
long as they are unoccupied. Similarly, an electron initially accommodated on an acceptor or donor level could
be excited to the CB.

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 Solid State Physics

causes the emission of a photon with energy equal to the energy difference between
the initial and final levels occupied by the electron. Instead, as far as non-radiative
recombinations are concerned, all or part of the energy lost by the electron during
the transition to a lower energy level is released in the form of phonon emission.
Non-radiative processes, therefore, increase the crystal lattice temperature.
The common feature of all non-equilibrium mechanisms is that the law of mass
action is no longer fulfilled since either n enh > n i2 in the injection and photo-
generation cases or n enh < n i2 if extraction or recombination occurs. A new
statistics, other than the equilibrium one, is needed: this is an advanced topic of
semiconductor theory [1, 2, 4] which we outline only qualitatively by introducing the
carrier generation rate G and the carrier recombination rate R: they, respectively,
count the number of carriers generated or destroyed per unit time in the unit volume
because of non equilibrium mechanisms19. It is a difficult task of solid state theory to
provide a microscopic expression for them: we assume they are known. Obviously,
the net rate of variation of the carrier populations with respect to equilibrium is
given by the difference (G − R), which could be either positive or negative provided
that a larger or smaller number of carriers is, respectively, found because of
injection, generation, extraction and recombination mechanisms. These phenomena
are reflected in a modification of the continuity equations for the electron and hole
currents which must now be written as follows
⎧ ∂n e 1
⎪ electron current:
⎪ = Ge − R e + ∇JCB
⎨ ∂t e
(9.40)
⎪ ∂nh 1
⎪ hole current: = G h − Rh + ∇JVB,
⎩ ∂t e
where the two density current vectors possibly contain either the drift or diffusion
contributions as explained in section 9.2.7.

9.4 Optical absorption

9.4.1 Conceptual framework
The notions developed in the previous chapters allow us to model a semiconductor
as a system composed by an electron gas (which determines its electrical conduction
properties) penetrated by a phonon gas (which determines its vibrational and
thermal properties). If this system is embedded into an electromagnetic (em)
radiation field, we must add to such a physical model also a photon gas component.
Investigating the optical properties of a semiconductor is equivalent to studying the
mutual interactions among electrons, phonons and photons. They give rise to two
opposite optical phenomena: the semiconductor can (i) absorb photons from the
radiation field, a process resulting in the generation of electron–hole pairs which sum
to those already existing for ordinary thermal excitation, or (ii) host events of
electron–hole recombination, with corresponding loss of energy by emission of

19
We understand that there exist different generation Ge,h and recombination R e,h rates for electrons and holes.

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 Solid State Physics

photons and/or phonons. In the first case we speak of absorption, in the second of
emission of electromagnetic radiation.
Ultimately, understanding the physics of energy exchange processes among
electrons, phonons and photons is a key issue both for the most complete character-
isation of the physical properties of a semiconductor and for mastering the principles
of operation of optoelectronic devices. In our tutorial introduction we will only
address absorption, since it is comparatively more direct to access. Emission
phenomena (inter-band emission as well as photo- and electro-luminescence) are
treated elsewhere [16].

9.4.2 Phenomenology of optical absorption

When an electromagnetic radiation is directed on a semiconductor, reflection,
transmission and absorption are empirically observed. Maxwell equations, through
the laws of Snell and Fresnel [16–18], make reason of the first two phenomena. In
order to characterise absorption we need, instead, the development of a microscopic
theory able to account for any single possible conversion process of photons into
excitations of the electron and/or phonon system. To this end, it is useful to consider
the prototypical situation shown in figure 9.11 in which an optical absorption
experiment is conceptually illustrated: the surface of a semiconductor in equilibrium
at room temperature is enlightened by a monochromatic radiation at normal
incidence, measuring how its intensity decreases inside the illuminated sample.
The physical process that determines the attenuation of the radiation—that is,
absorption—is governed by the energy subtraction mechanisms operated by the
medium at damage of the radiation.
So, let Φphoton be the flux of photons (that is, the number of photons per unit of
time incident on the exposed surface of the semiconductor) each of energy ℏω
(where, of course, ω is their angular frequency, equal for all provided that the em
radiation is monochromatic). If the incident intensity is I0 = ℏωΦphoton , then inside
the material it decreases in proportion to the number of absorbed photons. This
number, in turn, varies with the depth z with respect to the surface: the deeper the
layer, the less is the amount of received photons, as a consequence of all the
absorption processes occurring in shallower layers. Therefore, we can indicate by
I (z ) the intensity of em radiation at a depth z from the surface. Moreover, the

z dz

I(z)

Figure 9.11. Conceptual representation of an optical absorption experiment at normal incidence in a

semiconductor. I (z ) is the intensity of the absorbed electromagnetic field.

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 Solid State Physics

amount of energy absorbed by an inner layer depends on its thickness20. By

combining these simple observations, we can write the amount of energy absorbed
by a layer of thickness dz at a distance z from the surface as
dI (z ) = −α I (z ) dz , (9.41)
where α is known as the absorption coefficient of the material21. The negative sign
that appears in this equation indicates that we are actually calculating a decrease in
the number of photons. In essence, α quantifies the effectiveness of the photon
absorption process by the crystal. By integration equation (9.41) we immediately
obtain the Beer law
I (z ) = I0 exp( −αz ), (9.42)
which we will use as the theoretical basis to interpret an optical absorption
experiment.
Before going into the detail of the microscopic model for α, we must observe that
it depends on the frequency of the incident em radiation. Although the theoretical
demonstration of this result goes beyond the level of treatment that we are
developing22, we can make account of this empirically verified important fact by
discussing the main phenomenological results under the following assumptions:
• the radiation field is weak enough not to alter the pre-existing band structure of
the semiconductor. In other words, we will assume that the only effect of the
radiation field consists in promoting transitions between states (which remain
unchanged, as compared to the case in which the semiconductor is not
irradiated);
• the vibrational spectrum contains infrared-active optical phonons, that is, such
vibrational modes are capable of generating oscillating electric dipoles that
mate with the em field in the infrared spectral region. The coupling results in
absorption of infrared photons23;
• there exists a population of free carriers that ensure continuous background
absorption at all frequencies. These are the carriers created in VB or in CB by
thermal excitation (or even by the same photo-excitation). Since, because of
this, both bands become only partially filled, it is possible to promote free
carrier transitions between levels within the same band. Therefore, through
these transitions it is possible to absorb photons of any energy.

As a last preliminary remark, we stress that the true characteristics of a typical

semiconductor band structure make different types of absorption transitions

20
It is intuitive that a thick layer will absorb more photons than a thinner layer.
21
A simple dimensional analysis shows that it is measured in units [m−1].
22
It is in fact necessary to describe the effects of the radiation field on the crystalline electron levels by quantum
time-dependent perturbation theory. In this way the transition probability for electrons and phonons to change
their quantum state is calculated, obtaining the absorption coefficient [6, 16].
23
As already extensively discussed in chapter 3, not all semiconductors enjoy this capability: in elemental
semiconductors optical vibration modes do not generate oscillating electric dipoles, given the fact that the two
basis atoms are equal and, therefore, equally charged (to the first order).

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 Solid State Physics

possible, namely: (i) the inter-band transitions, corresponding to direct promotion of

electrons from the VB to the CB; (ii) transitions between impurity levels and the
closest band, such as electron excitations from the VB to acceptor levels or from
donor levels to the CB; (iii) intra-band involving transitions between states lying in
the same band. The synopsis of absorption processes is reported in figure 9.12, left.
The typical absorption spectrum of a semiconductor is shown qualitatively in
figure 9.13. First of all we observe that α (ω ) is represented in arbitrary units because
its quantitative estimation can only be obtained by thoroughly calculating the
transition probabilities for any single intra-band and inter-band transition, as well as
for any other one occurring between band levels and impurity states or by lattice
absorption [6]. In addition, those that in figure 9.13 are schematically displayed as
single peaks, in fact appear in a real experimental spectrum as multi-peak structures,
each one corresponding to a specific transition of the family of transitions active in
the spectral region considered.

9.4.3 Inter-band absorption

We will develop our arguments considering the model case of an intrinsic semi-
conductor in thermal equilibrium at room temperature and in approximation of
non-degenerate parabolic bands. We will focus on the spectral region from near

Figure 9.12. Left: Classification of the typical absorption transitions in a model compensated semiconductor at
T > 0 K, where both donor and acceptor impurities are present (green lines). The final and initial position of
the excited electrons are shown by a full and empty dot, respectively. In the latter case the situation is
interpreted as a hole generation. Right: the corresponding CB and VB eDOS. A larger number of photons with
energy ℏω1 is absorbed since the joint density of states is larger. The colour gradient in the CB and VB bands
indicates the number of states which, at the different energies, are available to serve as starting or ending point
of the electron transition: the darker the colour, the higher such a number.

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 Solid State Physics

Figure 9.13. Schematic representation of the optical absorption spectrum (red line) of a semiconductor. The
intensity and the shape of each peak are purely indicative.

infrared to ultraviolet. Despite its simplicity this model contains all the fundamental
physics necessary for a conceptually rigorous treatment of optical absorption,
allowing us to extend the main results of the model to all semiconductors. Under
these assumptions, the only possible absorption process is associated with the
electron transitions between VB and CB states since: (i) the intrinsic character of
the semiconductor excludes transitions involving impurity states; (ii) the number of
free carriers (electrons in CB or holes in VB) is very small, as the assumed
temperature value sets a freeze-out condition (see figure 9.7) and, therefore, the
continuous background absorption (dashed line in figure 9.13) due to intra-band
transitions is negligible; (iii) absorption processes involving phonons are not possible
in the selected spectral region, well above the energy scale of lattice vibrations.
As already anticipated, the absorption process of em radiation by a material
medium is ultimately due to individual photon–electron interactions24. Since they
occur in the absence of dissipative phenomena25, both the total energy and the total
momentum of the electron–photon pair are conserved during the absorption process.
If the radiation is a monochromatic wave of angular frequency ωphoton and the
photon wavevector is k photon , then we can impose

24
Yet another tacit assumption: we are assuming that only one-photon absorption processes occur. Two(or
more)-photon processes are indeed possible, but definitely much less frequent and challenging to describe [16,
19, 20].
25
In other words, the photon–electron interaction does not involve external actions to the particle pair.

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 Solid State Physics

E2 − E1 = ℏωphoton and ℏk2 − ℏk1 = ℏk photon, (9.43)

where the initial and final electron states have, respectively, energy E1 and E2 and the
corresponding wavefunctions have wavevector k1 and k2. The momentum con-
servation constraint lends itself to an important remark: for any radiation frequency
in between the infrared and the ultraviolet regions, the photon wavelength λ photon is
always λ photon ⩾ 250 nm on the other hand, the typical de Broglie wavelength λ e of
the matter wave describing a crystalline electron is comparable to the lattice
constant a0, so that λ e ∼ 1 nm . Therefore, always verified is the condition
λ photon ≫ λ e . This simple order-of-magnitude estimates clearly shows that the photon
wavevector is always negligibly small as compared to its electron counterpart26. This
leads to an important practical consequence: we can to a very good approximation
always set
ℏk2 − ℏk1 = ℏk photon ∼ 0 → Δk = 0, (9.44)
which dictates that one-photon inter-band transitions can only be vertical, that is, they
occur with no change of electron momentum.
We eventually observe that the physics of optical absorption transitions depends
on the underlying band structure which (i) determines the energy positioning of a
pair of initial and final electron states and (ii) associates with any such pair a specific
wavevector, fulfilling the conservation law in the form given in equation (9.43). In
order to proceed further we must, therefore, consider separately the case of a direct
gap semiconductor form the case of an indirect gap one.

Direct transitions
Let us consider a direct gap semiconductor, such as gallium arsenide, under the
above simplifying assumptions. The first important characteristic of intra-band
processes that take place under irradiation is that no absorption of photons with
energy ℏωphoton < Egap can actually take place. Let us consider an electron initially
placed anywhere in VB with energy E1; then, the absorption of a photon of energy
ℏωphoton < Egap would promote it to an energy level E2 = E1 + ℏωphoton . This level,
however, would inevitably end up falling below the bottom of the CB, that is in the
forbidden gap. We can describe this situation by saying that this absorption process is
prohibited because the energy E2 of the final state does not correspond to any allowed
crystalline electron state27. Ultimately, the semiconductor is transparent to the
electromagnetic radiation component at this frequency. The physical situation is
shown schematically in figure 9.14. This result can be summarised by stating that
α (ω ) = 0 for any ω < Egap /ℏ.

26
More explicitly: for a photon in the indicated spectral range we have k photon = 2π /λ photon < 0.025 nm−1.
Instead, for a crystalline electron we have k e ∼ 2π /a 0 ∼ 101 − 102 nm−1. Therefore, k photon is always some
good order-of-magnitude smaller than k e .
27
A much less phenomenological approach would require the application of time-dependent perturbation
theory, eventually providing a zero probability for such a transition [16, 19, 20].

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 Solid State Physics

Egap

Figure 9.14. These direct optical transitions do not occur since the final state lies in the forbidden gap.

The occurrence of intra-band transitions has a dramatic dependence on the density

of the electron states: the absorption, in fact, will be more or less intense due to the
greater or lesser number of pairs of initial/final states available to host the
corresponding electron transition. The finite temperature eDOS across the forbidden
gap is represented in figure 9.12(right), where the eDOS given in equations (9.3) and
(9.4) are reported, respectively, weighted by the probability that their levels are
empty (CB states must host the promoted electrons) and occupied (VB states must
release electrons), respectively. From figure 9.12(right) it is immediate to conclude
that the absorption will be low for photons with energy ℏωphoton ∼ Egap , given the
fact that the number of pairs of full/empty states available, respectively, in proximity
of the VB top and CB bottom is low; on the other hand, as ωphoton increases the
number of pairs accordingly increases and the absorption of photons becomes more
and more likely. The situation is described by stating that near the gap the absorption
coefficient depends on the joint density of states in VB and CB. Here, the actual α (ω )
dependence directly derives from the parabolic bands approximation
1/2
(
α(ω) ~ ℏω − Egap ) , (9.45)

which indicates the more straight method to determine Egap in direct gap semi-
conductors: regardless of practical experimental setup28, just measure α (ω ) over a
suitable range of frequencies and plot its square α 2(ω ) as a function of the energy of
the absorbed photons. This is done in figure 9.15: the intercept of the linear
extrapolation of the experimental data for α 2(ω ) with the energy axis directly
provides the semiconductor energy gap.
We conclude by observing that the momentum conservation constraint makes
intra-band transitions between electronic states with different wave vectors prohibited,
even if energetically admissible. In figure 9.16 we illustrate this concept: the

28
Not at all an easy task, indeed.

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 Solid State Physics

Figure 9.15. Room temperature absorption in InAs. It is reported the square magnitude α 2 of the coefficient
for a direct estimation of Egap = 0.35 eV through equation (9.45). Experimental data (blue dots) are taken
from [21].

∆k = 0
∆k = 0

Figure 9.16. Schematic representation of allowed and forbidden optical transitions in a direct gap
semiconductor.

transition on the left in the picture is allowed, because it is vertical; conversely, the
transition on the right is clearly prohibited because Δk ≠ 0 and, therefore, it would
imply a variation of electron wavevector. Ultimately, the two conservation laws
reported in equation (9.43) act in practice as selection rules that discriminate between
physically possible absorption processes and prohibited ones.

