Advances in Physics

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Raman spectroscopy of graphene and carbon

nanotubes

R. Saito , M. Hofmann , G. Dresselhaus , A. Jorio & M. S. Dresselhaus

To cite this article: R. Saito , M. Hofmann , G. Dresselhaus , A. Jorio & M. S. Dresselhaus (2011)
Raman spectroscopy of graphene and carbon nanotubes, Advances in Physics, 60:3, 413-550,
DOI: 10.1080/00018732.2011.582251

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Advances in Physics
Vol. 60, No. 3, May–June 2011, 413–550

REVIEW ARTICLE

Raman spectroscopy of graphene and carbon nanotubes

R. Saitoa , M. Hofmannb , G. Dresselhausc , A. Joriod , and M.S. Dresselhausb,e *
a Department of Physics, Tohoku University, Sendai 980-8578, Japan; b Department of Electrical
Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA 02139-4307,
USA; c Francis Bitter Magnet Laboratory, Massachusetts Institute of Technology, Cambridge, MA
02139-4307, USA; d Departamento de Física, Universidade Federal de Minas Gerais, Belo Horizonte -
MG, 30123-970, Brazil; e Department of Physics, Massachusetts Institute of Technology, Cambridge, MA
02139-4307, USA
(Received 28 June 2010; final version received 5 April 2011 )

This paper reviews progress that has been made in the use of Raman spectroscopy to study
graphene and carbon nanotubes. These are two nanostructured forms of sp2 carbon materi-
als that are of major current interest. These nanostructured materials have attracted particular
attention because of their simplicity, small physical size and the exciting new science they have
introduced. This review focuses on each of these materials systems individually and compara-
tively as prototype examples of nanostructured materials. In particular, this paper discusses the
power of Raman spectroscopy as a probe and a characterization tool for sp2 carbon materials,
with particular emphasis given to the field of photophysics. Some coverage is also given to the
close relatives of these sp2 carbon materials, namely graphite, a three-dimensional (3D) mate-
rial based on the AB stacking of individual graphene layers, and carbon nanoribbons, which
are one-dimensional (1D) planar structures, where the width of the ribbon is on the nanometer
length scale. Carbon nanoribbons differ from carbon nanotubes is that nanoribbons have edges,
whereas nanotubes have terminations only at their two ends.

PACS: 78.30.-j Raman Spectroscopy; 61.48.Gh Structure of graphene; 61.48.De Structure of

nanotubes; 71.38.-k Polarons and electron-phonon interactions.
Keywords: resonance Raman spectroscopy; carbon nanotubes; graphene nanoribbons;
electron–phonon interaction

Contents PAGE

1. Basic concepts 417

1.1. Overview of sp2 carbon materials 417
1.1.1. Graphite 417
1.1.2. Fullerenes 418
1.1.3. Carbon nanotubes 419
1.1.4. Graphene 420
1.1.5. Nanoribbons 421
1.1.6. Raman spectroscopy for sp2 carbon materials 421
1.2. Various photo-physical processes 423
1.2.1. Valence-to-conduction band transitions and excitons 423
1.2.2. Optical transitions including impurity levels 424
1.2.3. Optical phonons 425
1.2.4. Free carrier processes 425

*Corresponding author. Email: millie@mgm.mit.edu

ISSN 0001-8732 print/ISSN 1460-6976 online
© 2011 Taylor & Francis
DOI: 10.1080/00018732.2011.582251
http://www.informaworld.com
414 R. Saito et al.

1.3. Interactions for optical processes 425

1.3.1. Photon-induced electron–phonon interaction 425
1.3.2. Electron–phonon interaction in infrared absorption 426
1.3.3. Dipole–phonon interaction in non-radiative decay 426
1.3.4. Dipole–photon interaction in photoluminescence 427
1.3.5. Rayleigh scattering 427
1.3.6. Brillouin scattering 427
1.3.7. Raman scattering 427
1.3.8. First- and higher-order Raman processes 428
1.4. Characteristics of the Raman effect 428
1.4.1. Raman spectra and the Raman excitation profile 429
1.4.2. Incident and scattered resonance conditions 429
1.4.3. Stokes and anti-Stokes Raman processes 429
1.4.4. The Raman spectra and spectral width: Lorentzian lineshape 430
1.4.5. The Breit–Wigner–Fano (BWF) lineshape 431
1.4.6. The effect of defects on spectral broadening 432
1.4.7. The Raman resonance window 432
1.5. Raman measurements of low-dimensional materials 433
1.5.1. Cutting lines and van Hove singularities of the density of states 434
1.5.2. Dimensionality and the resonance Raman effect 434
1.5.3. Coherence time and length in Raman processes 435
2. Experimental progress of Raman spectroscopy and related optics 435
2.1. Electrons and phonons in graphene 436
2.1.1. The hexagonal crystal structure of graphene 436
2.1.2. The electronic structure and optical transitions 436
2.1.3. Phonons and the el–ph interaction in graphene 437
2.2. Electrons and phonons in 1D carbon nanostructures 438
2.2.1. 1D carbon structures 438
2.2.2. π bands and phonons in carbon nanotubes and nanoribbons 438
2.2.3. Optical transitions 438
2.2.4. Type I and II semiconductor nanotubes 438
2.2.5. 1D exciton, exciton–photon and exciton–phonon interactions 439
2.3. The optical measurement techniques 440
2.3.1. Light absorption 440
2.3.2. Resonance Rayleigh scattering 441
2.3.3. Photoluminescence excitation spectra 442
2.3.4. Electro-luminescence 443
2.3.5. Infrared absorption spectroscopy 444
2.3.6. Coherent phonon spectroscopy 444
2.4. Raman spectroscopy of sp2 carbons 445
2.4.1. Historical background 445
2.4.2. Raman spectra of graphite and graphene: G- and G -bands 446
2.4.3. First-order RBM, G+ and G− Raman spectra of SWNTs 446
2.4.4. The defect-induced Raman spectral features: D- and D -bands 447
2.5. Laser Raman scattering measurements 447
2.5.1. The Raman setup 447
2.5.2. Polarized and micro Raman measurements 447
2.5.3. Confocal Raman spectroscopy and Raman imaging 448
2.5.4. Characterization of the sample edges and the imaging of defects 448
2.5.5. Resonance Raman spectroscopy 449
2.5.6. The Raman excitation profile 449
 Advances in Physics 415

2.5.7. The Kataura plot 449

2.6. Other measurement techniques related to Raman spectroscopy 450
2.6.1. Surface-enhanced Raman spectroscopy 450
2.6.2. Surface and interference enhanced Raman spectroscopy 451
2.6.3. Near-field enhanced Raman spectroscopy 451
2.6.4. Tip enhanced Raman spectroscopy 451
2.6.5. Simultaneous atomic force microscopy, near-field Raman and PL imaging 452
2.6.6. Coherent anti-Stokes Raman spectroscopy (CARS) 452
2.7. Kohn anomaly in graphene and carbon nanotubes 453
2.7.1. Kohn-anomaly of the G-band of graphene 453
2.7.2. Kohn anomaly of bi-layer graphene 453
2.7.3. Kohn anomalies of SWNTs 454
2.8. Classification of Raman processes 455
2.8.1. First-order Raman process 455
2.8.2. Two-phonon second-order Raman process 455
2.8.3. One-phonon and one-elastic second-order Raman process 456
2.8.4. Double resonance Raman spectra 456
2.8.5. Dispersive behavior of the phonon energy in DR Raman processes 457
2.8.6. The inter-valley double resonance Raman scattering processes 458
2.8.7. Forward and backward scattering 458
2.8.8. DR q circles in 2D graphene 460
2.8.9. Dispersive behavior of the G - and G∗ -band 461
2.8.10. Double resonance, overtone and combination modes 462
2.9. Summary 462
3. Calculation method of resonance Raman spectra 464
3.1. Overview of calculations reviewed in this section 464
3.1.1. Raman scattering and phonon energy dispersion 465
3.1.2. Electronic energy bands 465
3.1.3. The double resonance process 465
3.1.4. Electron–photon and electron–phonon interactions 465
3.1.5. Excitons 466
3.1.6. Resonance window and the Kohn Anomaly 466
3.2. Tight-binding calculation for phonons 466
3.3. Simple tight-binding calculation for the electronic structure 468
3.3.1. Extended tight-binding calculation for the graphene electronic structure 471
3.4. Calculations of matrix elements 471
3.4.1. The electron–photon matrix element 471
3.4.2. Electric dipole vector for graphene 472
3.4.3. Calculation of the electron–phonon interaction 474
3.5. Calculation of excitonic states 477
3.5.1. The Bethe–Salpeter equation for exciton states 477
3.5.2. Exciton–photon matrix element 479
3.5.3. The exciton–phonon interaction 479
3.6. The resonance Raman process 480
3.6.1. Matrix elements for the resonance Raman process 480
3.6.2. Matrix elements for double resonance Raman scattering 480
3.6.3. Resonance window width 481
3.6.4. G-band intensity for semiconducting SWNTs 483
3.6.5. G-band intensity for metallic SWNTs: The Kohn Anomaly 484
4. Raman spectra of graphene 487
4.1. The G-band and G -band intensity ratio 488
416 R. Saito et al.