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 Solid State Physics

Indirect transitions
It is now time to consider the case of an indirect gap semiconductor, such as silicon,
always under the same simplifying hypotheses as before. As a starting point of our
discussion, we can assume that also in this case the Egap value is an absorption
threshold for photons: in fact, the same arguments developed previously are still
valid. However, in order to exceed this threshold it is now necessary to supply to the
electron not only the minimum amount of energy sufficient to pass from the VB to the
CB, but also the exact amount Δk of wavevector necessary to connect the VB top to
the CB bottom (which in this case are no more vertically aligned). Since the absorbed
photon does not carry momentum as imposed by equation (9.44), the transition can
take place only if accompanied by the absorption of a phonon with suitable wave vector
q = Δk . Of course, the phonon also carries a certain amount of energy ℏωphonon
(where ωphonon is the frequency of the corresponding vibrational mode), which
implies that the absorption threshold is lowered to Egap − ℏωphonon . In other words,
the electron must simultaneously absorb a photon and a phonon, respectively, from
the radiation and thermal bath29. The situation is schematically represented in
figure 9.17.
Indirect transitions are complex physical processes involving three-particles: an
electron, a photon, and a phonon30. Their occurrence rate is therefore much lower
than in the case of direct transitions [4]: indirect gap semiconductors are very poorly
efficient absorbers of electromagnetic radiation. We finally observe that also possible
is the indirect process involving the absorption of a large energy photon (that brings
the electron far above the CB bottom), followed by the emission of a phonon that, at

Figure 9.17. Schematic representation of an allowed optical transition in an indirect gap semiconductor.

29
The careful reader should have noticed that this is the second time—after discussing the electron–phonon
scattering in transport theory—we deal with phenomena falling beyond the strict adiabatic approximation
discussed in section 1.3.4.
30
Actually, things can be even more complicated than that, as more than two phonons can be actually
required. It is said that indirect transitions are processes of a higher order than direct ones.

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 Solid State Physics

the same time, allows (i) the excited electron to relax towards the CB bottom level
and (ii) the gain of that momentum component necessary to activate the non vertical
transition.
The α (ω ) dependence for indirect transitions can no longer be derived in an
elementary way as in the previous case. In fact, only an accurate perturbative
calculation [4] shows that this dependence is
2
α(ω) ~ ⎡⎣ℏω − Egap + ℏωphonon ⎤⎦ ,
( ) (9.46)

where for sake of simplicity we have assumed a one-phonon event. This result is
nicely consistent with experimental evidence, as reported in figure 9.18 in the case of
germanium. Thanks to equation (9.46) the energy gap value is determined by a
simple extrapolation, once that absorption shoulder generated by the phonon-
assisted mechanism at energies slightly lower than Egap is properly considered.

9.4.4 Excitons
Each inter-band transition generates an electron–hole pair. Since the two particles
are generated at the same point within the semiconductor and have opposite electric
charge, they can give life to a bound electron–hole system, referred to as exciton,
which (in some cases) is free to move with the host crystal. As shown graphically in
figure 9.19, we must distinguish between Wannier excitons and Frenkel excitons: the
former are formed in semiconductors and, in fact, are delocalised; the latter, instead,
are formed in insulating materials and are located near specific lattice sites, which

Figure 9.18. Room temperature absorption in Ge. It is reported the square root α of the coefficient for a
direct estimation of Egap = 0.66 eV through equation (9.46) by extrapolation (magenta dashed line), once that
tell-tale sign of phonon absorption (black full line) has been duly taken into account. Experimental data (blue
dots) are taken from [21].

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 Solid State Physics

Figure 9.19. Pictorial view of Wannier and Frenkel excitons in a model semiconductor whose atoms are
represented by the grey spheres.

act as exciton traps. We will only address the case of Wannier excitons, without
further specification.
In order to describe the basic physics of an exciton we can use a rather simple
model by extending the results known for the hydrogen atom [5] to the case of the
electron–hole bound pair. In doing so, we must have only two shrewdnesses. First of
all, we must use an effective exciton mass value μx properly defined as
1 1 1
= + , (9.47)
μx me mh
and then we must consider that the bound pair moves within a crystal and not in
vacuum: consequently, the intensity of the Coulomb electron–hole interaction will
be decreased, compared to that of the electron–proton pair of the hydrogen atom, by
a factor ϵr−1 provided by the relative permittivity of the semiconductor. By
introducing these corrections to the hydrogenic levels, we immediately get for the
exciton energy E x this result
μx 1
Ex = − H with n = 1, 2, 3, ⋯ , (9.48)
meϵr2 n2
where RH is the Rydberg constant for hydrogen. Breaking news: an exciton
represents a quantum system with a discrete energy spectrum. Typical values of the
binding energy of an exciton in its ground-state (corresponding to the n = 1 state in
equation (9.48)) fall in the range 10−3–10−2 eV. Therefore, at room temperature31 it
is very difficult to observe excitons in a semiconductor: we must cool the system at
very low temperatures, so that the thermal excitations become inefficient to dissolve
the electron–hole pair.

31
The energy of the thermal bath is of the order of kBT ∼ 0.026 eV at room temperature.

9-34
 Solid State Physics

The exciton size is defined by its electron–hole orbital radius

mϵ
rx = e r a H n 2 with n = 1, 2, 3, … , (9.49)
μx
where aH = 0.529 Å is the Bohr radius. Typical values in semiconductor materials
are in the range 1–10 nm.
Measurements provide evidence of some general trends in most elemental and
compound semiconductors (both type III–V and type II–VI): (i) it is easier to
observe excitons in direct gap semiconductors; (ii) the larger the forbidden energy
gap, the larger the exciton binding energy; (iii) as the forbidden energy gap increases,
the electron–hole orbital radius decreases. The first evidence is due to the fact that, in
order to form a bound pair, the electron and the hole must have the same velocity32.
In direct gap semiconductors this condition is fully verified by the zone-centre
transitions, i.e. those occurring at k = 0 (see for example the case of GaAs shown in
figure 8.8): vertical inter-band transitions occur between the VB top and the CB
bottom at Γ; this is tantamount to saying that the transitions interest states for both
of which the corresponding group velocities are zero. The last two characteristics are
instead due to the empirically observed fact that upon increasing the value of the
forbidden gap (i) the relative dielectric constant ϵr increases and (ii) the exciton
effective mass μx decreases.
Excitons affect the optical spectrum of a semiconductor in the region of energies
close to the absorption threshold, as shown in figure 9.20. We will discuss this
phenomenon in the particularly simple case of a direct gap semiconductor in which
the vertical transition takes place at the zone-centre: this is the case of GaAs. The
amount of energy stored in an exciton is given by the difference between the energy
required to form the electron–hole pair and the energy needed to permanently bind

Egap

Figure 9.20. Absorption spectrum (conceptual rendering) of a direct gap intrinsic semiconductor nearby the
absorption threshold. Blue full line: inter-band absorption; red lines: exciton absorption. Dashed magenta line:
the residual absorption subtracted by exciton effects. The label n is defined in equation (9.50).

32
If the two velocities were different, the two particles would inevitably end up moving away, thus weakening
their binding and eventually dissolving the pair.

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 Solid State Physics

the two particles. In our study-case, the first energy is equal to Egap , while the second
is given by equation (9.48). A simple relation for the discrete exciton spectrum E x,n is
accordingly calculated
μx 1
E x,n = Egap − 2
RH 2 with n = 1, 2, 3, … , (9.50)
m e ϵr n
showing that an exciton can in fact absorb photons at discrete energies, each smaller
than the energy gap. Similarly to atomic hydrogen spectroscopy, these processes
generate a discrete absorption spectrum that adds to the ordinary inter-band
spectrum and, in fact, lowers the intrinsic absorption threshold of the
semiconductor.

References
[1] Balkanski M and Wallis R F 1989 Semiconductor Physics and Applications (Oxford: Oxford
University Press)
[2] Cardona M and Yu P Y 2010 Fundamentals of Semiconductors (Heidelberg: Springer)
[3] Grundmann M 2010 The Physics of Semiconductors (Heidelberg: Springer)
[4] Seeger K 1989 Semiconductor Physics (Heidelberg: Springer)
[5] Colombo L 2019 Atomic and Molecular Physics: A Primer (Bristol: IOP Publishing)
[6] Wolfe C M, Holonyak N Jr and Stillman G E 1989 Physical Properties of Semiconductors
(Englewood Cliffs, NJ: Prentice-Hall)
[7] Wolfe C M, Stillman G E and Lindley W T J 1970 Appl. Phys. 41 3088
[8] Jacoboni C, Canali C, Ottaviani G and Alberigi Quaranta A 1977 Solid-State Electron. 20
77–89
[9] Sze S M 1981 Physics of Semiconductor Devices (New York: Wiley)
[10] Swendsen R H 2012 An Introduction to Statistical Mechanics and Thermodynamics (Oxford:
Oxford University Press)
[11] Callen H 1985 Thermodynamics and An Introduction to Thermostatistics (New York: Wiley)
[12] Dittman R and Zemansky M 1997 Heat and Thermodynamics 7th edn (New York: McGraw-
Hill)
[13] Jacobini C, Nava F, Canali C and Ottaviani G 1981 Phys. Rev. B 24 1014
[14] Reif F 1987 Fundamentals of Statistical and Thermal Physics (New York: McGraw-Hill)
[15] Pauling L 1970 General Chemistry (New York: Dover)
[16] Fox M 2001 Optical Properties of Solids (Oxford: Oxford University Press)
[17] Feynman R P, Leighton R B and Sands M 1963 The Feynman Lectures on Physics (Reading,
MA: Addison-Wesley)
[18] Kenyon I R 2008 The Light Fantastic—A Modern Introduction to Classical and Quantum
Optics (Oxford: Oxford Science Publications)
[19] Grosso G and Pastori Parravicini G 2014 Solid State Physics 2nd edn (Oxford: Academic)
[20] Bassani F and Pastori-Parravicini G 1975 Electronic States and Optical Transitions in Solids
(Oxford: Pergamon)
[21] Palik E D 1985 Handbook of the Optical Constants of Solids (San Diego, CA: Academic)

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 IOP Publishing

Solid State Physics

A primer
Luciano Colombo

Chapter 10
Density functional theory

Syllabus—Density functional theory is outlined on the basis of the fundamental

Hohenberg–Kohn theorem and the practical Kohn–Sham procedure. The theorem first
provides the mathematical background for replacing the total electron wavefunction
with the electron density as the key object for the quantum theory of a system of
interacting electrons subject to an external potential. Next, the Kohn–Sham procedure
effectively reduces the overwhelmingly complicated many-body electron problem to a
much simpler one corresponding to a fictitious system of non-interacting electrons,
characterised by the same density as the real one. The combination of theorem and
procedure leads to defining the exchange-correlation energy, whose calculation is here
developed under the leading local density approximation. The density functional theory
is nowadays referred to as the standard model for a fully parameter-free quantum
theory of the solid state.

10.1 Setting the problem and cleaning up the formalism

In chapter 1 we elaborated the most fundamental quantum mechanical formulation
of solid state theory, leading to the formidable total Schrödinger problem given in
equation (1.13), which is here reported for convenience
⎡ ℏ2 1 2 ℏ2 ⎤
⎢− ∑ ∇α − ∑ ∇i2 + Vˆne (r , R) + Vˆnn(R) + Vˆee (r)⎥Φ(r , R) = ET Φ(r , R), (10.1)
⎢⎣ 2 α Mα 2me i ⎥⎦

where the meaning of each symbol has been previously defined1. Thanks to the
adiabatic approximation (see section 1.3.4), the total wavefunction is cast in the
form of the product between the total ionic and electronic wavefunctions
Φ(r, R) = Ψn(R)Ψ (eR)(r) and this allows us to write two separate constitutive
equations for the electron and ion problems, respectively; in particular, the

1
We recall that in this equation r and R are the full set of positions of all electrons and all ions, respectively.

doi:10.1088/978-0-7503-2265-2ch10 10-1 ª IOP Publishing Ltd 2021

 Solid State Physics

eigenvalue problem for the electrons is set by the following equation (previously
labelled as equation (1.15))
⎡ ℏ2 ⎤
⎢−
⎢⎣ 2m e
∑ ∇i2 + Vˆne(r , R) + Vˆee(r)⎥⎥ Ψ(eR)(r) = Ee(R)Ψ(eR)(r), (10.2)
i ⎦

where Ψ (eR)(r) and Ee(R) are the total electron wavefunction and energy, respectively,
when ions are clamped in the R configuration. This equation contains a first
important conceptual achievement: in solid state physics, the electron eigenvalue
problem must be solved for each possible ion displacement pattern: this, of course,
reflects a very demanding task.
The estimation of the workload required to solve equation (10.2) is even more
daunting if we consider that no analytical solution is indeed conceivable and,
therefore, we must seek a numerical one. Basically, this requires that we partition the
real space into a grid of discrete points, so that the derivatives appearing in equation
(10.2) can be calculated by finite-difference methods. We better understand the key
issue of the numerical approach by considering the practical case of a crystal
containing Nc atoms in its unit cell of volume Vc . We know that such a volume is
3
Vc ∼ O(Å ) and, therefore, it is straightforward to admit that an accurate discretisa-
tion for the numerical calculation should consist in a grid of points at least spaced by
∼0.1 Å along any Cartesian direction. This implies that we need at least Np ∼ O(103)
points. Counting Nval valence electrons for each basis atom, the numerical calcu-
lation would require NpNcNval complex numbers. In the case of, say, silicon Nc = 2,
Nval = 4 so that O(1024) numbers should be calculated. Since the problem given in
equation (10.2) is typically translated into matrix operations2, it is clearly impossible
to perform them with arrays of O(1024) rank. The single-particle approximation was
elaborated precisely to bypass the numerical bottleneck just described. Here we are
instead going to follow a rather different approach, which will consist in replacing
the total electron wavefunction by the electron density as the key quantity of the solid
state theory.
In order to keep the formalism as clean as possible we are going to omit some
(annoying) details: at this stage of the discussion, the following simplifications of the
system of symbols should not generate any ambiguity. First of all, we will leave out
the indication of the functional dependence of Ψ (eR)(r) on the set of all electronic
coordinates r , as well as its parametric dependence on the ionic configuration R.
Also, since the adiabatic approximation has been operated and we are focussing just
on the electronic problem, there is no need to label either the wavefunction or the
energy. In short: Ψ (eR)(r) → Ψ and Ee(R) → E .
Another major simplification comes by switching from the SI units adopted so far
to the atomic units system where distances are measured in units of the Bohr radius

2
The electron energies and wavefunctions are numerically calculated respectively as the eigenvalues and
eigenvectors of the matrix representation of the Hamiltonian operator.

10-2
 Solid State Physics

a0 [1], masses in units of the electron mass me , and energies in units e 2 /(4πϵ0a 0 ) which
is referred to as Hartree. By this choice we get a twofold advantage: (i) atomic units
are very conveniently proportioned, so that most of the solid state properties turn
out to have almost unitary values (for instance, first nearest neighbours distances in
crystals are of the order of 1 a.u. of length) and (ii) the electron Schrödinger equation
assumes a very simple form, only depending on pure numbers3.
By combining all these changes in the notation, we can rewrite equation (10.2) in
the more compact form
⎡ ⎤
⎢− 1 ∑ ∇i2 + ∑ Vˆext(ri ) + ∑
1 ⎥
Ψ = E Ψ, (10.3)
⎢⎣ 2
i i i>j
∣ri − rj ∣ ⎥⎦

where it has been introduced the operator

Qα
Vˆext(ri ) = − ∑ , (10.4)
α
∣ri − Rα∣

describing the external potential Vext(ri) acting on each ith electron because of ionic
Coulomb attraction. This wording stresses the fact that this kind of electrostatic
interaction does involve the ions, which are objects external to the electron system.
Equation (10.3) is very elegant since it clearly indicates that the only parameters
needed to solve it are the Qα charges, namely: we only need to know which chemical
species are present in the crystal of interest in order to proceed with the solution of
the electron eigenvalue problem. For this reason, this way of proceeding is known as
a first-principles or ab initio method: the reader should appreciate the conceptual
difference with respect to the tight-binding method developed in appendix G, where
two- and three-centre energy integrals were considered disposable constants to fit.
The notation here presented will be used throughout this chapter. We made this
choice so as to match our formalism to the common practice [2–6]: as a matter of
fact, the density functional theory literature is prevalently written in atomic units
and makes use of the distinction between electron–ion interactions (external
potential term) and electron–electron interactions (Hartree term).

10.2 The Hohenberg–Kohn theorem

We start from a seemingly trivial statement: equation (10.3) provides evidence that
any change in E is associated with a change in the electron wavefunction Ψ.
Formally, this is tantamount to saying that the energy is a functional of Ψ this
mathematical concept is outlined in appendix I.
The key achievement of the resulting formalism—which is referred to as the
density functional theory (DFT)—is a theorem firstly proved by Hohenberg and
Kohn in 1964 which states that the ground-state energy EGS of an electron–ion
system4 is uniquely defined by a functional of the corresponding electron density ρGS

3
This means that it can be solved once for all, regardless the adopted unit system.
4
Throughout this chapter we assume spinless electrons.

10-3
 Solid State Physics

E GS = F [ρGS ], (10.5)
where, interestingly enough, ρGS is a function of just three variables: those needed to
specify the space position where such a density must be evaluated. We remark that,
for any quantum state, the electron density is defined as

ρ(r) = N ∫ Ψ 2 d r2d r3d r4⋯d rN , (10.6)

where N is the total number of electrons so that

∫ ρ(r)d r = N , (10.7)

which corresponds to a standard normalisation condition.

The proof of the Hohenberg–Kohn theorem proceeds in two steps. We first
observe that by changing the ionic configuration R → R′ we correspondingly change
′ or, in other words, we affect the Schrödinger
the external potential Vext → Vext
problem cast in equation (10.3). Therefore, we conclude that in any quantum state
the external potential uniquely determines the total electron wavefunction and so the
electron density. We now introduce an arbitrary assumption by supposing that the
very same ground-state electron density ρGS can be found for two unalike ionic
configurations, respectively, generating the external potentials Vext and Vext ′ with
′ . By setting
Vext ≠ Vext
⎡ ⎤
1 1 ⎥
Hˆ = ⎢ − ∑ ∇i2 + ∑ Vˆext(ri ) + ∑ ∣
⎢⎣ 2 ri − rj ∣ ⎥⎦
i i i>j (10.8)
= Tˆ + ∑ Vˆext(ri ) + VˆHartree
i

⎡ ⎤
1 ′ (ri ) + 1 ⎥
Hˆ ′ = ⎢ − ∑ ∇i2 + ∑ Vˆext ∑
⎢⎣ 2 ∣ri − rj ∣ ⎥⎦
i i i>j (10.9)
= Tˆ + ′ (ri ) + VˆHartree,
∑ Vˆext
i

we can write two eigenvalue problems

Hˆ Ψ = E Ψ and Hˆ ′Ψ′ = E ′Ψ′ , (10.10)
where Ψ ≠ Ψ′ and E ≠ E ′. For further convenience we have introduced the total
electron kinetic energy operator T̂ and the Hartree operator V̂Hartree associated to the
electron–electron Coulomb interaction energy. By using the eigenfunction Ψ of the
first Schrödinger problem appearing in equation (10.10), we can write5

5
This proof is based on the standard variational principle of quantum mechanics.

10-4
 Solid State Physics

⎡ ⎤
∫ ⎢ − 1 ∑ ∇i2 + ∑
Ψ*
1 ⎥
Ψ d r1⋯d rN
⎢⎣ 2
i
∣r − rj ∣ ⎥⎦
i>j i
⎡ ⎤
+ ∫
Ψ* ⎢∑ Vˆext(ri )⎥ Ψ d r1⋯d rN = E
⎢⎣ i ⎥⎦
(10.11)
⎡ ⎤
1 1 ⎥
∫ Ψ* ⎢ − ∑ ∇i2 + ∑
⎢⎣ 2 ∣r − rj ∣ ⎥⎦
Ψ d r1⋯d rN
i i>j i
⎡ ⎤
+ ∫
Ψ* ⎢∑ Vˆext
⎢⎣ i
′ (ri )⎥ Ψ d r1⋯d rN > E ′ ,
⎥⎦

where the second inequality holds since Ψ is not the ground-state eigenfunction of
Hˆ ′. These two relations can be reformulated in terms of the ground-state electron
density by taking into account the definition given in equation (10.6)

⎡ ⎤
1 1 ⎥
∫ Ψ* ⎢ −
⎢⎣ 2
∑ ∇i2 + ∑
∣ri − rj ∣ ⎥⎦
Ψ d r1⋯d rN + ∫ ρGS (r)Vext(r) d r = E
i i>j
(10.12)
⎡ ⎤
1 1 ⎥
∫ Ψ* ⎢ −
⎢⎣ 2
∑ ∇i2 + ∑ ∣ri − rj ∣ ⎥⎦
Ψ d r1⋯d rN + ∫ ′ (r) d r > E ′ .
ρGS (r)Vext
i i>j

By combining these results we get

E − E′ > ∫ ρGS (r)[Vext(r) − Vext′ (r)] d r. (10.13)

The same procedure could be repeated by first calculating the ground state energy E ′
with the eigenfunction Ψ′ and then by calculating the expectation value of Ĥ in the
same state; this leads to

E′ − E > ∫ ρGS (r)[Vext′ (r) − Vext(r)] d r. (10.14)

By summing equations (10.13) and (10.14) we obtain the mathematical contra-

diction 0 > 0: indeed an intriguing result! Since all calculations are correct, the only
possible logical conclusion is that our initial assumption that two unalike ionic
configurations correspond to the very same ground-state electron density is false.
In summary, the Hohenberg–Kohn theorem states that the ground-state energy of
an electron system is uniquely determined by the corresponding ground-state electron
density which only depends on three space coordinates. Through the use of a
functional of the electron density, this beautiful formal result dramatically reduces
the workload of the theory6 by reducing the N-electron problem—defined by 3N

10-5
 Solid State Physics

space coordinates—to a different problem ruled by just three space coordinates. It is

worth noting another important issue: since the knowledge of the ground state
energy enables the calculation of many other properties, the formulation of the
theorem can be meaningfully extended up to understand that the ground state
electron density uniquely determines the properties of such a state7.
In conclusion, we notice that the theory we developed so far does not provide any
practical information about the functional F [ρGS ]: we still lack any information
about its mathematical form and the practical way to calculate it. This task will be
developed in the next section.

10.3 The Kohn–Sham equations

In order to work out the electron density functional, we can combine equation (10.5)
with equation (10.12)(top)

F [ρ ] = ∫ ρ(r)Vext(r) d r + ∫ Ψ* ( Tˆ + VˆHartree ) Ψ d r1⋯d rN , (10.15)

and notice that the dependence upon the electron density is explicit in the first term
describing the external potential, while it is only implicit8 in the kinetic energy and
Hartree terms, thus making the mathematical problem really very convoluted.
The problem is solved by the Kohn–Sham procedure [2, 3] which relies on
suitable independent-electron wavefunctions obtained by introducing a fictitious
system of non-interacting electrons characterised by the same density as the real one.
We are developing this procedure in the following, omitting most of the mathemat-
ical subtleties which are addressed in more advanced books [4, 5].
The key idea is threefold: (i) first, we introduce new single-particle Kohn–Sham
orbitals {φKS,i (r)} i =1,2,…,N to be determined in the following and write

2
ρ(r) = ∑ φ KS,i (r) , (10.16)
i

for the electron density of the corresponding fictitious system of independent

electrons; (ii) next, we calculate its kinetic and Hartree contributions to the total
energy, as well as the contribution due to the external potential; (iii) finally, we add a
new term to the total energy functional F [ρ ] accounting for the difference between
the fictitious independent-electron system described by the Kohn–Sham orbitals and
the real one described by a many-body multi-electron wavefunction. This new term,
hereafter indicated by E xc[ρ ], is cast in the form of a (still unknown) functional of the

6
More precisely: the workload associated with the numerical implementation of the theory.
7
We stress that this result is only valid for the ground state, since its energy is the lowest possible one: a feature
which has been explicitly exploited in our formal proof of the Hohenberg–Kohn theorem. For a generic
quantum state other than the ground one, a similar conclusion cannot be drawn.
8
More precisely, it is the wavefunction itself to be a functional of the density: Ψ[ρ ].

10-6
 Solid State Physics

electron density and it is referred to as the exchange-correlation energy.

This procedure leads to

F [ρ ] = ∫ ρ(r)Vext(r) d r
1
−
2
∑∫ *
φKS, i (r)∇ φ KS,i (r) d r
2

i (10.17)
1 ρ(r)ρ(r′)
+
2
∫ ∫ ∣r − r′∣
d rd r′

+ E xc[ ρ ],

where the first three contributions on the right-hand side define the total energy of an
idealised system of independent electrons9. The last term instead carries all the
physical information needed to pass from the idealised to the real system: its is
customary to set E xc[ρ ] = E x[ρ ] + Ec[ρ ] and treat separately the exchange and the
correlation contributions, as outlined in the following.
According to the Hohenberg–Kohn theorem, the ground-state electron density
ρGS (r) is that which minimises the energy E = F [ρ ]: this statement is formalised by a
variational principle, that is by setting to zero its functional derivative (see appendix I)
δF [ρ ]
= 0. (10.18)
δρ ρ=ρGS

By using the chain rule for the derivative of a composite function, we can write

δF [ρ ] δF [ρ ] δρ δF [ρ ]
= = φ KS,i = 0, (10.19)
*
δφKS, δρ *
δφKS, i δρ
i ρ=ρGS ρ=ρGS ρ=ρGS

which represents an interesting conclusion: the functional derivative of the total

energy with respect to the KS orbitals is zero in the ground-state. In order to proceed
we must request such orbitals to be orthonormal

∫ φKS,
*
i (r)φ KS,j (r) d r = δij , (10.20)

so that the KS electron density provided in equation (10.16) is properly normalised.

As usual, this constraint to equation (10.19) is enforced (i) by introducing Lagrange
multipliers ξij , (ii) by defining the constrained energy functional F ′
⎡ ⎤
F′ = F − ∑ ξik⎣⎢∫ *
φKS, i (r)φ KS,j (r) d r − δij ⎥
⎦, (10.21)
ij

9
Once again, we remark that they correspond to the external potential (electron–ion Coulomb interaction
term), the total kinetic energy, and the Hartree (electron–electron Coulomb interaction term) contributions,
respectively.

10-7
 Solid State Physics

and (iii) by eventually imposing its minimum condition

δF ′[ρ ]
= 0. (10.22)
*
δφKS, i ρ=ρGS

This leads to

δF [ρ ]
∑ ξij φ KS,j = *
δφKS,
j i ρ=ρGS

1 2
=− ∇ φ KS,i
2
δ ⎡ ⎤
+
δρ ⎢
⎣∫ ρ(r)Vext(r) d r⎥⎦ φ KS,i (r) (10.23)

1 δ ⎡ ρ(r)ρ(r′) ⎤
+ ⎢
2 δρ ⎣
∫ ∫ ∣r − r′∣
d rd r′⎥ φ KS,i (r)
⎦
δE xc[ρ ]
+ φ KS,i (r),
δρ

where we have made use of equation (10.19). By calculating the functional

derivatives (see appendix I)
δ ⎡ ⎤
∫
⎢
δρ ⎣
ρ(r)Vext(r) d r⎥⎦ = Vext(r)
(10.24)
δ ⎡ ρ(r)ρ(r′) ⎤ ρ(r′)
∫ ∫
⎢
δρ ⎣ ∣r − r′∣
d rd r′⎥ = 2
⎦
∫ ∣r − r′∣
d r′ ,

we obtain
⎡ 1 ρ(r′) δE xc ⎤
∑ ξij φ KS,j = ⎢ − ∇2 + Vext(r) +
⎣ 2
∫ ∣r − r′∣
d r′ + ⎥ φ (r),
δρ ⎦ KS,i
(10.25)
j

which translates into a set of single-particle Kohn–Sham equations

⎡ 1 2 ⎤
⎢⎣ − ∇ + Vext(r) + VHartree(r) + Vxc(r)⎥⎦ φ KS,i (r) = Eiφ KS,i (r), (10.26)
2
where the new term Vxc(r) is defined as

δE xc[ρ ]
Vxc(r) = , (10.27)
δρ ρ=ρGS

and it is called exchange-correlation potential.

The achievement represented by the equations (10.26) is more conceptual than
practical, since we do not really know the exchange-correlation potential yet and this

10-8
 Solid State Physics

information is needed in order to provide the exact ground-state density and energy
of the real many-body electron system. Without this correction we still operate at the
level of the independent electron approximation (although at a higher erudition with
respect to the tight-binding theory): something we definitely want to improve. For
this reason, we need to develop some knowledge about the way to construct E xc[ρ ]:
this is the aim of the next section. Here we can nevertheless anticipate a major
conceptual breakthrough: provided that the exchange-correlation problem has been
solved in some way or another, the set of equations (10.26) plays a key role in the first-
principles theory of the solid state for a twofold reason, namely: (i) they allow us to
construct the electron density as shown in equation (10.16); and (ii) they make it
possible to calculate many ground-state physical properties of crystalline systems,
through the prediction of its total energy EGS = F [ρGS ]. This is often recognised by
addressing these equations as the Kohn–Sham theory of the solid state.

10.4 The exchange-correlation functional

The construction of the most accurate exchange-correlation functional for a many-
body electron system is an active and lively topic of contemporary solid state theory.
Presenting a full account of the various ongoing efforts pointing in this direction is
surely beyond the scope of the present introductory chapter to DFT and we forward
the avid reader to more advanced textbooks [4–6]. Here we limit ourselves to
treating the problem at the lowest possible sophistication, known as the local density
approximation (LDA), which has historically represented a first widely-adopted
practical implementation of DFT and it still represents a meaningful benchmark.
The starting point is to approximate the real electron system as a homogeneous
electron gas (HEG), namely a fictitious system where the ion positive charges are
uniformly distributed in space—so as to form a positive background—and the
electron density is similarly uniform. This model allows us to focus just on electrons
and their mutual interactions, without taking into explicit consideration the role of
ions10. The exchange energy ExHEG of this model system is exactly obtained by a
standard quantum mechanical calculation [2]
3 ⎛ 3 ⎞1/3
E xHEG = − ⎜ ⎟ V [ρ HEG ] 4/3 , (10.28)
4 ⎝π ⎠
where ρHEG = N /V is the density of an N-electron homogeneous gas with volume V.
On the other hand, there is no exact result for the correlation energy EcHEG . A
numerical procedure has therefore become the common practice: the correlation
energy is obtained by the computed value of the HEG total energy (which is the
output of the numerical solution of its complete Schrödinger equation) from which
all the known contributions (kinetic, Hartree, and exchange energy) are subtracted.
A standard parametrisation formula is used for the result [3–5]

10
It is important to remark that the HEG is not the free electron gas discussed in chapter 7, since Coulomb
repulsions between electrons are now considered.