4.2. Layer number dependence of G -band 488

4.2.1. The number of graphene layers with AB stacking 488
4.2.2. Characterization of the graphene stacking order by the G spectra 490
4.3. D-band and G-band intensity ratio and other disorder effects 491
4.3.1. Ar+ ion bombardment on graphene 491
4.3.2. The D to G intensity ratio and the LD dependence 492
4.3.3. The D to G intensity ratio: the “local activation” model 493
4.3.4. The Local Activation Model and the Raman Integrated Areas 495
4.3.5. Modeling disorder effects in the Raman linewidths and frequency shifts:
the spatial correlation model for defects 497
4.3.6. Evolution of overtone and combination modes 500
4.3.7. Disorder and the number of layers 501
4.4. Edge phonon Raman spectroscopy 501
4.5. Polarization effects in graphene nanoribbons 503
5. Raman spectra of carbon nanotubes 504
5.1. The radial breathing mode and the Kataura plot 504
5.1.1. The RBM frequency 505
5.1.2. The RBM for double wall carbon nanotubes 506
5.1.3. The Raman excitation profile for the RBM 508
5.1.4. The RBM spectra of SWNT bundles 510
5.1.5. (n, m) dependence of RBM intensity – experimental analysis 511
5.1.6. The experimental Kataura plot 513
5.2. Exciton environmental effect 513
5.2.1. The effect of the dielectric constant κ on Eii 514
5.2.2. Screening effect: a general κ function 515
5.2.3. Effect of the environmental dielectric constant κenv on Eii 518
5.3. Splitting of the G mode 518
5.3.1. The G-band eigenvectors and curvature 518
5.3.2. The six G-band phonons – confinement effect 519
5.3.3. The diameter dependence of the G band phonon frequencies 522
5.4. Kohn Anomaly effect on the G-band and the RBM mode 524
5.4.1. G-band Kohn anomaly 524
5.4.2. Chemical doping and the G band 525
5.4.3. Substrate interaction and the G band 525
5.4.4. Theoretical approach to the Kohn Anomaly 525
5.4.5. RBM band and G-band Kohn anomaly 527
5.5. Double resonance effect and quantum confinement 528
5.5.1. The G -band in SWNT bundles 529
5.5.2. The (n, m) dependence of the G -band; phonon trigonal warping 530
5.6. Near-field Raman spectroscopy 533
6. Challenges of Raman spectroscopy in graphene and carbon nanotubes 533
6.1. The novelties of graphene 534
6.2. The novelties of carbon nanotubes 535
6.3. Near-field Raman spectroscopy and microscopy 536
6.4. Time-dependent Raman and coherent phonon spectroscopy 536
6.5. Conclusion and messages for the future 536
Acknowledgements 537
Notes 537
References 538
 Advances in Physics 417

This review of the Raman spectroscopy of graphene and carbon nanotubes starts with a brief
presentation of background material to set the stage for a discussion of the present status of
knowledge on this topic by providing a broad overview of the field in Section 1. This is followed,
in Section 2, by an overview discussion of experimental progress that has been made in recent years
using many experimental techniques. Advances in the theoretical understanding needed to interpret
the many new results that appear daily on the photo-physics of graphene and carbon nanotubes
are reviewed in Section 3. The detailed consideration of the Raman spectroscopy of graphene
and carbon nanotubes is presented in Sections 4 and 5, both individually and comparatively,
while Section 6 looks at future developments in this field. A textbook helpful for understanding
background material relevant to this review article has recently been published [1].
Hereafter in, Section 1, we provide background to Raman spectroscopy in general, giving
special attention to resonance Raman spectroscopy, with illustrations of the various photo-physical
phenomena, which are given in terms of graphene and carbon nanotubes. Raman measurements in
low-dimensional (one- and two-dimensional) systems are then discussed in general terms, giving
specific examples of the differences and similarities between the Raman spectra for graphene and
carbon nanotubes in relation to other sp2 carbons. Finally, we discuss what can be learned from the
Raman spectrum from one laser line in comparison to what can be learned from using a continuum
of laser lines.

1. Basic concepts
In this section, we introduce the basic concepts of Raman spectroscopy, starting with where they
stand within the broad picture of light–matter interactions and then considering their general char-
acteristics. The concepts of first-order and higher-order scattering processes, Stokes vs. anti-Stokes
processes, lineshapes, resonance and coherent processes are introduced, as well as the classical
treatment of the Raman effect. Subsequently, we present a historical description of the Raman
spectroscopy used to study and characterize graphitic materials, from graphite to single-walled
carbon nanotubes (SWNTs) and graphene. This brief introduction to the Raman spectroscopy of
sp2 nanocarbons should be useful to general readers who may not be so familiar with Raman
spectroscopy.

1.1. Overview of sp2 carbon materials

We start this section with a brief history of several important carbon materials.

1.1.1. Graphite
Carbon materials have been the objects of study and use for many years [2]. Three-dimensional
(3D) graphite is one of the longest known forms of pure carbon, naturally occurring on the surface of
the earth as a mineral (as, for example, in the mineral deposits of Ticonderoga (NewYork) graphite,
Ceylon (Sri Lanka) and the large graphite deposits in Minas Gerais, Brazil). Structurally, graphite
is a layered material, with individual graphene layers, as shown in Figures 1 and 2(a). These
individual graphene layers are stacked in the ABAB Bernal stacking order in the most common
form of graphite, as shown in Figure 2(b) to illustrate the relative in-plane atomic arrangements
of the A and B carbon atoms within the layer plane and in adjacent layers. Here, one type of
carbon atom (labeled A) is aligned on top of another A atom in the direction perpendicular to the
graphene layer, while the other type (the B carbon atom) is aligned in every other layer1 [2], so
that the graphene planes are arranged in the so-called ABAB Bernal stacking sequence, as shown
in Figure 2(c) [6], where a carbon atom is found on adjacent layer planes over the empty center of
a hexagon. This AB stacking order (Figure 2(c)) also applies to bi-layer graphene prepared by the
418 R. Saito et al.

Figure 1. STM image of graphite. Notice the different brightness for the A and B atoms (see footnote 1).
Reprinted from Carbon, 48(5), M.M. Lucchese et al. pp. 1592–1597 [3]. Copyright © (2010) Elsevier.

mechanical exfoliation method [7,8], while Figure 2(d) shows the stacking arrangement of trilayer
graphene. In graphite, layers 1 and 3 are crystallographically equivalent and are translated from
one another by the out-of-plane lattice parameter c = 0.670 nm (see Figure 2(d)), thus generating
hexagonal graphite. Another related crystalline arrangement is the less common form known as
rhombohedral graphite [9], which has an ABC stacking order, consisting of three layers and a
lattice constant of 1.005 nm.
Of all materials, graphite has the highest melting point (4200 K), the highest thermal conduc-
tivity (3000 W/mK) and a high room-temperature electron mobility (30, 000 cm2 /V s) [10,11]. 3D
graphite was synthesized for the first time in 1960 by Arthur Moore and co-workers [12–16] and
their high-temperature, high-pressure synthesis method yielded the material commonly known as
highly oriented pyrolytic graphite (HOPG). Graphite and its related carbon fibers [17–19] have
been used commercially for decades [20]. Carbon fiber applications range from use as conductive
fillers, and as mechanical structural reinforcements in composites (e.g., in the aerospace industry),
to their use as electrode materials for making steel, exploiting their good electrical conductivity
and in lithium ion battery applications exploiting their high resiliency [20,21].

1.1.2. Fullerenes
In 1985, the discovery of another unique sp2 carbon system took place, the observation of the
C60 fullerene molecule [22]. The fullerene molecule consists of 60 carbon atoms with mostly sp2
bonding and appropriate π bonding to form a closed surface with full icosahedral symmetry.
Because of topological restrictions, fullerenes, in general, have 12 pentagonal rings and any
numbers of hexagonal rings, thereby generating a large variety of Cn fullerene molecules. The C60
molecule with full icosahedral symmetry can be regarded as the first isolated carbon nanosystem.
Fullerenes stimulated and motivated a large scientific community into new research directions from
the time of their discovery up to the end of the twentieth century, but fullerene-based applications
 Advances in Physics 419

Figure 2. (a) Top view of the real space unit cell of mono-layer graphene showing the inequivalent carbon
atoms A and B and the graphene unit vectors a1 and a2 . (b) Top view of the real space bi-layer graphene
structure. The light/dark gray dots and black circles/black dots represent the carbon atoms in the upper and
lower layers, respectively, of bi-layer graphene (2-LG). (c) The unit cell and the x and y unit vectors of bi-layer
graphene and (d) the same as (c) but for trilayer graphene. (e) The reciprocal space unit cell showing the first
Brillouin zone with its high symmetry points and lines, such as the line T connecting  to K. (f) The Brillouin
zone for 3D graphite showing high symmetry points and axes. Here  is a high symmetry point along the
axis connecting points A and , and u is a general point in the KM plane. Adapted figure with permission
from A. Jorio et al. Spectroscopy in Graphene Related System, 2010 [1] Copyright © Wiley-VCH Verlag
GmbH & Co. KGaA; and with permission from L. M. Malard et al., Physical Review B 79, p. 125426, 2009
[4]. Copyright © (2009) by the American Physical Society; and with permission from Physics Reports 473,
L.M. Malard et al. pp. 51–87 [5]. Copyright © (2009) Elsevier.

remain sparse to date. In this review, we do not mention fullerenes much. See [23] for more
information on fullerenes.

1.1.3. Carbon nanotubes

Carbon nanotubes arrived actively on the scene in 1991 following the footsteps of the emergence
of the C60 fullerene molecule. Since their emergence, carbon nanotubes have evolved into one of
the most intensively studied materials, now being held responsible for co-triggering the nanotech-
nology revolution. The strong entry of nanotubes on the scene in 1991 was through a report of the
observation of multi-walled carbon nanotubes (MWNTs) on the cathode of a carbon arc used to
produce fullerenes [24]. Actually, carbon nanotubes had been identified in the 1970s in the core
structure of vapor-grown carbon fibers as very small carbon filaments [25–27] and carbon fibers
were reported even earlier in the 1950s in the Russian literature [28] (see Figure 3). However,
single-walled carbon nanotubes (SWNTs), the most widely studied carbon nanostructure in the
1995–2005 timeframe, were first synthesized systematically in 1993 [30,31]. Carbon nanotube
research took off at this point in time. The great interest in the fundamental properties of carbon
nanotubes and in their exploitation through a wide range of applications is due to their unique struc-
tural, chemical, mechanical, thermal, optical, optoelectronic and electronic properties [21,32,33].
The growth of an SWNT at a specific location and point in a given direction and the growth of a huge
420 R. Saito et al.

Figure 3. Early transmission electron microscopy images of carbon nanotubes [29]. The early reported
observations of nanotubes (a) in 1952 [28] and (b) in 1976 [25]. (c) Observation of SWNTs that launched
the field in 1993 [30,31] together with an example of their observation. Adapted figure with permission
from Carbon 14(2), A. Oberlin et al., pp. 133–135 [25]. Copyright © (1976) Elsevier; and with permission
from Carbon 44, M. Monthioux and V.L. Kuznetsov, pp. 1621–1623 [29]. Copyright © (2006) Elsevier; and
with permission from Macmillan Publishes Ltd. Nature [30]. Copyright © (1993); and with permission from
Macmillan Publishes Ltd. Nature [31], Copyright © (1993).

amount of millimeter-long nanotubes with nearly 100% SWNT purity (absence of other carbon
forms) have now been achieved [34], and further improvements in nanotube synthesis are evolving
rapidly at this time. Substantial success with the separation of nanotubes by their (n, m) structural
indices, metallicity (semiconducting or metallic) and by their length has been achieved by different
methods, especially by the density gradient approach of Hersam and Arnold [35]. Advances have
been made with doping either n-type or p-type nanotubes for the modification of their properties,
as summarized in reference [36,37]. Studies on nanotube mechanical properties [37,38], optical
properties [39–45], magnetic field-dependent properties [46], optoelectronics [47,48], transport
properties [49] and electrochemistry [50,51] have exploded, revealing many rich and complex
fundamental excitonic and other collective phenomena [21]. Nanotube-based quantum transport
phenomena, including quantum information applications, spintronics and superconducting effects,
have also been explored [49]. After more than a decade and a half of intense activity in carbon
nanotube research, more and more attention is now being focussed on the practical applications of
the many unique and special properties of carbon nanotubes [20]. Further background information
on the synthesis, structure, properties and applications of carbon nanotubes can be found in [1,21]
and some of these topics are further emphasized in the present review.

1.1.4. Graphene
The interest in a single atomic layer of sp2 carbon (called graphene; see Figure 1) goes back to
the pioneering theoretical work of Wallace in 1947 [52] which was for many years used as a
model system for all sp2 carbons. This very early work provides a framework for comparing the
structures of graphite, fullerenes, carbon nanotubes and other sp2 nanocarbons. The synthesis of
single-layer graphene was actually reported by Boehm in 1962 [53], but this early discovery was
neither confirmed nor followed up for many years.
More recently, mono-layer graphene was synthesized from nano-diamonds in 2001 [54] and
from SiC [55]. The material synthesized from nano-diamonds is generally a few-layer graphene
material but thin ribbon specimens of mono-layer thickness are also contained in such samples. In
the studies on these nanoribbons, emphasis was given to the properties of edges of the nanoribbons
and especially to the magnetic properties of these edges [54,56–59]. The graphene prepared from
heating SiC to 1300 ◦ C emphasized the 2D electron gas properties of this graphene in an electric
field, but did not especially focus on the number of layers [55].
The widespread study of graphene was launched by the preparation of mono-layer graphene
by Novoselov et al. [7], using a simple Scotch tape method to prepare and transfer mono-layer
graphene from the c-face of graphite to a suitable substrate such as SiO2 for the measurement
of the electrical and optical properties of mono-layer graphene [60]. The Novoselov and Geim
studies of transport in few-layer graphene in 2004 [7] led to a renewed interest in mono-layer and
 Advances in Physics 421

few-layer graphene and to an in-depth study of the unique properties of this material in the mono-
layer and bi-layer limit. Surprisingly, this very basic system, which had been conceptually utilized
by researchers over a period of many decades, suddenly appeared on the experimental scene,
demonstrating many novel physical properties that were not even imagined previously [60,61].
The discovery of these novel properties launched a rush into the study of graphene science in the
first decade of the twenty-first century, and culminating in the 2010 Nobel Prize in physics [62].
Besides the outstanding mechanical properties [63] (breaking strength ∼40 N/m, Young’s
modulus ∼1.0 TPa) and thermal properties [64,65] (room temperature thermal conductivity
∼3000 W m−1 K −1 [65]), the scientific interest in graphene was stimulated [66,67] by the
widespread report of the relativistic (massless) electronic properties of the conduction elec-
trons (and holes) in a single layer less than 1 nm thick, with a state-of-the-art mobility reaching
μ = 200, 000 cm2 /V s at room temperature for freely suspended graphene [66–71]. Other unusual
properties have been predicted and demonstrated experimentally, such as the minimum conductiv-
ity and the half-integer quantum Hall effect in mono-layer graphene [72] and the integer quantum
Hall effect in bi-layer graphene [73], ambipolar transport by either electrons or holes by the vari-
ation of a gate potential, operation as a transparent conductor [47,74], Klein tunneling [75–82],
negative refractive index and Veselago lensing [80], anomalous Andreev reflections at metal–
superconductor junctions [76,81–84], anisotropies under antidot lattices [85] or periodic potentials
[86], and a metal–insulator transition via hydrogenation of graphene [87]. Applications such as
fillers for composite materials, as super-capacitors, batteries, interconnects and field emitters are
being developed, although it is still too early to say to what extent graphene will be able to com-
pete with carbon nanotubes and other established materials systems in the applications world
[88]. Although nanotubes and graphene are both carbon-based nanostructures, they have different
properties related to the planar aspects of graphene and the tubular aspects of nanotubes, and this
basic difference should distinguish their optimal usage in applications.