10-9
 Solid State Physics

⎧ 0.002 rs ln rs − 0.116 rs + 0.031 1 ln rs − 0.048 if rs < 1

E cHEG = V ρ HEG × ⎨ (10.29)
⎩− 0.142 3 (0.333 4 rs + 1.052 9 rs + 1)−1 if rs ⩾ 1,

where rs = (3/4πρHEG )1/3.

The electron density in a real crystal is of course not the same as in a
homogeneous electron gas (simply because it cannot be as uniform) and, con-
sequently, it would be a too drastic approximation just using the same HEG
exchange and correlation energies throughout the system. Rather, we could divide
the crystal into infinitesimal volume elements d r and attribute each of them a
different local density value ρ(r) to be used in the HEG expressions for the exchange
and correlation energy functionals. The total exchange and correlation energies are
then obtained by integrating over the crystal volume. This is easy to show in the case
of the exchange term (which has the exact result reported in equation (10.28)) that
this way of proceeding leads to
1 3 ⎛ 3 ⎞1/3
E xLDA[ρ(r)] =
V
∫
V
E xHEG[ρ(r)] d r = − ⎜ ⎟
4 ⎝π ⎠
∫V [ρ(r)]4/3 d r, (10.30)

which provides the exchange energy functional in the local density approximation.
The corresponding exchange potential Vx(r) is obtained by a similar functional
derivation as in equation (10.27) (see appendix I)
δE xLDA ⎛ 3 ⎞1/3
VxLDA(r) = = −⎜ ⎟ [ρ(r)]1/3 , (10.31)
δρ(r) ⎝π ⎠

which represents a really important practical result: in order to calculate the

exchange potential in position r , we only need to know the electron density in that
point.
A similar conclusion is drawn for the correlation potential, although the algebra
is more complicated since EcHEG is not available in analytical form, as reported in
equation (10.29). It is anyway understood that ExcLDA = ExLDA + EcLDA .
Nowadays a number of improvements going beyond the local density approx-
imation are indeed available including, but not limited to, the more fundamental
description taking into account the electron spin or the more accurate treatment of
non-homogeneous electron densities (like, for instance, the gradient correction
approach) [2–5].

10.5 The practical implementation and applications

Everything is now ready to solve the equations (10.26) since we have
computable expressions for any energy term there contained. The problem is that
both the Hartree and the exchange-correlation terms do depend on the electron
density or, equivalently, on the very solutions of the Kohn–Sham equations, as
clearly suggested by equation (10.16). The solution to this impasse is offered by the
iterative method presented in section 1.4.1: some initial electron density is guessed
(for instance, ρ could be defined as the sum of the electron densities of isolated

10-10
 Solid State Physics

atoms) and the corresponding Kohn–Sham equations solved; next, a new density is
generated by using the just calculated Kohn–Sham orbitals and confronted with the
previously guessed one: if they differ above an agreed accuracy threshold, the new
density is used to write another set of Kohn–Sham equations to be again solved; this
loop is repeated until the electron densities of the nth and (n − 1) th iterations agree
to within the chosen accuracy.
Once that full convergence is eventually proclaimed, the resulting orbitals,
ground-state total energy and electron density can be used to calculate, in a totally
parameter-free fashion, a large variety of solid state properties, including (but not
limited to): equilibrium crystalline structures; cohesive energy and elastic moduli;
vibrational spectra and phonon properties (like, for instance, the lattice thermal
conductivity); the band structure and electron properties (like, for instance, the
electrical conductivity)11. For instance, the phonon dispersion relations reported in
figures 3.7–3.9 have been obtained by diagonalising a dynamical matrix whose
entries are the second derivatives of the crystal total energy E T [7] which, as
discussed in section 1.3.4, has been calculated as E T = F [ρGS (r)] + U (R). The same
DFT vibrational frequencies can be used to predict the heat capacity CVquantum(T )
according to equation (4.17) or the thermal conductivity κ l(T ) according to equation
(4.40).
In conclusion, DFT is nowadays looked upon as the ‘standard model’ for
condensed matter physics: as a matter of fact, it is the most widespread method
for calculating the quantum-mechanical properties of molecules, solids, nano-
structures, surfaces and even of liquids. In addition, all superior theoretical schemes
realistically applicable on a large scale (like, for instance, methods based on hybrid
functionals, or the time-dependent DFT) are based on the density functional theory
[2–6].

References
[1] Colombo L 2019 Atomic and Molecular Physics: A Primer (Bristol: IOP Publishing)
[2] Giustino F 2014 Materials Modelling using Density Functional Theory (Oxford: Oxford
University Press)
[3] Sholl D S and Steckel J A 2009 Density Functional Theory—A Practical Introduction (New
York: Wiley)
[4] Engel E and Dreizler R M 2011 Density Functional Theory (Berlin: Springer)
[5] Martin R M 2012 Electronic Structure—Basic Theory and Practical Methods (Cambridge:
Cambridge University Press)
[6] Parr R G and Yang W 1989 Density Functional Theory of Atoms and Molecules (Oxford:
Oxford Science Publications)
[7] Baroni S, de Gironcoli S, Dal Corso A and Giannozzi P 2001 Phonons and related crystal
properties from density-functional perturbation theory Rev. Mod. Phys. 73 515

11
On the other hand, since DFT is a ground-state theory it is unable to accurately predict the band gap of
insulators or any optical property, like the absorption coefficient. These properties can be calculated by using
time-dependent DFT or many-body techniques.

10-11
 Part IV
Concluding remarks
 IOP Publishing

Solid State Physics

A primer
Luciano Colombo

Chapter 11
What is missing in this ‘Primer’

A ‘Primer’ cannot be, by definition, a complete and in-depth introduction to any topic.
This is especially true for condensed matter physics, which is likely the most rich,
diverse, interdisciplinary, and fast-developing area of physical sciences1. This volume
is no exception and, therefore, it must be acknowledged that many important issues
have not been dealt with. We feel dutifully committed to mentioning them.
A concise list of missing arguments, as well as of topics which have not been
thoroughly treated, is the following:
1. crystallography treated by group theory,
2. Hartree–Fock theory for the many-body electron problem,
3. dielectric screening,
4. non-adiabatic phenomena,
5. a more detailed account for optical properties in metals and insulators,
6. magnetic field effects on the electron gas,
7. magnetic properties and magnetic ordering,
8. superconductivity,
9. surface physics,
10. physics of low-dimensional solid state systems.

Most of the above topics are covered by the textbooks quoted in the bibliography
appended to each chapter: for further information, we refer the reader to these manuals.
We observe, however, that their level is still introductory, although often more thorough
than found in this Primer. The cutting-edge research in modern condensed matter
physics is quite far away: at that level, it is preferable to deal directly with the scientific
literature published in specialised journals of solid state and/or materials physics.

1
An interest account for this is reported in the paper by Sinatra R, Deville P, Szell M, Wang D and Brabási A-L
2015 Nature Physics 11 971, where a thorough analysis based on Web of Science data reveals the growth and
multidisciplinarity of any physics subdiscipline over the last 100 years, or so.

doi:10.1088/978-0-7503-2265-2ch11 11-1 ª IOP Publishing Ltd 2021

Part V
Appendices
 IOP Publishing

Solid State Physics

A primer
Luciano Colombo

Appendix A
The periodic table of elements

By applying in conjunction the Pauli principle and the Hund rules [1], we can
determine the ground-state electronic configurations of all atoms and, accordingly,
draw the modern release of the Mendeleev periodic table of elements. It is reported
in figure A.1, where the color shadowing represents the progressive filling of the
atomic s-, p-, d-, and f-shells.

Figure A.1. The periodic table of the elements.

doi:10.1088/978-0-7503-2265-2ch12 A-1 ª IOP Publishing Ltd 2021

 Solid State Physics

Reference
[1] Colombo L 2019 Atomic and Molecular Physics: A Primer (Bristol: IOP Publishing)

A-2
 IOP Publishing

Solid State Physics

A primer
Luciano Colombo

Appendix B
Alloys, polycrystals, and quasi-crystals

B.1 Alloys
Let us consider a diamond lattice and imagine randomly occupying its sites with Si
and Ge atoms. By setting x the concentration of Si atoms, with the constraint that
x ⩽ 1, it turns out that the corresponding concentration of lattice sites occupied by
Ge atoms must be equal to (1 − x ). The resulting crystal structure is neither
diamond-like (because we have two different chemical species) nor zincblende-like
(because the two species are not orderly distributed). Therefore, we will refer to this
new atomic architecture as SixGe1−x alloy. A similar situation is found if we start
with a zincblende lattice and imagine occupying (i) the anionic sublattice with As
atoms and (ii) the cationic sublattice by a random distribution of Al atoms in
concentration x and Ga atoms in concentration (1 − x ). Once again, the constraint
x ⩽ 1 must hold. In this case the structure is in part ordered (the underlying lattice is
fcc with a two-atom basis) and in part random (the chemical distribution is random,
even if it respects the stoichiometry of a typical III–V semiconductor). We will refer
to this new atomic architecture as AlxGa1-xAs alloy. The two alloys are pictorially
represented in figure B.1.
Metal alloys also exist, sometimes referred to as bulk metallic glasses. They are
classified in three most common structures: (i) substitutional alloy (a bcc or fcc
lattice is randomly decorated by two or more metal species); (ii) interstitial alloy
(where a number of interstitial defects are found within a normal metal lattice,
corresponding to chemical species other than the host one); and (iii) substitutional–
interstitial alloy (where the two previous situations are both found). Interstitial
defects can be either metallic or non-metallic, like e.g. in the mostly relevant case of
steel: an alloy made by iron, silicon, and carbon. In general, the resulting alloy has
properties that differ, even largely, from those of the pure metals, such as increased
strength or hardness. This is the realm of physical metallurgy [1, 2].
The lattice constant of an alloy a 0alloy is predicted by the Vegard empirical law
which in the simple case of a binary mixture reads as

doi:10.1088/978-0-7503-2265-2ch13 B-1 ª IOP Publishing Ltd 2021

 Solid State Physics

Figure B.1. A SixGe1−x alloy (left) and a AlxGe1−xAs alloy (right). See figure 2.12 for the corresponding
diamond and zincblend lattice, respectively.

a 0alloy = x a 0a + (1 − x ) a 0b + ξ x(1 − x ), (B.1)

where a 0a and a 0b are the lattice constants of the two elemental solids made with the
same atoms found in the alloy1. The last term ξ x(1 − x ) (where ξ is a pure
phenomenological constant to be determined, case by case, in an empirical way) is
often omitted, so approximating the variation to a linear law: in many cases this is a
more than good approximation.

B.2 Polycrystals
Polycrystals are solid objects made by an assembly of randomly oriented finite
crystals (usually referred to as crystallites or grains). Crystallites have typical
dimensions ranging from the nm to the μm length scale which correspond to
nanocrystalline or microcrystalline systems, respectively. Interface regions between
facing grains usually have high structural disorder: here, grain boundaries are
defined by the misalignment among crystallites, generating a non-crystalline bond-
ing interface environment or, possibly, even a thin amorphous layer.

B.3 Quasi-crystals
Quasi-crystals are atomic architectures with symmetry properties in between a
crystal and a liquid. They are experimentally obtained by rapidly cooling binary
molten mixtures (typically, inter-metallic alloys). The intermediate crystalline-liquid
character of their crystal structure is exploited by the twofold feature that (i) it lacks
translational invariance, but (ii) it contains a regular pattern of similar structural
units.
The most effective way to visualise a quasi-crystal is by following the Penrose
tiling construction shown in figure B.2. Two rhombic building blocks are chosen,
respectively with a pair of internal angles of (108°, 72°) and (144°, 36°). These
rhombuses are combined as shown in the figure, so as to generate a structure that,
although lacking translational invariance, presents a regular repetition of equally
oriented decagons. This regular patter defines families of parallel lines, always

1
With reference to figure 2.22 they are the Si and Ge lattice constants for the alloy SixGe1−x or the AlAs and
GaAs lattice constants for the alloy AlxGa1−xAs.

B-2
 Solid State Physics

Figure B.2. A two-dimensional quasi-crystal obtained by combining the two structural units shown as shaded
rhombic building blocks. The red decagon represents the structural unit which is found with regular
orientation, so that periodic parallel lines (shown in blue colour) are found. Note that similar families of
parallel planes are found in the quasi-crystal, forming a 72° angle to each other.

forming a 72° angle to each other. The resulting lattice displays a five-fold rotational
symmetry, which is indeed forbidden in normal two-dimensional lattices, as
discussed in section 2.3.2.
The three-dimensional counterpart of the quasi-crystal shown in figure B.2 is
obtained by placing atoms with icosahedral symmetry2. This structure is obtained
provided that a tetrahedral arrangement of atoms is used as basic building block in
the attempt to realise a close-packed structure: the icosahedron is obtained by letting
20 tetrahedra to share a common vertex. Slight distortions of such icosahedra must
be allowed to obtain the three-dimensional quasi-crystal, thus preventing long-range
translational invariance. X-ray diffraction provides experimental evidence that some
inter-metallic alloys display a three-dimensional five-fold symmetry.

References
[1] Abbashian R, Abbashian L and Reed-Hill R E 2009 Physical Metallurgy Principles 4th edn
(Stanford, CT: Cengage Learning)
[2] Smallman R E and Ngan A H W 2007 Physical Metallurgy and Advanced Materials 7th edn
(Oxford: Butterworth-Heinemann)

2
We recall that icosahedrons are solids with six C5 rotation axes (Schoenflies nomenclature).

B-3
 IOP Publishing

Solid State Physics

A primer
Luciano Colombo

Appendix C
Essential thermodynamics

C.1 Basic definitions

Let us consider a condensed matter system and agree to describe it according to
classical physics. By assigning the individual positions and momenta of its
elementary constituents1, we define a microstate of the system. As easily understood,
the number of microscopic degrees of freedom needed to set a specific microstate is
really very high. On the other hand, any macroscopic configuration of the same
system considered as a whole can be precisely defined by a comparatively much
smaller number of thermodynamic parameters, like for instance its volume V or the
number of moles nk for each kth chemical species found in the system (more
thermodynamic parameters are rigorously defined below). We will address such a
configuration as a macrostate. The key idea underlying thermodynamics is to replace
the description of a system based on microstates with a representation of its physics
based on macrostates. In essence, thermodynamics is a sophisticated averaging
procedure of microscopic degrees of freedom [1–3].
In general, a macrostate can be realised in a myriad of different micro-modes2. If
we name Ω such a number and set [3]
S = kB ln Ω , (C.1)
we introduce an operative definition of the system entropy S which is, therefore, a
direct estimation of all the possible microstates compatible with the assigned set of
thermodynamic parameters. The equilibrium state is defined as the state with
maximum entropy; in other words, it represents the macrostate with the maximum
number of micro-realisations. When a system is at equilibrium, no time variation of
its thermodynamic parameters is observed. However, we remark that it undergoes

1
We could also define a set of generalised coordinates and conjugated momenta: our reasoning is unaffected by
this choice.
2
Just think about a mole of a monoatomic gas in a container with rigid and fixed walls: each possible
macrostate can be obtained by several choices of the particle positions within the container.

doi:10.1088/978-0-7503-2265-2ch14 C-1 ª IOP Publishing Ltd 2021

 Solid State Physics

microscopic evolution: basically, it can explore all those different microstates which
correspond to the same set of thermodynamic parameters defining the state of
equilibrium.
Under the constrain which limits our theory to equilibrium states only, it is
nevertheless possible to consider changes of thermodynamic state caused by some
mechanical, chemical, or thermal action occurring so slowly that the system can
reach the equilibrium at each intermediate step. In this case we speak of quasi-static
transformations which, by construction, are a sequence of equilibrium states. In
general, a transformation is ‘slow enough’ to become quasi-static if it occurs over a
time span much larger than the typical relaxation time of the system.