1.1.5. Nanoribbons
Graphene nanoribbons are of particular interest for introducing a bandgap into graphene-related
systems. Bandgaps are needed for many electronics applications of nanomaterials. Since graphene
can be patterned using, for example, high-resolution lithography [55,89], nanocircuits with
graphene–nanoribbon interconnects can be fabricated. Many groups are now fabricating devices
using graphene and also graphene nanoribbons, which have a long length and a small nanoscale
width, and where the ribbon edges play an important role in both determining their electronic
structure and exhibiting unusual spin polarization properties [56]. Nanoribbons of small widths
exhibit 1D behavior analogous to carbon nanotubes, but have a high density of electronic states
at the Fermi level for the case of well-defined zigzag edges. This high density of electronic states
allows us to experimentally distinguish zigzag nanoribbons from armchair and chiral nanoribbons
which do not exhibit this property. While lithographic techniques have limited resolution for the
fabrication of small width nanoribbons (<20 nm wide), chemical [90] and synthetic [91] methods
have been employed successfully, including the unzipping of SWNTs as a route to produce carbon
nanoribbons [92,93]. Carbon nanoribbons have been shown to be especially sensitive for the study
of edge structures and edge properties and edge reconstruction effects [94].

1.1.6. Raman spectroscopy for sp2 carbon materials

Raman spectroscopy is the inelastic scattering of light, usually associated with the emission (Stokes
process) or absorption (anti-Stokes process) of phonons [95]. By knowing the energy shift of the
scattered light relative to the incident light, which yields Raman spectra in cm−1 (see Figure 4),
422 R. Saito et al.

Figure 4. Raman spectra from several sp2 nanocarbon and bulk carbon materials. From top to bottom:
crystalline mono-layer graphene, HOPG, an SWNT bundle sample, damaged graphene, single-wall carbon
nanohorns (SWNH). The most intense Raman peaks are labeled in a few of the spectra [1,96]. Note that
some authors call the G’ by 2D and the G” by 2D’ [97]. Reprinted with permission from M.S. Dresselhaus
et al., Nano Letters 10, pp. 953–973, 2010 [96]. Copyright © (2010) American Chemical Society.

we can obtain the phonon frequency which is useful for identifying the origin of an unknown
structure of a newly discovered molecule or of a new material in chemistry [98]. Among all
possible phonon modes for sp2 carbons, only a limited number of phonons are Raman-active
modes (namely those with A, E1 and E2 symmetry for carbon nanotubes, and E2g for graphite)
[99–101]. A common metric used to characterize the defect density in a material is the ratio of the
intensities of the disorder-induced D-band to the symmetry-allowed G-band ratio (ID /IG ) [102].
Study of the D- and G-band modes by Raman spectroscopy (see Figure 4) yields information
about the crystal structure of the material and about many of its interesting physical properties.
Raman spectroscopy for the various sp2 carbon materials (see Figure 4) has been mainly
used for sample characterization and these different carbon materials exhibit characteristic dif-
ferences related to the small differences in their structures. The fundamental sp2 carbon material
is mono-layer graphene which has the simplest and most fundamental spectrum showing the two
Raman-allowed features that appear in all sp2 carbon materials – the first-order G-band and the
second-order symmetry-allowed G -band,2 where the symbol G is used to denote “graphitic.” The
next most commonly observed feature is the D-band that is a defect-activated Raman mode. The
D-band occurs at about 1350 cm−1 at 2.41 eV laser excitation energy (Elaser ) and is highly disper-
sive as a function of Elaser (see Section 2.8.9). Since the graphite melting temperature is very high
(over 4200 K) and since no actual carbon materials are defect-free, the D/G-band intensity ratio
(ID /IG ) provides a sensitive metric for the degree of disorder in sp2 carbon materials over a wide
temperature range. In the case of fullerenes, a special Raman-active phonon mode related to the
vibrations of a pentagonal ring (1469 cm−1 ) is particularly sensitive for understanding the molec-
ular structure [23]. In the case of carbon nanotubes, it is common to survey unknown samples
using Raman spectroscopy to check for the presence of nanotubes in the sample by observing the
cylindrical-specific, Raman-active-mode radial breathing mode (RBM) in which atoms around the
circumference of a single wall carbon nanotube (SWNT) are vibrating in a breathing mode in the
radial direction (see Figure 4) [32,100,103–110]. This vibrational mode is unique to carbon nan-
otubes and serves to sensitively identify their presence in a given sample. Since the RBM frequency
 Advances in Physics 423

ωRBM is inversely proportional to the nanotube diameter dt , we can thus estimate the diameter
distribution of the nanotubes that are contained in a given sample [111]. When we observe an
isolated nanotube, we can use its Raman spectrum to obtain its detailed structure, which involves
identification of the spatial orientation, the diameter and the chiral angle of the nanotube, as well
as the nanotube (n, m) chirality assignment. This (n, m) assignment is based on the concept of
the resonance Raman effect (see Section 1.5.2). In the case of graphene [112–114], the inten-
sity ratio (IG /IG ) and the lineshape of the G -band (along with other indicators) can be used for
identifying the number of graphene layers (see Section 4.2.1). For graphene ribbons, we can use
Raman spectroscopy to study the edge structure of the ribbons [94,115] to yield information about
the structure and properties of graphene ribbons (see Section 4.4). For all these reasons, Raman
spectroscopy is very sensitive for the characterization of sp2 carbons.
Since most carbon materials are not soluble in water, Raman spectroscopy is useful as an
in situ, non-contact, non-destructive measurement tool that can be used at room temperature
and under ambient operating conditions, as well as for freely suspended carbon nanotubes and
graphene samples. Combining the continuing advances in optical techniques with new theoretical
developments that are rapidly developing, Raman spectroscopy studies of graphene and carbon
nanotubes have provided a great deal of information about their solid-state properties, which are
the main subjects of this review, including their behavior as a function of temperature, pressure
and Fermi energy [1].

1.2. Various photo-physical processes

In this section, various optical processes that are useful for the characterization of nanostructures
are very briefly discussed. The Raman effect [116,117] is briefly mentioned in this section, but is
further discussed in Sections 1.4–1.5.
When focusing light into a material (molecule or solid), part of the energy just passes through
the sample (by transmission), while the remaining photons interact with the system through light
absorption, reflection, light emission or light scattering. The amount of light that will be trans-
mitted, as well as the details for all the light–matter interactions, are determined by the electronic
and vibrational properties of the material. Furthermore, different phenomena occur when focusing
light with different energy photons into a given material [95,118], because different photon ener-
gies will be related to the different optical transitions occurring in the medium. As an example of
the richness of light–matter interactions, an overview of various optical absorption mechanisms
for a semiconducting material is shown in Figure 5. Using this figure as a guide, examples are
given for the many different effects that might occur when light interacts with a material. Starting
from the high-energy side of the diagram moving to mechanisms important at lower energies,
Figure 5 shows the absorption coefficient as a function of the energy of the photon which yields
the absorption spectra and enumerates the dominant photo-physical processes that are involved in
the various photon energy regimes in Figure 5 [95].

1.2.1. Valence-to-conduction band transitions and excitons

A photon (having an energy in the 1–5 eV range) can be absorbed by an electron making a transition
from the valence band to the conduction band, as shown in Figure 5. Such a transition generates a
free electron in the conduction band, leaving behind a “hole” in the valence band, where we use
the nomenclature for electrons and holes as used in semiconductor physics [119].
Photons with an energy smaller than the energy gap can generate a transition to an exciton
level, thereby creating an exciton which corresponds to an electron bound to a hole through the
Coulomb interaction. Although excitonic levels in model semiconducting systems such as GaAs
424 R. Saito et al.