C.2 Internal energy

We name internal energy U of a thermodynamic system its total energy content; it
corresponds to the work needed to form such an aggregate. The internal energy is a
state function since its value is uniquely defined by the thermodynamic parameters
defining the macrostate; we write
U = U(S ,V ,{nk}), (C.2)
where {nk } represents in compact form the set of numbers defining how many moles
for each k = 1, 2, 3, … different chemical species form the system. Upon
differentiation
∂U ∂U ∂U
dU =
∂S
dS +
∂V
dV + ∑∂ dnk , (C.3)
V ,{nk } S ,{nk } k
nk V ,S ,{nj }j ≠k

we get the definition of the following thermodynamic parameters [1, 2]

∂U
temperature: T=
∂S V ,{nk }

∂U
pressure: P =− (C.4)
∂V S ,{nk }

∂U
chemical potential: μk = ,
∂nk V ,S ,{nj }j ≠k

which are known as intensive parameters or state variables; on the other hand,
internal energy, entropy, volume, and mole numbers are extensive parameters.
Through these definitions we can rewrite equation (C.3) in a more compact and
elegant way
d U = T dS − P dV + ∑ μk dnk , (C.5)
k

which basically represents an energy balance since the quantities −PdV , TdS, and
μk dnk , respectively, represent the mechanical, non-mechanical, and chemical work
exchanged during a quasi-static infinitesimal transformation. The exchanged

C-2
 Solid State Physics

non-mechanical work is referred to as the exchanged heat and it is formally written

as dQ = TdS .
At equilibrium the thermodynamic parameters are such that U is minimum.
Furthermore, they obey restrictions that can be easily obtained by considering the
simple case of a monoatomic system made by two sub-systems which we label by
‘1’ and ‘2’. We consider three different situations where, respectively, the two sub-
systems can (i) exchange just heat, or (ii) exchange heat and mechanical work, or (iii)
exchange heat and mass. Then, it is easy to prove that in the three cases at
equilibrium we have (i) T1 = T2 , or (ii) T1 = T2 and P1 = P2 , or (iii) T1 = T2 and μ1 = μ2 ,
where of course T1,2 , P1,2 , and μ1, 2 are the temperature, pressure, and chemical
potential of the two sub-systems. These results prove that the formal definitions
provided in equations (C.4) agree with their phenomenological counterparts.
Equation (C.2) is a fundamental thermodynamic equation since it contains any
information about the system3. Intensive parameters are related by equations of
state: a simple example is given by PV = nRT , namely the equation of state for a
monoatomic ideal gas. While useful for many applications, we must nevertheless
understand that each equation of state only contains a limited amount of information.
As a matter of fact, the full set of equations of state valid for a system is able to give
the same knowledge provided by its fundamental equation U = U(S ,V ,{nk }).

C.3 Thermodynamic potentials

Extensive and intensive parameters have been so far used as independent and
derived quantities, respectively. In many circumstances, however, it is more
appropriate to use intensive parameters as independent variables (since it is much
easier to measure them) and to replace the internal energy by some new thermody-
namic potential.
Formally, this is done by calculating the Legendre transformation LX [U] of the
internal energy with respect to a suitable intensive parameter X. Accordingly, we
define [1–3]
Helmholtz free energy: LT [U] = U − TS = F(T ,V ,{nk}) (C.6)

enthalpy: LP [U] = U + PV = H(S , P,{nk}) (C.7)

Gibbs free energy: LT ,P [U] = U − TS + PV = G(T, P ,{nk}), (C.8)

representing, respectively, the work exchanged quasi-statically during a transforma-
tion occurring at constant temperature (Helmholtz free energy), at constant pressure
(enthalpy), and at constant temperature and pressure (Gibbs free energy).
The three thermodynamic potentials allow us to define the equilibrium condition
when the system is constrained. For instance, if the system is coupled to a heat
reservoir (that is, its temperature is kept constant), then the equilibrium state is that

3
It is also possible to choose entropy S = S (U,V ,{nk }) as the state function of the system. This allows for a
different formal development of thermodynamics.

C-3
 Solid State Physics

with minimum Helmholtz free energy. On the other hand, if the system is kept at
constant pressure (that is, it undergoes the mechanical action of a pressure reservoir)
then the equilibrium is defined by the minimum of the enthalpy. Finally, the
minimum of the Gibbs free energy defines the equilibrium state of a system coupled
to the double action of a heat and a pressure reservoir.

C.4 Some thermodynamic materials properties

Through the derivatives of the internal energy reported in equations (C.4), we have
defined the relevant thermodynamic parameters of temperature, pressure, and
chemical potential. Other physical properties of solids can be similarly defined. In
the following, we will always assume that the numbers of moles are kept constant: for
brevity, we will omit formally indicating this constraint.
It is known experimentally that, in general, a solid increases its volume upon
heating. The thermal expansion coefficient β defines the fractional volume increase
per unitary increment of temperature at constant pressure4
1 ∂V 1 ∂V ∂P 1 ∂P
β= =− = , (C.9)
V ∂T P V ∂P T ∂T V B ∂T V

where in the right-hand side of the above equation the bulk modulus has been
introduced
∂P
B = −V , (C.10)
∂V T

with B > 0 since (∂P /∂V )T is negative for condensed matter systems; its inverse
function
1 ∂V
αT = B −1 = − , (C.11)
V ∂P T

is referred to as the isothermal compressibility: it provides the fractional volume

decrease per unitary increment of pressure at constant temperature.
More robust experimental evidence is that, by quasi-statically supplying heat to a
thermodynamic system, we generally observe an increase of its temperature. We
define the amount of heat we need to supply in order to produce a unitary increase of
the temperature, while keeping the volume constant, as
∂U ∂S ∂Q
CV (T ) = =T = , (C.12)
∂T V ∂T V ∂T V

4
In deriving this equation we make use of a nontrivial mathematical identity. Let us consider a function
f = f (x,y,z ) under the constraint f (x,y,z ) = constant. Then, the variables (x,y,z ) are not independent since the
constraint defines an implicit relationship among them. Accordingly, their partial derivatives satisfy a number
of relations, among which (∂x /∂y )z = −(∂z /∂y )x /(∂z /∂x )y = −(∂x /∂z )y (∂z /∂y )x that we used to obtain
equation (C.9). We also use the identities (∂x /∂y )z = 1/(∂y /∂x )z and (∂x /∂y )z = (∂x /∂w )z /(∂y /∂w )z .

C-4
 Solid State Physics

and we refer to this quantity as the constant-volume heat capacity of the system.
CV (T ) is an extensive property, while its corresponding intensive property is referred
to as the constant-volume specific heat and it is calculated dividing CV (T ) by the
system mass5. Similarly we define the constant-pressure heat capacity as
∂U ∂S ∂Q
CP(T ) = =T = . (C.13)
∂T P ∂T P ∂T P

For any real crystal we have [1, 2]

β2
CP(T ) = CV (T ) + TV
αT
(∂P / ∂V V )2
= CV (T ) − T (C.14)
∂P / ∂V T
⎛ ∂V ⎞2 ∂P
= CV (T ) − T ⎜ ⎟
⎝ ∂T P ⎠ ∂V T

that is CP(T ) > CV (T ): we need to supply a larger amount of heat in constant-

pressure conditions since part of the dispensed thermal energy is spent in mechanical
work.

References
[1] Callen H 1985 Thermodynamics and An Introduction to Thermostatistics (New York: Wiley)
[2] Dittman R and Zemansky M 1997 Heat and Thermodynamics 7th edn (New York: McGraw-
Hill)
[3] Goodstein D L 1985 States of Matter (New York: Dover)

5
Sometimes the specific heat is defined per unit of volume, instead of per unit of mass.

C-5
 IOP Publishing

Solid State Physics

A primer
Luciano Colombo

Appendix D
Force constant models for lattice dynamics

In this appendix we briefly describe some force constant models which have been
extensively used in lattice dynamics.
In the phenomenological approach, the force constant models are basically used
to calculate the entries of the dynamical matrix, whose eigenvalues eventually
provide the vibrational frequencies. The most relevant common feature is that they
exploit a number of empirical interactions, each depending on unknown parameters
that must be fitted on some experimental information.
The actual form of such empirical interactions is guessed according to the
character of the interatomic bonding found within the crystal of interest. Quite a
few different analytical forms for such interactions have been proposed over the
years, including (but not limited to): pairwise or many-body potentials, as well as
central or bond-angle dependent potentials [1–3]. However, in one way or another,
under the harmonic approximation they are reduced to a system of effective springs
connecting either massive point-like ions or massless objects mimicking the charge
distribution of the valence electrons1. The common feature of empirical force fields is
that they are short-ranged: interactions only extend to just the nearest coordination
shells. For this reason the force constant models are commonly labelled as first-next-
neighbours or second-next-neighbours (and so on …) interactions models.
Normally, the more extended the interactions, the more accurate is the calculation
of the vibrational spectrum. The price to pay is the increasing number of fitting
parameters, a feature that makes the resulting dynamical model definitely not a
fundamental one (and, therefore, poorly transferable to systems unalike the fitted
one).
In the case of ionic crystals, long-range Coulomb interactions are added to short-
range ones. Ions are considered as point-like objects (and, therefore, no atomic
polarisation effects are usually included at this level) with an effective dynamical
charge to be empirically determined [2, 4]. The evaluation of the corresponding

1
The object representing the electron charge distribution is assumed to be massless in order to enforce the
adiabatic approximation.

doi:10.1088/978-0-7503-2265-2ch15 D-1 ª IOP Publishing Ltd 2021

 Solid State Physics

electrostatic forces requires laborious lattice sums: indeed a very tricky task, which is
typically solved by the Ewald construction [5].
Before outlining the main physical features of some popular force constant
models, it is important to remark that the modern theory of lattice dynamics no
longer relies on such empirical models. Rather, a more fundamental and superior
first principles approach is followed: a fully quantum mechanical calculation of the
crystal total energy is performed, from which force constants are eventually
obtained. This approach requires the more advanced theory outlined in chapter
10, as well as a nontrivial numerical implementation based on high-performance
computing algorithms [6]. For this reason empirical force constant models are
nowadays still useful as pedagogical tools: they allow a very intuitive approach to
the calculation of the vibrational properties of a crystal, easily translated into a low-
complexity computational scheme.

D.1 The rigid ion model

By far the most simple approach is based on the rigid ion model, which over the years
has been extensively applied to metals as well as to ionic crystals. In the first case just
central pair interactions between ions are used, up to a pre-assigned order of
neighbours (in fact the range of interactions works as an additional free parameter of
the model), as shown in figure D.1(top).
In view of its simplicity, the rigid ion model has been occasionally adopted to
calculate vibrational frequencies in covalently bonded solids. In this case both
Coulomb and short-range forces are needed, the latter both central and non-central
(to stabilise, for instance, the tetrahedral coordination in crystals with a diamond-
like structure).
The next level of increasing sophistication consists in guessing a lattice potential
energy depending both on ionic displacements and bond angle distortions. The two
sets of force constants, respectively, describe bond-stretching and bond-bending
vibrations. This solution is referred to as the valence force field.

D.2 The shell model

A significant step forward in accuracy is represented by the shell model, existing in
different versions for ionic and covalent crystals. Here an ion is described as a non-
polarisable core (formed by the nucleus and the core electrons) around which
valence electrons are modelled as a massless shell. Each ion core is coupled to its
shell by isotropic springs, as shown in figure D.1(bottom, left). Cores and shells have
different electric charge, so that their sum corresponds to the net charge of the ion.
The new key feature of this model is that ion polarisation is allowed by means of
the relative core–shell displacements. Similarly to the rigid ion case, core–core short-
range interactions are also considered. In addition, shell–shell and core–shell
interactions between nearest neighbouring ions are included, each described by a
suitable spring. The accuracy is significantly improved in the breathing shell model
version, where radial deformability of the electron shell is allowed as well (this is not
shown in figure D.1).

D-2
 Solid State Physics

Figure D.1. Schematic representation of the system of short-range effective harmonic springs for a two-ion
crystal (ion A and ion B), as predicted by the rigid ion model (top; it is shown the case of interactions up to the
second-next-neighbours), in the shell model (bottom, left) and in the bond charge model (bottom right; it is
shown the case of an elemental covalent solid). Long-range Coulomb interactions can be added, if requested.

Long-range Coulomb interactions are included in this scheme by considering each

core–shell pair as a point-like object and, therefore, by performing lattice sums
similarly to the case of the rigid ion model.

D.3 The bond charge model

The shell model can hardly be applied to covalent crystals, where the valence charge
is shared between nearest neighbouring ions rather than split among them (see
figure 2.22). In these crystals the maximum of the valence charge is located
somewhere along the bond. More specifically, this maximum is found at the
midpoint of the bond in elemental covalent crystals (like diamond, silicon or
germanium), while it is slightly shifted towards the anion in such a way as to divide
the bond in two segments whose lengths are found in the 3:5 or 2:6 ratio in III–V or
II–VI materials, respectively. In figure D.1(bottom, right) the case of an elemental
crystal is shown.
The valence charge is treated as a massless bond charge (bc) which is coupled to
neighbouring ions and other bc’s by effective springs. This mimics the incomplete
screening of ionic charges and yields effective non-central short-range interactions
between anions and cations. This is crucially important in order to stabilise the
tetrahedral coordination.

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 Solid State Physics

Long-range Coulomb interactions are included also in this scheme by considering

each ion and bc as a point-like object and, therefore, by performing lattice sums
similarly to the case of the rigid ion model.