Figure 5. Photo-physical mechanisms operative for various regions of the electromagnetic spectrum as
photons in various energy ranges interact with materials. Reprinted figure with permission from A. Jorio et
al. Spectroscopy in Graphene Related Systems, 2010 [1]. Copyright © Wiley-VCH Verlag GmbH & Co.
KGaA.

have excitonic levels of a few meV below the bandgap, the excitonic levels in carbon nanotubes are
much larger (on the order of a few hundred meV), emphasizing the greater importance of excitonic
effects in low-dimensional nanosystems. Optical absorption in the case of carbon nanotubes was
predicted by Ando to be excitonic as far back as 1997 [120] and the importance of excitons
in nanotubes was demonstrated experimentally by the two photon experiments carried out by
the Heinz group in 2005 [121] and a similar result was independently obtained by the Berlin
group [122]. These experiments confirm theoretical concepts that excitonic effects would be
enhanced in low-dimensional systems. Although excitonic effects were found to be much larger
in semiconducting nanotubes than in metallic nanotubes, excitons nevertheless have been found
in these works to also dominate optical absorption processes in metallic nanotubes.

1.2.2. Optical transitions including impurity levels

Impurities can form states within the bandgap of semiconducting materials. If the impurity atom
has more valence electrons than the atom it replaces, then this impurity will act as an electron donor
making the nanotube an n-type semiconductor. However, if the impurity atom has fewer electrons,
then it will behave like an electron acceptor giving rise to a p-type semiconductor. Light can be
absorbed, generating electronic transitions from the valence band to such donor impurity levels,
or optical transitions can be made by taking electrons from an acceptor level to the conduction
band of a semiconductor. For semiconducting carbon nanotubes, typical photon energies for these
impurity levels are in the 10–100 meV range about a band edge, but they usually occur over an
energy range smaller than the energy gap.
Optical transitions to conduction band impurity levels or from valence band shallow acceptor
levels can also be responsible for light absorption. The corresponding photon energy would be
significantly lower than the stronger excitations from the dominant bright allowed state. In mate-
rials, such as carbon nanotubes, the lowest energy transitions however involve dark exciton states,
thereby lowering the intensity of these transitions.
 Advances in Physics 425

1.2.3. Optical phonons

Optical absorption by phonons typically occurs when the incident photon energy coincides with
the energy of optical phonons, corresponding, in general, to 50 meV to 0.2 eV for first-order
processes. These absorption processes generally occur in the infrared energy range and play a
major role in the field of infrared spectroscopy for both molecules and solid-state systems.
In sp2 carbon systems, the harmonics and combination modes of symmetry-allowed optical
phonons are clearly observed in the Raman spectra through double resonance (DR) processes.
Thus phonon-related effects can be observed over a photon energy range up to ∼350 meV for the
observation of phonon-related effects. It is noted that Raman processes and infrared absorption
processes generally occur at different photon energies because, in general, different phonons are
Raman- and infrared-active for high symmetry materials such as sp2 carbons.

1.2.4. Free carrier processes

Free carriers provide another mechanism for optical absorption. Free carriers are dominant in
metallic systems and are present in doped semiconducting systems through electrons and/or
holes. These carriers can also absorb light, usually occurring over a broad energy range from 1
to 10 meV. Other free carrier processes not shown explicitly in Figure 5 also occur. In a higher
energy region (1–10 eV), collective excitations of electrons also occur, giving rise to plasmon
absorption. At much higher energies in the ultraviolet and X-ray range, transitions from core
levels also take place, generating photo-excited electrons, which can be observed by ultraviolet
photo-electron spectroscopy (UPS) and X-ray photo-electron spectroscopy (XPS), depending on
the energy of these deeper levels. By measuring the momentum and energy of photo-electrons for a
given momentum and energy, information can be obtained about the momentum and energy of the
incident electron in the valence energy band, by the angle-resolved photo-emission spectroscopy
(ARPES) technique, which has been widely used for observing electron energy dispersion in
single layer and multi-layer graphene [123–126].

1.3. Interactions for optical processes

In this section, we focus on the different phenomena occurring through light–matter interactions
[95] (see Figure 6). The potential energy profile of an atom is modified by the electric field of the
incident photon, so that the energy and momentum of the incident light are changed in a scattering
process and a resulting photoluminescence or fluorescence process that frequently occurs after
absorption by an atom gives important information about this interaction. However, typical times
for the occurrence of such photoluminescence or fluorescence processes are much longer (in ns or
ms, respectively) than for typical light scattering events (in ps). We now briefly review the various
light–matter interactions.

1.3.1. Photon-induced electron–phonon interaction

Two kinds of light scattering processes are the elastic and inelastic scattering processes, which
are called the Rayleigh and the Raman scattering, respectively. In the case of Rayleigh scatter-
ing, only the direction of the light is changed with no change in photon energy, while in the
case of Raman scattering, either phonon creation or annihilation occurs in the scattering process.
When the electric field associated with the photon interacts with an atom, the electron and the
ion core move in opposite directions to each other to form a dipole and this oscillating dipole
interacts with the incident photon to generate the scattered light. The coupling of the dipole
with the electric field of the photon creates a phonon which is described by an electron phonon
426 R. Saito et al.

Figure 6. The light–matter interaction, showing the most commonly occurring processes. The waved arrows
indicate incident and scattered photons. The vertical arrows denote photon-induced transitions between
(a) vibrational levels and (b–e) electronic states. Curved arrow segments indicate electron–phonon (el–ph)
(hole–phonon) scattering events. In (e), the shortest vertical arrow also indicates an el–ph transition in Raman
scattering. In (d, e), the processes are resonant if the incident (or scattered) photon energy exactly matches the
energy difference between initial and excited electronic states. When far from the resonance window where
resonance occurs, the optical transition is called a virtual transition. The intensity for resonance Raman
scattering can be much larger for vertical processes that are resonant than those that are not resonant [1,95].
Reprinted figure with permission from A. Jorio et al. Spectroscopy in Graphene Related Systems, 2010 [1].
Copyright © Wiley-VCH Verlag GmbH & Co. KGaA.

(el-ph) interaction that occurs in the Raman scattering process. In this process the electron is
excited to a virtual state, for which the stable geometry of the chemical bond is no longer iden-
tical to that of the ground state. This perturbation generates a force resulting in atomic motion,
thereby providing a quantum explanation for the el–ph interaction. If the incident photons are
introduced through a sufficiently short light pulse with a duration comparable to the frequency
of the phonon, then all atoms start to move at the same time as a result of the el–ph interaction,
thereby creating a coherent motion of the atoms which can be detected by a second pulse of light
at a frequency at which the material is transparent. This sensitive technique is known as coherent
phonon spectroscopy.

1.3.2. Electron–phonon interaction in infrared absorption

When the photon energy matches the energy for allowed phonon creation, the photon can transfer
energy directly to create an acoustic or optical phonon (see Figure 6(a)). This resonance process
is called infrared (IR) absorption, since the phonons that are created have energies corresponding
to IR photon frequencies. IR-active phonon modes are phonon modes that have the symmetry
of a vector [1], which describes the vibration of an oscillating dipole moment of the ions in the
material. The dipole moment is directly coupled to the electric field associated with the incident
and scattered photons. We describe this interaction by an electromagnetic interaction perturbation
Hamiltonian between the vibrating atoms and the photons that induce the vibration. Since all the
atoms in nanocarbon materials are neutral carbon atoms, the dipole moment for IR absorption is
generated by photo-excited electrons created by the optical electric field.

1.3.3. Dipole–phonon interaction in non-radiative decay

For higher photon energies, the photons are absorbed by exciting an electron–hole (e–h) pair, as
shown in Figure 6(b). The photo-excited electron (or hole) then loses energy to electrons in the
 Advances in Physics 427

bottom (top) of the conduction (valence) energy band by creating multiple phonons of different
frequencies through el–ph coupling, in which the phonons are selected such that the initial and
final electronic states satisfy both energy and momentum conservation requirements. In the case
of a metal, such photo-excited electrons (together with their holes) will decay down (up) to
the ground states without emitting a photon, and such processes are called non-radiative decay
process as shown in Figure 6(b). In graphene or metallic carbon nanotubes, non-radiative decay
that generates heat frequently occurs and can have a special character because of their linear E(k)
dispersion relation.

1.3.4. Dipole–photon interaction in photoluminescence

If the material has an energy gap between the occupied (valence) and unoccupied (conduction)
bands, the photo-excited electron quickly (on a ps timescale) decays to the bottom of the conduction
band by an el–ph process and then to its ground state by emitting a photon with the bandgap energy
on a ns timescale, in a light emission process (see Figure 6(c)). This emission process is called
photoluminescence, and the energy decrease from the incident photon to the scattered photon is
generally called a Stokes process because of the decreased energy of the emitted photon.3 In order
to get photoluminescence emission, an interaction between the dipole moment of the atom with
the electric field of the photon is essential in which the dipole-selection rule must be satisfied for
the electronic excited states. When the energy gap is smaller than the optical phonon energy, no
photoluminescence occurs, but phonon emission can occur by an inter-band el–ph interaction. In
this case, the quantum efficiency for photon generation by the photo-excited electrons is suppressed
significantly.