References
[1] Brüesch P 1982 Phonons: Theory and Experiment vol I (Berlin: Springer)
[2] Srivastava G P 1990 The Physics of Phonons (Bristol: Adam Higler)
[3] Finnis M 2003 Interatomic Forces (Oxford: Oxford University Press)
[4] Böttger H 1983 Principles of the Theory of Lattice Dynamics (Berlin: Akademie)
[5] Kittel C 1996 Introduction to Solid State Physics 7th edn (Hoboken, NJ: Wiley)
[6] Baroni S, de Gironcoli S, Dal Corso A and Giannozzi P 2001 Phonons and related crystal
properties from density-functional perturbation theory Rev. Mod. Phys. 73 515

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 IOP Publishing

Solid State Physics

A primer
Luciano Colombo

Appendix E
Quantum statistics

E.1 Identical particles

Two particles are identical provided they have the very same characteristics (i.e. the
same intrinsic physical properties like, for instance, the mass, the charge, the spin,
and so on). Identical particles can be distinguished, if treated according to classical
mechanics, by taking into consideration their positions. On the other hand,
according to quantum mechanics we cannot assign a well defined position to any
particle and this prevents us distinguishing among them: in short, in quantum
mechanics identical particles are also indistinguishable.
Let us then consider the Hamiltonian operator Ĥ describing a set of N identical
and indistinguishable particles: it is invariant upon interchanging any two particles.
However, the same operation could differently affect its eigenfunction
Φ(r1, r2, … , rN ) describing the state of the system. This is full of consequences
worthy of consideration.
Let Pˆμ↔ν the permutation operator which acts on the eigenfunctions of Ĥ by
swapping the particles labelled by the indices μ ↔ ν . Because of the above
invariance, we have
[Hˆ , Pˆμ↔ν ] = 0, (E.1)

and, therefore, Φ(r1, r2, … , rN ) is a wavefunction of either Ĥ and Pˆμ↔ν . It is easy to

prove that
Pˆμ↔νΦ(r1, r2 , ⋯ , rN ) = ±Φ(r1, r2 , ⋯ , rN ), (E.2)

or, equivalently: any wavefunction describing a set of identical and indistinguishable

particles is either symmetric or antisymmetric under particle interchange:
Pˆμ↔νΦ = +Φ symmetric wavefunction, (E.3)

Pˆμ↔νΦ = −Φ antisymmetric wavefunction. (E.4)

doi:10.1088/978-0-7503-2265-2ch16 E-1 ª IOP Publishing Ltd 2021

 Solid State Physics

The symmetry property of a wavefunction represents an intrinsic property of the

specific set of particles considered since it cannot be altered by any external action.
Identical and indistinguishable particles are said to be fermions or bosons if their
collective quantum states are described by antisymmetric or symmetric wave-
functions, respectively [1–4].
The antisymmetric character of the fermion wavefunction has a very important
consequence which we will derive under the assumption that a system of N fermions
can be treated within a single-particle approximation; in other words, we will assume
that the complete problem
Hˆ Φ(r1, r2 , … , rN ) = ET Φ(r1, r2 , … , rN ), (E.5)

can be reduced to N different single-particle problems

hˆi ϕni (ri ) = Eiϕni(ri ), (E.6)

where i = 1, 2, … , N labels the fermions and ni is a suitable set of quantum

numbers describing the single-particle state with energy Ei. Of course we have
ET = ∑i Ei .
In order to exploit its antisymmetric character we can write the total wave-
function in the form of a Slater determinant
φn1(r1) φn1(r2) ⋯ φn1(rN )
1 φn2(r1) φn2(r2) ⋯ φn2(rN )
Φ(r1, r2 , … , rN ) = . (E.7)
N! ⋯ ⋯ ⋯ ⋯
φnN (r1) φnN (r2) ⋯ φnN (rN )

This form of Φ(r1, r2, … , rN ) naturally embodies the anti-symmetric character of the
state function, since by swapping two columns the Slater determinant will change
sign. In addition, we remark that a determinant vanishes if two rows are equal. This
implies that a system of identical and indistinguishable fermions cannot occupy a
many-body state where two single-particle states are equal. This statement is
commonly referred to as Pauli principle; it states that it is impossible to have two
fermions with the same set of quantum numbers.
The rationalisation of a large number of different experimental investigations
leads to the conclusion that electrons, protons, and neutrons are fermions, while
photons and phonons are bosons [5]. The fermion or boson character of an
elementary particle is ultimately determined by its spin: fermions have half odd
integer spin ℏ/2, 3ℏ/2, … while bosons have integer spin 0, ℏ, ….
In quantum statistics [6, 7] the ultimate goal is to find the equilibrium state of a
system of fermions or bosons. In this appendix we adopt the single-particle picture
and assume a discrete energy spectrum, where single-particle states can possibly be
degenerate. The idea is to calculate at first the number of different and distinguish-
able ways to distribute the particles among the available energy levels; next, the most
probable partition is determined as that with the maximum number of different and

E-2
 Solid State Physics

distinguishable realisations; eventually, the equilibrium state is identified with such a

most probable distribution.
We will separately consider the case of fermions and bosons. Since we are mainly
interested in the physics of electrons and phonons, we will impose that the total
number of fermions is conserved, while the number of bosons can vary.

E.2 Fermi–Dirac statistics

Let us consider a system of fermions at temperature T. We name by ϵi the energy of
the ith single-particle state with degeneracy di. The number of different and
distinguishable ways of placing ni ⩽ di fermions on this level is
di (di − 1)(di − 2) ⋯ (di − ni + 1) di!
= . (E.8)
ni ! n i !(d i − n i )!

Accordingly, the product

di!
Pfermion = ∏ n i !(d i − n i )!
(E.9)
i

provides the total number of different and distinguishable ways of distributing

n1, n2 , n3, … fermions on the levels with energy E1, E2, E3, … and degeneracy
d1, d2, d3, …, respectively. The equilibrium distribution for the fermion system is
found by maximising Pfermion under the two constraints

N= ∑ ni ET = ∑ n i Ei , (E.10)
i i

corresponding to the conservation of the number of particles and system total

energy, respectively. The standard way to proceed is maximising the function
ln Pfermion instead of Pfermion , since we can take profit from the Stirling formula
ln x! = x ln x − x which is valid for x ≫ 1. The two conservation conditions are
compensated by two Lagrange multipliers. This leads to
di
n i (T ) = = nFD(Ei , T ), (E.11)
⎡
exp⎣(Ei − μc ) / kBT ] + 1

which is known as the Fermi–Dirac distribution law [6, 7]; μc is referred to as the
chemical potential of the fermion system and it is itself a function of the temperature
μc = μc (T ). It is easy to prove that at zero temperature we find ni = di for all states
with energy up to E = μc , while for all states with energy E > μc we have ni = 0. In
other words, at zero temperature all levels with energy below the chemical potential
are fully occupied, while those with energy higher than the chemical potential are
totally empty. The zero-temperature value of the chemical potential μc (T = 0) = EF
is known as the Fermi energy.

E-3
 Solid State Physics

E.3 Bose–Einstein statistics

Let us consider a system of bosons at temperature T. We name Ei the energy of the
ith single-particle state with degeneracy di. Since in this case there is no limit in the
number of particles that can be accommodated on a single level, the number of
different and distinguishable ways to arrange ni bosons on this level is
(ni + di − 1)!
. (E.12)
ni !(di − 1)!
Accordingly, the product
(ni + di − 1)!
Pboson = ∏ ni !(di − 1)!
, (E.13)
i

provides the total number of different and distinguishable ways of distributing

n1, n2 , n3, … bosons on the levels with energy E1, E2, E3, … and degeneracy
d1, d2, d3, …, respectively. The equilibrium distribution for the boson system is
found by maximising Pfermion under the only constraint

ET = ∑ n i Ei , (E.14)
i

since, by direct reference to phonons and photons, we assumed that the number of
particles is not conserved in this case1. We can proceed similarly to the case of
fermions2 obtaining
di
n i (T ) = = n BE(Ei , T ), (E.15)
exp [Ei / kBT ] − 1
which is known as the Bose–Einstein distribution law [6, 7].

References
[1] Sakurai J J and Napolitano J 2011 Modern Quantum Mechanics 2nd edn (Reading, MA:
Addison-Wesley)
[2] Miller D A B 2008 Quantum Mechanics for Scientists and Engineers (New York: Cambridge
University Press)
[3] Griffiths D J and Schroeter D F 2018 Introduction to Quantum Mechanics 3rd edn
(Cambridge: Cambridge University Press)
[4] Bransden B H and Joachain C J 2000 Quantum Mechanics (Upper Saddle River, NJ: Prentice-
Hall)

1
In our discussion we will treat only two kinds of boson particles, namely phonons and photons. In both cases
the elementary processes we will be interested in generate as well as annihilate particles: for instance, by
increasing/decreasing the crystal temperature we create or annihilate phonons; the same situations are found
when considering anharmonicity, that is phonon–phonon interactions; finally, the absorption/emission of
electromagnetic radiation by a crystalline solid is described in the quasi-classical picture in terms of photon
annihilation/generation, respectively.
2
In this case, however, just one single Lagrange multiplier is needed, providing the Boltzmann factor 1/kBT .

E-4
 Solid State Physics

[5] Eisberg R and Resnick R 1985 Quantum Physics of Atoms, Molecules, Solids, Nuclei, and
Particles 2nd edn (Hoboken, NJ: Wiley)
[6] Glazer M and Wark J 2001 Statistical Mechanics—A Survival Guide (Oxford: Oxford
University Press)
[7] Swendsen R H 2012 An Introduction to Statistical Mechanics and Thermodynamics (Oxford:
Oxford University Press)

E-5
 IOP Publishing

Solid State Physics

A primer
Luciano Colombo

Appendix F
Fermi–Dirac integrals and Sommerfeld
expansion

At finite temperature the free electron gas is described by the Fermi–Dirac

distribution law1
1
nFD(E , T ) = , (F.1)
1 + exp[(E − μc)/ kBT ]

providing the probability that the quantum state at energy E is occupied. In this
law μc is the chemical potential which is itself a function of temperature; in
particular, at T = 0 K we have μc = EF , where EF is known as Fermi energy; for
most metals EF ∼ 10 4 K (see table 7.2) and, therefore, in practice we always have
T ≪ TF = EF /kB.
When developing the quantum physics of the free electron gas, we often meet
integrals of the form
+∞
I (E ) = ∫0 f (E ) nFD(E , T ) dE , (F.2)

where f (E ) is a generic function of the energy E: for instance, in section 7.3.2 two
different forms of f (E ) are used to calculate the total number of electrons and the
energy of the gas, respectively. These integrals belong to the family of Fermi–Dirac
integrals and do not benefit from a general analytical expression. Their calculation
proceeds through an expansion2 [2, 3] which relies on the assumption that
μc ∼ EF ≫ kBT for any temperature of physical interest. This assumption will be
justified later by the results we will obtain.

1
In this appendix we assume spin-resolved energy levels and, therefore, no indication of the degeneracy degree
is reported (see appendix E for more details).
2
The asymptotic form of the Fermi–Dirac integrals was originally published in [1].

doi:10.1088/978-0-7503-2265-2ch17 F-1 ª IOP Publishing Ltd 2021

 Solid State Physics

The calculation starts by splitting the integral given in equation (F.2) in two parts
μc +∞
I (E ) = ∫0 f (E ) nFD(E , T ) dE + ∫μ c
f (E ) nFD(E , T ) dE . (F.3)

Next, the Fermi–Dirac distribution law is conveniently rewritten as

1 −1
nFD(E , T ) = = 1 − {1 + exp[ −(E − μc )/ kBT ]} , (F.4)
1 + exp[(E − μc)/ kBT ]

and the I (E ) is accordingly recast in the form of a sum of three terms

μc
I (E ) = ∫0 f (E ) dE
μc
− ∫0 f (E ) {1 + exp[ −(E − μc )/ kBT ]} dE
−1
(F.5)
+∞
+ ∫μ c
f (E ) {1 + exp[(E − μc )/ kBT ]} dE ,
−1

which can be separately evaluated by introducing the integration variables

ξ = −(E − μc )/kBT in the second integral and ζ = +(E − μc )/kBT in the third
integral so that
μc
I (E ) = ∫0 f (E ) dE
μc /k BT
− kBT ∫0 f (μc − kBTξ ) [1 + exp(ξ )]−1 dξ (F.6)
+∞
+ kBT ∫0 f (μc + kBTζ ) [1 + exp(ζ )]−1 dζ.

We now consider the upper limit of the second integral: by assuming that μc ∼ EF,
we obtain that μc /kBT ∼ EF /kBT ≫ 1 at any temperature T of physical interest.
Therefore, to a very good approximation we can extend the upper limit of such an
integral up to infinity. This procedure leads to rewriting equation (F.6) in the
following new form
μc
I (E ) = ∫0 f (E ) dE
+∞ (F.7)
+ kBT ∫0 ⎡⎣f (μ + kBTξ ) − f (μ − kBTξ )⎤⎦[1 + exp(ξ )]−1 dξ.
c c

The function f (μc + kBTξ ) is further expanded in powers3 of ξ

3
This is a subtle point: basically, we are assuming that this function is analytical and well-behaved nearby
E = μc . While this is indeed the case for a free electron gas, small corrections are likely expected in more
realistic situations where, for instance, the electronic density of states could be somewhere singular or show
rapid variations in this energy range.

F-2
 Solid State Physics

+∞
1 (n)
f ( μc + kBTξ ) = ∑ f ( μc ) (kBT )n ξ n, (F.8)
n=0
n!

where f (n) (μc ) indicates the nth order derivative of the f function calculated at μc .
This leads to4
μc
I (E ) = ∫0 f (E ) dE
+∞ +∞ (F.9)
1
+2 ∑ (2m + 1)!
f (2m+1) (μc ) (k BT )2(m+1) ∫0 ξ 2m+1 [1 + exp(ξ )]−1 dξ .
m =0

The integrals appearing in this equation can be calculated exactly [4], eventually
providing the general expression
μc
π 2 (1) 7π 4 (3)
I (E ) = ∫0 f (E ) dE +
6
f (μc ) (kBT )2 +
360
f (μc ) (kBT )4 + ⋯ , (F.10)

for the Fermi–Dirac integrals in the Sommerfeld expansion.

In order to accept this general result, we must verify the assumption μc ∼ EF : this
check is carried out in section 7.3.2.

References
[1] Sommerfeld A 1927 Z. Physik 47 1
[2] Ashcroft N W and Mermin N D 1976 Solid State Physics (London: Holt-Saunders)
[3] Grosso G and Pastori Parravicini G 2014 Solid State Physics 2nd edn (Oxford: Academic)
[4] Gradshteyn I S and Ryzhik I M 1980 table of Integrals Series, and Products (New York:
Academic)

4
When comparing equation (F.9) to equation (F.7), it must be understood that m = (n − 1)/2 since, by
replacing equation (F.8) in equation (F.7) we easily recognise that the second integral vanishes for any even
value of n.

F-3
 IOP Publishing

Solid State Physics

A primer
Luciano Colombo

Appendix G
The tight-binding theory

G.1 From atomic orbitals to Bloch sums

In the spirit of the single-particle picture described in section 1.4.1, the many-body
electron problem is reduced to the problem of one electron moving in the average
field Vcfp due to the other valence electrons and to the ions (clamped at their
equilibrium position). So as not to burden the formalism too much, we will omit any
explicit reference to the actual ionic configuration R in the electron wavefunction
and energy symbols. The reduced quantum one-electron Hamiltonian operator
reads as

ℏ2 2
Hˆ = − ∇ + Vˆcfp(r), (G.1)
2m e

whose eigensolution Φnk(r) is provided by the Schrödinger equation

Hˆ Φnk(r) = En(k)Φnk(r), (G.2)

where k is the electron wavevector, n is the band index, and En(k) is the one-electron
energy. Since the Hamiltonian Ĥ is invariant upon lattice translations, the one-
electron wavefunction must obey the Bloch theorem (see section 6.3)

Φnk(r + Rl) = ei k·RlΦnk(r), (G.3)

where Rl is a lattice vector. The tight-binding method consists in writing the

crystalline wavefunction as a linear combination of atomic orbitals (LCAO)
{φα lb(r) = φα(r − Rl − Rb)}, where consistently with the notation used elsewhere in
this Primer Rb will indicate the ion positions within the basis, while the label α will
stand for the full set of quantum numbers defining the atomic orbital. The LCAO
expansion must of course fulfil the Bloch theorem and, therefore, we preliminarily
introduce the Bloch sums

doi:10.1088/978-0-7503-2265-2ch18 G-1 ª IOP Publishing Ltd 2021

 Solid State Physics

1
φαBloch
bk (r) = ∑ ei k·R φα(r − Rl − Rb),
l
(G.4)
N l

where N is the number of unit cells contained in the region which the periodic Born–
von Karman boundary condition is applied to, so that
1
Φnk(r) = ∑ B˜nαbφαBloch
bk (r)
Nb αb
(G.5)
1
= ∑e i k·Rl
B˜nαbφα(r − Rl − Rb),
NNb α lb

where Nb is the number of atoms in the lattice basis. The B̃ coefficients are set to
Bnαlb(k) = ei k·Rl B˜nαb, (G.6)
for further convenience.
Unfortunately, it is impossible to straightforwardly implement the tight-binding
method so formulated because of the very large number of integrals that should be
computed, corresponding either to scalar products between two Bloch sums or to
matrix elements of the Hamiltonian. This is basically due to the fact that the {φα lb(r)}
basis set is non orthogonal, since atomic wavefunctions centred on different lattice
positions in principle are not orthogonal.
We can remove the non-orthogonality problem by introducing a new set of
Löwdin orthogonalized orbitals. To this aim, let us preliminarily define the overlap
integral between two atomic orbitals as
Sα′ l ′ b′,αlb = 〈φα′ l ′ b′(r)∣φαlb(r)〉 − δα′ αδ(l ′ b′)(lb), (G.7)
and recast the Schrödinger problem in matrix notation
  n (k) = En(k) ( +  )  n (k), (G.8)
where  is the overlap matrix whose entries are defined in equation (G.7), while  is
the Hamiltonian matrix whose elements are defined as
Hα′ l ′ b′,αlb = 〈φα′ l ′ b′(r)∣Hˆ ∣φαlb(r)〉, (G.9)

and the Dirac notation has been used. Obviously, the entries of the column vectors
 n (k) are given in equation (G.6), while  is the unit matrix.
By now introducing a new set of  n(k) = {Cnα lb(k)} coefficients
 n (k) = ( +  )−1/2  n(k), (G.10)
equation (G.8) is easily recast in the following form
 L  n(k) = En(k) n(k), (G.11)
where the label L reminds that a Löwdin matrix transformation
 L = ( +  )−1/2  ( +  )−1/2 , (G.12)

G-2
 Solid State Physics

has been executed. The crystalline wavefunction is now accordingly written as

1
Φnk(r) = ∑ Bnαlb(k)φα(r − Rl − Rb)
NNb α lb
1
= ∑ ∑ ( +  )−αlb,1/2α′ l′ b′ Cnα′ l′ b′(k) φα(r − Rl − Rb) (G.13)
NNb α lb α ′ l ′ b ′
1
= ∑ Cnα′ l ′ b′(k)ψα′(r − Rl ′ − Rb′),
NNb α ′l ′b ′

where the new Löwdin orbitals have been introduced

ψα′ l ′ b′(r) = ψα′(r − Rl ′ − Rb′) = ∑( +  )−αlb,1/2α′ l′ b′ φα(r − Rl − Rb). (G.14)
α lb

The Löwdin orbitals are mutually orthogonal (the proof is trivial) and have the same
symmetry properties (that is the same s- or p- or d-character) of the original non-
orthogonal atomic wavefunctions which they are derived from.

G.2 The two-centre approximation

The formal solution of the eigenvalue problem leads to the following secular
problem

∑⎡⎣Hα′ l′ b′,αlb − En(k)δα′ αδ(l′ b′)(lb)⎤⎦Cnαlb(k) = 0, (G.15)

α lb

where the matrix elements of the Hamiltonian are calculated by using the Löwdin
orbitals as
Hα′ l ′ b′,αlb = 〈ψα′ l ′ b′(r)∣Hˆ ∣ψαlb(r)〉. (G.16)

In principle, provided that the Löwdin orbitals are known as a result of an atomic
physics calculation, these terms can be computed exactly, thus providing a
parameter-free theory for the calculation of the band structure of a solid.
However, this approach appears of little sense1 since it is rather cumbersome.
Some further handling of the theory is therefore worthy at this stage, aimed at
transforming the present formal theory into a practical tool. Two strategies are in
fact possible: either we can choose suitable approximations which allow us to focus
just on the leading terms and to disregard other computationally-intensive (but less
important) contributions; or we can introduce empirical features into the present
theory so to avoid expensive calculations.
By assuming that the crystal field potential Vcfp(r) is the sum of spherical potentials
Vlb(r) located at each (Rl + Rb) lattice site, we can rewrite equation (G.1) as

1
The evaluation of the merits and limitations of the LCAO approach represents a classical topic of quantum
chemistry [1–3].

G-3
 Solid State Physics

ℏ2 2
Hˆ = − ∇ + Vˆlb(r) + ∑ Vˆl ′ b′(r) + ∑ Vˆl ″ b″(r), (G.17)
2m e l ′ b ′≠ lb l ″ b ″≠ l ′ b ′ ,lb

and accordingly separate three different contributions

ℏ2 2
Hα′ l ′ b′,αlb = 〈ψα′ l ′ b′(r)∣− ∇ + Vˆlb(r)∣ψαlb(r)〉
2m e
+ 〈ψα′ l ′ b′(r)∣ ∑ Vˆl ′ b′(r)∣ψαlb(r)〉 (G.18)
l ′ b ′≠ lb

+ 〈ψα′ l ′ b′(r)∣ ∑ Vˆl ″ b″(r)∣ψαlb(r)〉,

l ″ b ″≠ l ′ b ′ ,lb

which are separately handled. The first intra-atomic term is easily computed thanks
to the orthogonality of the Löwdin orbitals
ℏ2 2
〈ψα′ l ′ b′(r)∣− ∇ + Vˆlb(r)∣ψαlb(r)〉 = Eαδα′ αδ(l ′ b′)(lb), (G.19)
2me
while the remaining two contributions are called, respectively, two-centre and three-
centre energy integrals for obvious reasons.
It is quite reasonable to assume that three-center integrals are comparatively
smaller than two-centre ones, because of the localized character of the Löwdin
orbitals. A two-center approximation is accordingly introduced, assuming that three-
centre energy integrals can be simply disregarded. This drastic guess was originally
introduced by Slater and Koster in 1954 [4] to formally treat the Hamiltonian matrix
elements as if we had a sort of ‘diatomic molecule’ system. By introducing the
distance vector
t = Rl ′ b′ − Rlb, (G.20)
a sort of ‘molecular axis’ between any pair of atoms is naturally set with two
spherical coordinates systems centred on the two atomic positions connected by such
a vector and the z-axes parallel to t , as indicated in figure G.1. In this double frame
of reference, the angular parts of the orbitals ψα l ′ b ′(r) and ψα lb(r) have the form of a

Figure G.1. Definition of the two spherical coordinates systems for the equivalent ‘diatomic molecule’ formed
by atoms at Rl ′ b ′ (left) and Rlb (right), respectively. Any point P is defined by different polar angles θ and θ′,
but by a single azimuth angle since φ = φ′ by construction.

G-4
 Solid State Physics

spherical harmonic function, both depending on the same azimuth angle φ.

Therefore, it turns out that the angular functions depending on such an azimuth
angle combine so that the matrix element of the Hamiltonian becomes proportional
2π
to ∫ ei (m−m ′)φdφ, where m and m′ are the magnetic quantum numbers of the two
0
orbitals: this implies that it is zero unless m = m′ and, by adopting the same notation
as in atomic physics, the σ , π , d , … symbols are used to label the non-vanishing
matrix elements corresponding to m = m′ = 0, 1, 2, …, respectively. The radial and
θ-dependent angular parts of the same integrals are commonly indicated as ss, or sp,
or pp, … according to the actual symmetry of the two involved orbitals. The resulting
complete labelling for the two-centre hopping integrals is (ssσ ), or (spσ ), or (ppσ ),
and so on. The practical application of this procedure is schematically illustrated in
figure G.2 in the case of s- and p-like orbitals.
In order to pass from the Slater–Koster ‘diatomic molecule’ scheme to a full
crystalline picture, any pair of spherical harmonic functions (defined with respect the
t axes) should be rotated so as to match the same Cartesian frame of reference used
to span the crystal lattice. This is done by means of the direction cosines (lˆ , mˆ , nˆ )
defining each t vector with respect to such a frame. The procedure has been worked
out once and for all for any possible pair of orbitals [4], and its result is summarised
in table G.1. It should be noted that all hopping integrals do depend on the
interatomic distance between the two selected atomic centres: therefore, one should
calculate different values for such quantities between first-, second-, third-, … nearest
neighbours.

G.3 Calculating the hopping energy integrals

Despite the two-centre approximation, the computational effort required to com-
pute the Hamiltonian matrix elements still remains a formidable task: in order to set
up a practical tool, we need to further reduce the complexity of the tight-binding
method. A key step in this direction is to consider the two-centre integrals as
disposable constants to fit on the basis of some simplifying approximations.
First of all, only close-neighbours interactions are typically taken into account.
This is tantamount to stating that, among all two-centre integrals, those

Figure G.2. Overlaps between s- and p-like orbitals within the Slater–Koster ‘diatomic molecule’ scheme. The
bold arrow indicates the distance vector t = Rl ′ b ′ − Rlb between two atoms. The standard labelling for the
two-centre hoppings is shown.

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 Solid State Physics

Table G.1. Two-centre energy integrals. The (lˆ , mˆ , nˆ ) direction cosines define each t vector with respect to
unique crystalline frame of reference. The Löwdin orbitals are symbolized according to the standard notation
valid for usual atomic orbitals.

Orbital pair Orbital symmetry Energy integral

s–s s, s (ssσ )
s–p s, px lˆ (spσ )

lˆ (ppσ ) + (1 − lˆ )(ppπ )
2 2
p–p px , px
px , py ˆ ˆ (ppπ )
ˆlmˆ (ppσ ) − lm
px , pz ln ˆ ˆ (ppπ )
ˆ ˆ (ppσ ) − ln

s–d s, dxy ˆ ˆ (sdσ )

3 lm
3 (lˆ − mˆ 2)(sdσ )
1 2
s , d x 2− y 2 2
[nˆ 2 − 2 (lˆ + mˆ 2)](sdσ )
1 2
s, d3z 2−r 2
p–d px , dxy 3 lˆ mˆ (pdσ ) + m(1 − 2lˆ )(pdπ )
2 2

px , dyz 3 lmn ˆ ˆ ˆ (pdπ )

ˆ ˆ ˆ (pdσ ) − 2lmn
3 lˆ nˆ (pdσ ) + nˆ (1 − 2lˆ )(pdπ )
2 2
px , dzx
3 lˆ (lˆ − mˆ 2)(pdσ ) + lˆ (1 − lˆ + mˆ 2)(pdπ )
1 2 2
px , d x 2−y2 2
3 mˆ (lˆ − mˆ 2)(pdσ ) + mˆ (1 + lˆ − mˆ 2)(pdπ )
1 2 2
py , d x 2−y2 2
3 nˆ (lˆ − mˆ 2)(pdσ ) + nˆ (lˆ − mˆ 2)(pdπ )
1 2 2
pz , d x 2−y2 2
lˆ [nˆ 2 − 2 (lˆ + mˆ 2)](pdσ ) − 3 ln
ˆ ˆ 2(pdπ )
1 2
px , d3z 2−r 2
mˆ [nˆ 2 − 2 (lˆ + mˆ 2)](pdσ ) − 3 mn
1 2
py , d3z 2−r 2 ˆ ˆ 2(pdπ )
nˆ [nˆ 2 − 2 (lˆ + mˆ 2)](pdσ ) − 3 nˆ (lˆ + mˆ 2)(pdπ )
1 2 2
pz , d3z 2−r 2
d–d dxy, dxy 3lˆ mˆ 2(ddσ ) + (lˆ + mˆ 2 − 4lˆ mˆ 2)(ddπ ) + (nˆ 2 + lˆ mˆ 2)(ddδ )
2 2 2 2

dxy, dyz ˆ 2 ˆ 2 ˆ
3lmˆ nˆ (ddσ ) + lnˆ (1 − 4mˆ )(ddπ ) + lnˆ (mˆ − 1)(ddδ )
2

3lˆ mn ˆ ˆ (1 − 4lˆ 2 )(ddπ ) + mn

ˆ ˆ (lˆ 2 − 1)(ddδ )
2
dxy, dzx ˆ ˆ (ddσ ) + mn
3ˆ
lmˆ (lˆ ˆ ˆ (mˆ 2 − lˆ 2 )(ddπ ) + 1 lm
ˆ ˆ (lˆ 2 − mˆ 2)(ddδ )
2
dxy, d x 2−y2 2
− mˆ 2)(ddσ ) + 2lm 2
dyz , d x 2−y2
3
2
ˆ ˆ (lˆ 2
mn ˆ ˆ [1 + 2(lˆ 2 − mˆ 2)](ddπ )
− mˆ 2)(ddσ ) − mn
ˆ ˆ [1 + 21 (lˆ 2 − mˆ 2)](ddδ )
+ mn
3 ˆ ˆ2
dzx, d x 2−y2 2
nl ˆ ˆ [1 − 2(lˆ 2 − mˆ 2)](ddπ )
ˆ (l − mˆ 2)(ddσ ) + nl
ˆ ˆ [1 − 1 (lˆ 2 − mˆ 2)](ddδ )
− nl 2
dxy, d3z 2−r 2 ˆ ˆ [nˆ 2 − 1 (lˆ 2 + mˆ 2)](ddσ ) − 2 3 lmn
3 lm ˆ ˆ ˆ 2(ddπ )
2
1
+ 2 3 lmˆ ˆ (1 + nˆ 2)](ddδ )
dyz , d3z 2−r 2 ˆ ˆ [nˆ 2 − 2 (lˆ + mˆ 2)](ddσ ) + 3 mn
3 mn
1 2
ˆ ˆ (lˆ 2 + mˆ 2 − nˆ 2)(ddπ )
1
− 3 mn
2
ˆ ˆ (lˆ 2 + mˆ 2)(ddδ )
dzx, d3z 2−r 2 ˆ ˆ [nˆ 2 − 1 (lˆ 2 + mˆ 2)](ddσ ) + 3 ln
3 ln ˆ ˆ (lˆ 2 + mˆ 2 − nˆ 2)(ddπ )
2
1
− 3 ln ˆ ˆ (lˆ + mˆ 2)(ddδ )
2
2
3 ˆ2
− mˆ 2)2(ddσ ) + [lˆ + mˆ 2 − (lˆ − mˆ 2)2](ddπ )
2 2
d x 2− y 2 , d x 2− y 2 4
(l
+ [nˆ 2 + 4 (lˆ − mˆ 2)2)](ddδ ))
1 2