1.3.5. Rayleigh scattering

In the Rayleigh scattering process, a photon is virtually4 absorbed by a material and the oscillating
electric field of the photon just shakes the electrons. In this case, the electrons just scatter that
energy back to another photon having the same energy as the incident photon. When the incident
and scattered photons have the same energy, the scattering process is said to be “elastic” and is
named Rayleigh scattering (see Figure 6(d)). Elastic scattering occurs by an interaction of the
electric field of the photons with the crystal or atomic potential and it can occur even for the case
of real absorption, in which the Rayleigh scattering intensity is enhanced significantly, and such
a process is called the resonance Rayleigh scattering [44].

1.3.6. Brillouin scattering

In solid materials, a further distinction is made between the inelastic scattering by acoustic phonons
(called Brillouin scattering) and by optical phonons (called Raman scattering). Brillouin scattering
occurs by an el–ph interaction with acoustic phonons or with any other low-energy excitation such
as a magnon. The concept of Brillouin scattering does not apply to molecular systems for which the
acoustic phonon would represent a translation of the molecule. Since Brillouin scattering generally
appears in a lower energy region than that for an optical phonon, a special experimental set-up
based on low-frequency instrumentation is required for observing Brillouin scattering, namely
a set-up that very strongly suppresses the presence of the Rayleigh signal. When the Brillouin
scattering is stimulated by the electric field of the laser light itself, a strong Brillouin signal is
obtained which is known as stimulated Brillouin scattering.

1.3.7. Raman scattering

The inelastic scattering of light is called the Raman effect, named in honor of the discoverer of the
Raman effect in 1927, commonly attributed to Sir Chandrasekhara Venkata Raman (1888–1970),
428 R. Saito et al.

an Indian scientist who was awarded the Nobel Prize in Physics in 1930 for his work on “the
scattering of light and for the discovery of the effect named after him”. In the Raman process,
an incident photon with energy Ei = Elaser and momentum ki = klaser reaches the sample and is
scattered, resulting in a photon with a different energy Es and a different momentum ks . For energy
and momentum conservation,

Es = Ei ± Eq and ks = ki ± q, (1)

where Eq and q are the energy and momentum change during the scattering event induced by
electromagnetic excitation of the medium. The quantities Eq and q can be considered to be the
energy and the momentum of the phonon.
The Raman process which emits (absorbs) a phonon is called a first-order Raman process. In
order to recombine an electron at ks with a hole at ki , the wavevector q should be almost zero.
Thus, only phonons near the  point (zone-centered phonon) in the phonon dispersion relation can
be Raman-active modes. However, when we consider second-order Raman processes in which
two scattering events are involved, the restriction for q = 0 is relaxed. Further, the photo-excited
electrons in sp2 carbons are located in k space near the hexagonal corners of the 2D Brillouin zone
(BZ), named the K and K  points, where the states of lowest energy are located. Here, there are
two possibilities for q  = 0 scattering: intra-valley (K → K, K  → K  ) and inter-valley scattering
(K → K  , K  → K). We will show (see Section 2.8.4) that a double resonance Raman process
involving excitations near the K and K  points in the 2D BZ yields a large Raman signal.

1.3.8. First- and higher-order Raman processes

The order of the Raman process is given by the number of scattering events that are involved in
the Raman process. The most usual case is the first-order Stokes Raman scattering process, where
the photon energy excitation creates one phonon in the crystal with a very small momentum
(q ≈ 0). If two, three or more scattering events occur in the Raman process, then the process
is called second-, third-, or higher-order, respectively. The first-order Raman process gives the
basic quantum of vibration, while higher-order processes give very interesting information about
overtones and combination modes. In the case of overtones, the Raman signal appears at nEq
(n = 2, 3, . . .) and the Raman signal from combination modes appears at the sum of the different
phonon energies (Eq1 + Eq2 , etc). An interesting point in the higher-order Raman signal in a solid
material is that the restriction for q ≈ 0 in a first-order Raman scattering process is relaxed. The
photo-excited electron at k can be scattered to k + q and can go back to its original position at k
after the second scattering event by scattering a phonon with wavevector −q, which allows the
recombination of the photo-excited electrons with their corresponding holes. The probability for
selecting a pair of q and −q phonons is usually small and not very important for solids. However, we
see in Section 2.8.4–2.8.10 that, under special resonance conditions (the DR condition) commonly
occurring in sp2 nanocarbons, we can expect a clear Raman signal from q  = 0 scattering events.

1.4. Characteristics of the Raman effect

Next, we very briefly define in this section a number of the important characteristics of the
Raman effect that will be commonly used throughout this article. We first discuss the Raman
spectra giving special emphasis to low-dimensional systems, thereby introducing the concept of
the Raman excitation profile which relates the laser frequencies over which resonant processes
take place (Section 1.4.1 and 1.4.2). This leads to a discussion of Stokes and anti-Stokes spectra
and the difference between the two (Section 1.4.3), followed by a discussion of the spectral width
of particular phonon features in the Raman spectra (Section 1.4.4–1.4.6). Finally, we discuss
 Advances in Physics 429

in Section 1.4.7, the characteristics of the Raman excitation profile, particularly the resonance
window width.

1.4.1. Raman spectra and the Raman excitation profile

When we use one laser energy and observe the intensity of the scattered photons as a function
of the shift of energy in cm−1 from the incident light (the Raman shift), this plot is called a
Raman spectrum, from which we determine the phonon energy and its spectral width. The Raman
spectrum is observed by passing the scattered light through a monochromator which divides the
light entering the instrument into the scattered light on the lower energy side relative to the incident
light (the so-called Stokes Raman spectra) and the scattered light on the higher energy side (the
so-called anti-Stokes Raman spectra). At the zero energy shift, we generally see a strong Rayleigh
signal which can be eliminated by using a so-called notch filter5 or a (triple) monochromator.6
When we have an experimental laser system where the photon energy can be tuned (or changed),
the Raman intensity for a low-dimensional sp2 carbon system such as carbon nanotubes will show a
sharp maximum at the resonance energy where the laser energy matches that of the excited states
of the nanotubes. This resonant enhancement of the Raman intensities is called the resonance
Raman effect. We denote the measurement of the Raman spectral intensity as a function of the
laser energy as the Raman resonance window or the Raman excitation profile. The resonance
window thus defined is normally expressed as the energy width at full-width at half-maximum
(FWHM ) intensity of the Raman intensity in the Raman excitation profile (which is a plot of the
Raman intensity vs. Elaser in units of eV).

1.4.2. Incident and scattered resonance conditions

In the resonance excitation profile for an individual carbon nanotube, sharp maxima in the intensity
are observed when the laser energy Elaser matches an optical transition energy Eii , For a Raman-
active phonon with an energy Eq , we expect two resonance conditions: Elaser = Eii and Elaser =
Eii + Eq (for Stokes), which we call the incident and scattered resonance conditions, respectively.
In the incident resonance condition, the initial photon absorption becomes a “real” absorption
process, while in the scattered resonance condition, the final photon emission becomes a “real”
emission process. For all Raman-active phonons, the resonance energy for the incident resonance
is commonly Eii , while the resonance energies for the scattered phonons depend on Eq . When
the resonance window width in the Raman excitation profile is larger than Eq , we can see only
one resonance peak. In the case of carbon nano-materials, a typical resonance window width is
50 meV and thus we can see two distinct peaks for optical phonons (0.2 eV), but not for RBM
phonons (10–20 meV) in the case of SWNTs. It is important to note that the scattered resonance
condition for the anti-Stokes Raman process is given by Elaser = Eii − Eq .

1.4.3. Stokes and anti-Stokes Raman processes

The Stokes (S) and anti-Stokes (aS) processes for phonon scattering exhibit different Raman
intensity behaviors from each other because the phonon number in the Stokes process is increased
from n to n + 1 and this phonon creation process can always be carried out. However in the
anti-Stokes process, a phonon is annihilated so that for the process to progress, phonons must be
present to be annihilated and this may not be possible if the phonon energy is large compared
to kB T where kB is the Boltzmann constant. The average number of available phonons n at a
430 R. Saito et al.

temperature T with energy Eq is given by the Bose–Einstein distribution function

1
n= . (2)
eEq /kB T − 1
Of particular significance, room temperature (300 K) corresponds to 25.85 meV = 208.5 cm−1 .
For carbon nanotubes the energies of the RBM phonons are comparable in energy to room tem-
perature while the G-band and the D-band phonons are of much higher energies than the room
temperature energy. Of particular significance is the relative magnitude of Eq relative to kB T . Most
Raman spectra are conveniently taken at room temperature (300 K),
As stated above, the probability for the S and aS processes differs because in the Stokes process
the system goes from n phonons to n + 1, while in the anti-Stokes process (aS) the system goes
from n + 1 to n. Using time reversal symmetry, the matrix elements for the transition n → n + 1
(S) and n + 1 → n (aS) are the same, and the intensity ratio between the S and aS signals IS /IaS
from one given phonon can be obtained by
IS n+1
∝ = eEq /kB T . (3)
IaS n
When we use Equation (3) for evaluating Eq or T , we must be careful not to be close to the resonance
conditions for either the S or aS process. If Elaser corresponds to the resonance condition for
Elaser = Eii for the incident photon, then all phonons will be in resonance in the Raman scattering
process. But for Elaser = Eii ± Eq = ES only the scattered light is in resonance and although each
phonon will be in resonance in the Raman spectrum of the nanotube, each phonon will be in
resonance at a different ES value.

1.4.4. The Raman spectra and spectral width: Lorentzian lineshape

A Raman spectrum is a plot of the scattered intensity IS as a function of Elaser − ES (Raman shift,
see Figure 7), and the energy conservation relation given by Equation (1) is a very important
aspect of Raman spectroscopy. The Raman spectra will show peaks at a phonon energy ±Eq , and
in Figure 7 the energy of the Stokes process is shown at positive energy, while the anti-Stokes
process is shown at negative energy in Figure 7. Thus, in the spectrometer (grating) which divides
the scattered light into different directions, the anti-Stokes signal appears in the opposite position
relative to the Stokes signal when measured from the central Rayleigh signal.
The Raman lineshape contains a wealth of information about the electrons and phonons for the
various sp2 nanocarbon systems. The Raman spectrum for semiconducting tubes exhibits a peak
intensity I(ω) at a phonon energy of Eq displaced from Elaser . An expression for the frequency
dependence of the phonon excitation can be obtained using a model based on a harmonic oscillator
damped by some other interactions (similar to a mass-spring system inside a liquid). Therefore,
the shape of the Raman peak will be the response of a damped harmonic oscillator with an
eigenfrequency ωq that is forced by an external field oscillating with a frequency ω. Considering
the damping frequency to be given by q , the intensity I(ω) of a forced damped harmonic oscillator
model of the phonon excitation is a Lorentzian curve
I0 1
I(ω) = (4)
πq (ω − ωq )2 + q2

in the limit where the frequency ωq q , where q is the damping term.7

The full width at half maximum intensity (Raman spectral width) is given by FWHM= 2q .
The center of the Lorentzian lineshape gives the natural phonon frequency ωq , and q is related to
 Advances in Physics 431

Figure 7. Schematics showing the Rayleigh line (at 0 cm−1 ) and the Raman spectrum. The Rayleigh intensity
is always much stronger and it has to be filtered out for any meaningful Raman experiment. The Stokes process
(positive frequency peaks) are usually stronger than the anti-Stokes process (negative frequency peaks) due to
phonon creation/annihilation statistics. Reprinted from Carbon, 48(5), M.M. Lucchese et al. pp. 1592–1597
[3]. Copyright © (2010) Elsevier.

the damping or the energy uncertainty or the phonon lifetime. The damping of the amplitude (as
characterized by q ) is observed as Elaser is tuned (and thus as the scattered light energy is varied).
The damping of I(ω) thus provides information on the phonon lifetime, t. The uncertainty
principle Et ∼  gives an uncertainty in the value of the phonon energy, as measured in the
Raman spectrum, which corresponds to the spectral FWHM of 2q . Therefore, q is the inverse of
the lifetime for a phonon, and Raman spectra in this way provide information on phonon lifetimes.
There are two origins for the finite phonon lifetime: the anharmonic potential and the el–ph
interaction, each of which are discussed below. This is followed by a further discussion of the
lineshapes I(ω) observed for phonons.
(i) Anharmonic potential
Anharmonicity of the inter-atomic potential for the phonon occurs for large r far from the
potential minimum. In this regime, the wave vector q of the phonon is no longer a good quantum
number and phonon scattering occurs by emitting a phonon (third-order process) or by phonon–
phonon scattering (fourth-order anharmonicity). Anharmonicity gives the main contribution to
the thermal expansion process (third-order process) and to the thermal conductivity (fourth-order
process).
(ii) Electron–phonon interaction
Another possible interaction is the el–ph interaction in which a phonon excites an electron in
the valence band to the conduction band or scatters a photo-excited electron to other unoccupied
states. The former el–ph process works for electrons in the valence band, while the latter el–ph
process works for electrons in excited states. Thus, the origins of the finite lifetime of the phonon
are different from each other for the case of electron and hole excitation, and is one mechanism
for breaking the symmetry between electrons and holes in graphene.

1.4.5. The Breit–Wigner–Fano (BWF) lineshape

In specific cases, the Raman spectra can deviate from the simple Lorentzian lineshape in
Equation (4). One obvious case is when the feature is actually composed of more than one phonon
432 R. Saito et al.

contribution. Then the Raman peak will be a convolution of several Lorentzian peaks, depending
on the frequency and weight of each phonon contribution.
One case of importance occurs when the lattice vibration couples to free electrons, as occurs
in graphene or metallic nanotubes when an el–ph interaction takes place. In this case, additional
line broadening and even distorted (asymmetric) lineshapes can result, and this effect is known as
the Kohn anomaly (KA) [127]. In cases where phonons are coupled to the continuum excitation
spectra of free electrons, the Raman peak may exhibit a so-called Breit–Wigner–Fano (BWF)
lineshape, given by [128,129]

[1 + (ω − ωBWF /qBWF BWF )]2

IBWF (ω) = I0 , (5)
1 + [(ω − ωBWF / BWF )]2

where 1/qBWF is a measure of the interaction of a discrete level (the phonon) with a continuum
of states (the electrons). Here ωBWF is the BWF peak frequency at the maximum intensity I0 , and
BWF is the frequency half width-half maximum for the intensity profile of the BWF peak. Such
effects are observed in certain metallic sp2 carbon materials and are discussed later, in connection
with metallic carbon nanotubes.

1.4.6. The effect of defects on spectral broadening

In a perfect system, the one-phonon Raman intensity I0 (ω) associated with a vibrational mode of
wavevector q0 and frequency ω(q0 ) is well described by a Lorentzian function

1
I0 (ω) ∝ , (6)
[ω − ω(q0 )]2 + [0 /2]2

as described above. A disordered distribution of point defects as would be produced by ion implan-
tation, however, will scatter phonons and will also add a contribution to the FWHM by coupling
phonons of wavevector q0 to those of wavevector q0 + δq [130]. In the limit of low levels of
disorder, the coupling will be most effective for small δq, so the phonon wave packet in k-space
can be described by a Gaussian function exp[−(q − q0 )2 Lpc 2
/4] centered at q0 and having a width
proportional to 1/Lpc ≈ δq. Therefore, in real space Lpc is a measure of the phonon coherence
length, which should also be a good measure of the average distance between point defects. Then,
the Raman intensity for the disordered graphene I(ω) can be written as [130–134]
 
 W (q) exp −(q − q0 )2 Lpc
2
/4
I(ω) ∝ d2q , (7)
BZ [ω − ω(q)]2 + [0 /2]2

where the integral is taken over the 2D BZ of graphene and W (q) is a weighting function that
describes the wavevector dependence of the el–ph coupling for the Raman process.