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 Solid State Physics

1 2 1 2 2
d x 2−y2, d3z 2−r 2 2
3 (lˆ − mˆ 2)[nˆ 2 − 2 (lˆ + mˆ 2)](ddσ ) + 3 nˆ 2(mˆ 2 − lˆ )(ddπ )
1 2
+ 4 3 (1 + nˆ 2)(lˆ − mˆ 2)(ddδ )
1 2 2
d3z 2−r 2, d3z 2−r 2 [nˆ 2 − (lˆ + mˆ 2)]2 (ddσ ) + 3nˆ 2(lˆ + mˆ 2)(ddπ )
2
3 2
+ 4 (lˆ + mˆ 2)2(ddδ )

corresponding to a distance between atom (Rl ′ − Rb ′) and atom (Rl − Rb) larger
than a given cutoff are set to zero. This approximation is obviously justified by the
localised character of the basis set functions (a true characteristic for both atomic
orbitals and Löwdin orbitals) and greatly reduces the number of parameters to fit: in
many applications, just first-next-neighbours interactions are considered. Next, a
minimal basis set is used for the LCAO expansion. This means that we only consider
those orbitals whose energy is close to the energy of the electronic states of the
crystalline system we are interested in. In many applications, this procedure basically
amounts to considering only those orbitals with energy in the range containing the
topmost valence states and lowest conduction states. Such an approximation reduces
both the number of parameters to fit and the order of the secular problem to be
numerically solved.
These approximations obviously reflect in some important limitation of the tight-
binding theory, namely (i) we can explore just a part of the energy spectrum of the
crystal and (ii) we have no knowledge about the one-electron wavefunction. In fact,
by introducing empirical parameters to replace the Hamiltonian matrix elements, we
lose any chance to actually perform the LCAO expansion: accordingly, we cannot
compute any physical property directly depending on Φnk(r). This is, for example,
the case of the electron density which obviously is quite an important quantity in
solid state physics. Above all, however, the method turns out to be semi-empirical.
While the reassuring prefix ‘semi’ stands for the fact that the resulting model is firmly
rooted in a rigorous quantum treatment of crystalline electronic states, the
‘empirical’ character implies that its accuracy, reliability and transferability need
to be proved from case to case. In any case, the tight-binding theory still has a great
pedagogical value and it is really useful for quick-and-easy band-structure
calculations.
The fitted two-centre integrals are determined on a suitable database of physical
information obtained either from experiments or by more sophisticated first-principles
calculations. Slater–Koster parameters are nowadays available for a large variety of
elemental as well as compound materials, as for instance summarised in [5, 6].

G.4 Tight binding at work

As a showcase application of the tight-binding theory, let us consider the specific
case of silicon (a crystal with diamond structure and a two-atom basis) and let us
proceed under the two-centre and first-next-neighbours approximations. If we just

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 Solid State Physics

Table G.2. Two-centre integrals according to different sp3 parametrizations for silicon (units of eV).

Ref. [1] Ref. [2] Ref. [3] Ref. [4]

es −5.250 −6.535 −5.200 −4.478

ep +1.200 +1.760 +1.200 +2.552
(ssσ ) −2.038 −1.820 −1.940 −1.913
(spσ ) +1.745 +1.960 +1.750 +2.594
(ppσ ) +2.750 +3.060 +3.050 +4.464
(ppπ ) −1.075 −0.870 −1.080 −1.275

Ref. [1]: Kwon I, Biswas R, Wang C Z, Ho K M, and Soukoulis C M 1994 Phys. Rev. B 49 7242.
Ref. [2]: Goodwin L, Skinner A J, and Pettifor D G 1989 Europhys. Lett. 9 701.
Ref. [3]: Chadi D J and Cohen M L 1975 Phys. Status Solidi B 68 405.
Ref. [4]: This is the set of ‘universal tight-binding parameters’ discussed in [5].

focus on the topmost valence states, we can adopt a minimal sp3 basis-set for each
atom of the basis. The corresponding energy integrals have been fitted by many
authors and some parametrizations are reported in table G.2 where we have
introduced the notation es and ep for the on-site integrals corresponding to ss and
pp orbital pairs, respectively.
The two-centre energy integrals of the crystalline diamond structure are easily
computed using tables G.1 and G.2 and by taking into account that the needed
cosine directors are defined for the four atomic pairs formed by the first atom in
position (0, 0, 0)a 0 and the second atom in positions (1/4, 1/4, 1/4)a 0 ,
(1/4, −1/4, −1/4)a 0, ( −1/4, 1/4, −1/4)a 0 , and ( −1/4, −1/4, 1/4)a 0, respectively.
The lattice constant of silicon has been named a0. The Bloch phase factors
exp(ik · ti ) with i = 1, … , 4 are calculated with position vectors corresponding to
the four first-next-neighbours of the pivot atom placed at the origin. In summary, the
Hamiltonian matrix to be diagonalised for each k value within the 1BZ in order to
get the valence band dispersions EVB(k) is

⎡ es 0 0 0 Essg0 Espg1 Espg2 Espg3 ⎤

⎢ ⎥
⎢ 0 ep 0 0 − Espg1 E px ,pxg0 E px ,pyg3 E px ,pyg2 ⎥
⎢ 0 0 ep 0 − Espg2 E px ,pyg3 E px ,pxg0 E px ,pyg1 ⎥
⎢ ⎥
⎢ 0 0 0 ep − Espg3 E px ,pyg2 E px ,pyg1 E px ,pxg0 ⎥
ˆ ⎢ ⎥
H (k) = ⎢ E g * − Espg1* − Espg2* − Espg3* es 0 0 0 ⎥
(G.21)
ss 0
⎢ ⎥
⎢ Espg1* E px ,pxg0* E px ,pyg3* E px ,pyg2* 0 ep 0 0 ⎥
⎢ ⎥
⎢ Espg2* E px ,pyg3* E px ,pxg0* E px ,pyg1* 0 0 ep 0 ⎥
⎢ ⎥
⎢⎣ Espg3* E px ,pyg2* E px ,pyg1* E px ,pxg0* 0 0 0 ep ⎥⎦

where the e’s and E’s terms are, respectively, the on-site and two-centre energy
integrals reported in the right column of table G.1. The k -dependence enters through
the geometrical g-factors given by

G-8
 Solid State Physics

g0(k) = exp(i k · t1) + exp(i k · t2) + exp(i k · t3) + exp(i k · t 4)

g1(k) = exp(i k · t1) + exp(i k · t2) − exp(i k · t3) − exp(i k · t 4)
(G.22)
g2(k) = exp(i k · t1) − exp(i k · t2) + exp(i k · t3) − exp(i k · t 4)
g3(k) = exp(i k · t1) − exp(i k · t2) − exp(i k · t3) + exp(i k · t 4)

In figure G.3 the valence bands for crystalline silicon are reported, as obtained
from the four different sets of empirical tight-binding parameters reported in
table G.2, where the on-site terms have been shifted so as to align at zero energy
the top valence state at Γ. This is a standard choice which is allowed since the total
width of the valence band is rather governed by the difference ∣es − ep∣ than by their
absolute values.
While the basic features are equally provided by the four sets of parameters (like
for instance, their general topology and degeneracy), some details vary from case to
case. Nevertheless, it is important to stress that the semi-empirical tight-binding
method has not the ambition to provide an extreme degree of accuracy. Actually, it is
not realistic to pretend so, in view of the important approximations we adopted.
Nevertheless, it provides in any case a more than decent description of valence states.

Figure G.3. The valence band dispersions of crystalline silicon calculated with the tight-binding parameters
given in table G.2. The blue dots show the position of the top of the valence band (VB): the on-site terms have
been shifted so as to align it at zero energy.

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 Solid State Physics

Finally, it is important to remark that one should be very heedful in comparing

the values appearing in table G.2: while there is a surely meaningful trend in the
variation of the different two-centre integrals upon orbital symmetry (which is
common to all sets of parameters), it is hard to attach any clear physical connotation
to such numbers.

References
[1] Szabo A and Ostlund N S 1996 Modern Quantum Chemistry (New York: Dover)
[2] Jensen F 2006 Introduction to Computational Chemistry 2nd edn (Hoboken, NJ: Wiley)
[3] Simons J 2003 An Introduction to Theoretical Chemistry (Cambridge: Cambridge University
Press)
[4] Slater J C and Koster G F 1954 Simplified LCAO method for the periodic potential problem
Phys. Rev. 94 1498
[5] Harrison W A 1980 Electronic Structure and the Properties of Solids (New York: Dover)
[6] Papacostantopoulos D A 2015 Handbook of the Band Structure of Elemental Solids 2nd edn
(New York: Springer)

G-10
 IOP Publishing

Solid State Physics

A primer
Luciano Colombo

Appendix H
Stress and temperature effects on the energy gap

The intrinsic energy gap of an insulating material is affected by the variation of the
interatomic distances. This is especially clear in the tight-binding picture: the
elements of the one-electron Hamiltonian matrix and of the overlap matrices (see
equations (8.26) and (8.27), respectively) crucially depend on the relative distance
between all those neighbouring atoms that are included in the model. In turn,
interatomic distances are changed by deforming the lattice by means of some applied
stress or by increasing its temperature. Their net effect on Egap must be differently
treated.
Let us first consider the effect of a mechanical action which, for simplicity, we will
assume to be a simple hydrostatic stress. If V0 is the equilibrium volume and ΔV is
its stress-induced variation, then the quantity ϵ = ΔV /V0 quantitatively represents
the hydrostatic deformation, as extensively discussed in chapter 5. Upon either
compression or extension, the intrinsic energy gap value linearly changes with the
deformation ϵ since it is experimentally found [1] that
bottom bottom
E CB (ϵ ) = E CB (0) + a CB ϵ
top top
(H.1)
E VB (ϵ ) = E VB (0) + a VB ϵ ,

and therefore
Egap(ϵ ) = Egap(0) + (a CB − a VB) ϵ , (H.2)

where ECBbottom
(0) and E VB
top
(0) are the zero stress (that is, equilibrium) positions of the
minimum and maximum of the conduction and valence band, respectively, and
Egap(0) is the corresponding intrinsic gap.
The two coefficients a CB < 0 and aVB > 0 are referred to as the deformation
potentials for the two bands. In the simplified discussion here outlined the
deformation potentials are scalar quantities, while in fact they are tensors: a more
in-depth investigation can be found elsewhere [2–4]. A simple dimensional analysis
of the above equations shows that the deformation potentials have the dimensions of
an energy, while it is in general found that ∣a CB∣ ≠ ∣aVB∣. By adopting the parabolic

doi:10.1088/978-0-7503-2265-2ch19 H-1 ª IOP Publishing Ltd 2021

 Solid State Physics

Figure H.1. Hydrostatic stress effects on the energy gap of a direct gap material. It is assumed a CB < 0 and
a VB > 0 with ∣a CB∣ ≠ ∣a VB∣. Compression and expansion, respectively, correspond to a deformation ϵ < 0 and
ϵ > 0 , with ϵ = ΔV /V0 (where V0 is the undeformed crystal volume).

band approximation for a model direct gap material, this body of information is
conceptualised in figure H.1.
Finally, as for the temperature-dependence of the energy gap, we understand it is
ruled over by thermal expansion phenomena (see section 4.2.1). For most semi-
conductor materials the following phenomenological law [1] is found valid

(0) ξ1T 2
Egap(T ) = Egap − , (H.3)
ξ2 + T
where Egap
(0)
is the intrinsic gap measured at T = 0 K, while ξ1 and ξ2 are materials
specific empirical constants: typically, ξ1 = O(10−4) eV K−1 and ξ2 = O(102) K.

References
[1] Grundmann M 2010 The Physics of Semiconductors (Heidelberg: Springer)
[2] Cardona M and Yu P Y 2010 Fundamentals of Semiconductors (Heidelberg: Springer)
[3] Seeger K 1989 Semiconductor Physics (Heidelberg: Springer)
[4] Balkanski M and Wallis R F 1989 Semiconductor Physics and Applications (Oxford: Oxford
University Press)

H-2
 IOP Publishing

Solid State Physics

A primer
Luciano Colombo

Appendix I
Functionals and functional derivatives

In mathematics a function f = f (x ) is a rule where an input number x returns

another number f (x ). On the other hand, a functional is a rule where an input
function f (x ) returns a number F [ f (x )].
The functional derivative [1] of F [ f (x )] with respect to f (x ) is denoted by the
symbol δF /δf and it is defined through the equation

∫ δFf[ (fx()x)] η(x)dx = ddϵ F [ f (x) + ϵη(x)]ϵ=0 (I.1)

where η(x ) is an arbitrary function and ϵ a real number. Let us then consider some
functional derivative of physical interest; for direct use in chapter 10 we adopt the
atomic units. In any of the following worked examples let ρ(r) be the function
describing the electron density at position r .
The functional associated with the Coulomb interaction between an electron
charge distribution ρ(r) and a point-like nucleus sitting in position R α is written as
Qαρ(r)
VCoulomb[ ρ ] = ∫ ∣r − R α ∣
d r, (I.2)

where Qα is the nuclear charge. By applying the definition given in equation (I.1) we
calculate

δVCoulomb d ⎡ Qα[ ρ(r) + ϵη(r)] ⎤

∫ δρ(r)
η (r)d r = ⎢
dϵ ⎣
∫ ∣r − R α ∣
d r⎥
⎦ϵ=0
(I.3)
Qαη(r)
= ∫ ∣r − R α ∣
d r,

which leads to
δVCoulomb Qα
= , (I.4)
δρ(r) ∣r − R α ∣

doi:10.1088/978-0-7503-2265-2ch20 I-1 ª IOP Publishing Ltd 2021

 Solid State Physics

which is easily interpreted: the functional derivative of the Coulomb functional is the
ordinary electrostatic potential generated by the nucleus.
The functional associated with the Hartree energy of a system of interacting
electrons is written as
1 ρ(r)ρ(r′)
VHartree[ ρ ] =
2
∫∫ ∣r − r′∣
d rd r′ . (I.5)

By applying the definition given in equation (I.1) we calculate

δVHartree 1 d ⎡ [ ρ(r) + ϵη(r)] [ ρ(r′) + ϵη(r′)] ⎤

∫ δρ(r)
η(r)d r = ⎢
2 dϵ ⎣
∫ ∫ ∣r − r′∣
d rd r′⎥
⎦ϵ=0
(I.6)
ρ(r′)η(r)
= ∫ ∫ ∣r − r′∣
d rd r′ ,

which leads to
δVHartree ρ(r′)
δρ(r)
= ∫ ∣r − r′∣
d r′ , (I.7)

which, once again, shows that the functional derivative of the Hartree functional is
the electrostatic potential generated by the charge distribution.
Finally, let us consider the general power-law functional

F [ ρ] = ∫ ρ ξ (r)d r (I.8)

where ξ is a real number. By applying the definition given in equation (I.1) we

calculate
δF d ⎡ ⎤
∫ ρ(r)
η(r)d r = ⎢⎣
dϵ
∫ [ ρ(r) + ϵη(r)]ξ d r⎦⎥
ϵ=0
(I.9)
=ξ ∫ ρ ξ −1
(r)η (r)d r ,

which leads to
δF
= ξρ ξ−1(r). (I.10)
δρ(r)
This expression is used in chapter 10 to calculate the derivative of the exchange
energy functional in the local density approximation.

References
[1] Parr R G and Yang W 1989 Density Functional Theory of Atoms and Molecules (Oxford:
Oxford Science Publications)

I-2