1.4.7. The Raman resonance window

Here, we consider the Raman excitation profile or the Raman resonance window, including the
width of the resonance window γr . Both the lineshape and the width of the resonance window
are considered when obtaining information from the excitation profile. In principle, the Raman
intensity I(Elaser ) for a 1D or zero-dimensional carbon sp2 system shows a dependence on the
laser energy Elaser in the Raman excitation profile associated with resonances of Elaser with the
 Advances in Physics 433

incident Eii and scattered light Eii ± Eq according to

 2
 A 
I(Elaser ) =   , (8)
(Elaser − Eii − iγr )(Elaser − (Eii ± Eq ) − iγr ) 

where the resonant lineshape I(Elaser ) consists of two peaks at Elaser = Eii (incident resonance
condition) and Elaser = Eii ± Eq (the scattered resonance condition). Here γr is the FWHM width
discussed below. Experimentally it is not always possible to resolve the observed I(Elaser ) lineshape
into two peaks.
The FWHM width of each peak in the Raman excitation profile is the resonance window width
γr , and is related to the lifetime of an electron in its excited states. If the lifetime of the photo-
excited electron is finite, then the resonance condition in the Raman excitation profile may show
departures from Equation (8) which assumes γr Eq . The finite lifetime of the photo-excited
electron is subject to the uncertainty relation Et ∼ .
The photo-excited carrier can be relaxed from the excited states by several processes, each
occurring according to different time scales:

• The Coulomb interaction between electrons (10–100 fs).

• The electron–phonon interaction for all possible phonons (with lifetimes < 1 ps).
• The electron–photon interaction (with lifetimes < 1 ns).

When we consider the Coulomb interaction for a given electron, the other electrons should excite
the first electron to an unoccupied state. Thus, the Coulomb interaction depends on the metallicity
of the material. In the case of carbon nanotubes, the interaction between two excitons that is
relevant to this term is known as the Auger process. On the other hand, a photo-excited electron
has a definite lifetime for emitting any energy–momentum conserved phonon. Thus, the el–ph
interaction of the electron from a state k (a photo-excited state) to the energy–momentum conserved
k + q (phonon emitting electron state) is important. Note that γr (the resonance window width,
Eq. (8)) is physically different from q (the Raman spectral width, Eq. (4)).

1.5. Raman measurements of low-dimensional materials

The resonance Raman effect is especially important in low-dimensional nano-systems since the
density of states becomes singular (because of the presence of van Hove singularities in 1D
systems, see Figure 8) and the spectral energies become discrete. The latter effect is important
for decreasing the resonance window width since the number of energy–momentum conserved
intermediate states becomes small and thus the corresponding lifetime of the photo-excited carrier
becomes long. These are the reasons why we can observe Raman spectra even from a single
molecule or, for the case of carbon sp2 nanostructures, we can even see Raman spectra from an
individual carbon nanotube.

Figure 8. Typical electronic density of states for 3D, 2D, 1D and 0D systems.
434 R. Saito et al.

1.5.1. Cutting lines and van Hove singularities of the density of states
When the 2D sheets of graphene are rolled up to form 1D nanotubes, different subbands in the 1D
reciprocal space of the nanotube can be extended into the 2D reciprocal space of a single sheet of
the parent bulk layered material as a set of parallel equi-distant cutting lines [32,135,136]. This
procedure is shown in Figure 9(a) for states near the K point.
Figure 9(b) shows the electronic density of states (DOS) related to the nanotube electronic
band structure plotted schematically in Figure 9(a). Each of N cutting lines in Figure 9(a) (except
for the one that crosses the degenerate K point) gives rise to a local maximum in the DOS g(E) in
Figure 9(b), known as a (1D) van Hove singularity (vHS), given by
N   
2  ∂Eμ (k) −1
g(E) = δ[Eμ (k) − E] dk. (9)
N μ=1 ∂k

The four vHSs in Figure 9(b) are labeled by Ei(v) and Ei(c) for the electronic subbands in the valence
and conduction bands, correspondingly. The presence of vHSs in the DOS of 1D structures makes
these structures behave differently from their related 3D and 2D counterpart materials, as can be
seen in Figure 8. A finite density of states between the first singularities in the valence band and
conduction band for metallic nanotubes is shown in Figure 9(b).

1.5.2. Dimensionality and the resonance Raman effect

The electronic DOS profiles for systems of different dimensionality (3D, 2D, 1D, and 0D) are very
different from one another, as shown in Figure 8. The typical DOS dependence on energy near an

Figure 9. (a) The energy–momentum contours for the valence and conduction bands for a 2D system, with
each band obeying a linear dependence for E(k) and forming a degenerate point K where the valence and
conduction bands touch to define a zero gap semiconductor. The cutting lines of these contours denote the
dispersion relations for the 1D system derived from the 2D system. Each cutting line gives rise to a different
energy subband. The energy extremum Ei for each cutting line at the wave vector ki occurs at a van Hove
(v) (c)
singularity. The energies Ei and Ei for the valence and conduction bands and the corresponding wave
(v) (c)
vectors ki and ki at the van Hove singularities are indicated on the figure by the solid dots. (b) The 1D
density of states (DOS) for the conduction and valence bands in (b) corresponding to the E(k) dispersion
relations for the 1D subbands shown in (a) as thick curves. The DOS shown in (b) is for a metallic 1D system,
because one of the cutting lines in (a) crosses the degenerate Dirac point (the K point in the graphite Brillouin
Zone (BZ)). For a semiconducting 1D system, no cutting line crosses the degenerate point, thus resulting in
(v) (c)
a band gap opening up in the DOS between the van Hove singularities E1 and E1 [135].
 Advances in Physics 435

energy band extremum, g(E) is given by g ∝ (E − E0 )[(D/2)−1] , where D is an integer, denoting the
spatial dimension and D assumes the values 1, 2, and 3, respectively, for 1D, 2D, and 3D systems
[137]. The parameter E0 appearing in the density of states g(E) denotes the energy band minimum
(or maximum) for the conduction (valence) energy bands. For a 1D system, E0 would correspond
to the energy of a vHS in the DOS occurring at each subband edge, where the magnitude of the
DOS becomes very large. One can see from Figure 8 that 1D systems exhibit DOS profiles which
have some similarity to the case of 0D systems, with both 0D and 1D systems having very sharp
maxima at certain energies, in contrast to the DOS profiles for 2D and 3D systems, which show
a more monotonic increase with energy (see Figure 8). However, the 1D DOS is different from
the 0D DOS (δ function at each discrete energy level) in that the 1D DOS has a sharp threshold
and a decaying tail for each cutting line, so that the 1D DOS does not go to zero between the
sharp maxima, as the 0D DOS does (see Figure 8). This is even true for semiconducting nanotubes
which have a finite band gap and no occupied states between the first cutting lines in the valence
and conduction bands. The extremely high values of the DOS at the vHSs allow us to observe
physical phenomena for individual 1D nanostructures in various experiments, as discussed in
Section 1.1.6.

1.5.3. Coherence time and length in Raman processes

It is not trivial to define whether a real system is large enough to be considered as being effectively
infinite and therefore to exhibit a quasi-continuous phonon (or electron) energy dispersion relation.
Whether or not an explicit dispersion relation can be defined indeed depends on the process that is
under evaluation and the characteristics of this process. In the Raman process, we ask how long does
it take for an electron excited by the incident photon to decay? Considering this Raman scattering
time, what is the distance probed by an electron wave function? These issues are discussed in
condensed matter physics textbooks [119] under the concept of coherence. The coherence time
is the time the electron takes to experience an event such as a scattering process that changes its
state. Thus, the coherence length is the distance over which the electron maintains its quantum
state identity and its phase coherence. The coherence length is defined by the electron speed and
the coherence time, both of which can be measured experimentally. The Raman process is an
extremely fast process, and is in the range of femto-seconds (10−15 s). Considering the speed of
electrons in graphite and graphene (106 m/s), this electron speed gives a coherence length of the
order of nm. Interestingly, this number is much smaller than the wavelength of visible light. On the
other hand, this is a particle picture for the scattering process and consideration of both the particle
and wave aspects of electrons and phonons (as well as excitonic effects) are important for carbon
sp2 nanostructures. The study of such concepts is actually very interesting and important when
dealing with local processes induced by defects, as discussed later (Section 4.3) in this article.

2. Experimental progress of Raman spectroscopy and related optics

As shown in Figure 10, the use of photo-physical techniques has contributed a great deal to
our understanding of carbon nanostructures because of the large amount of information photo-
physical techniques have provided and the relative simplicity of many of these experimental
techniques. The resonance Raman effect and the strong el–ph coupling of carbon nanotubes and
graphene, together with the simplicity of sample preparation required for Raman experiments
when compared with many other techniques, elevate Raman spectroscopy to a special position.
Raman spectrometers are broadly available and generally easy to use. For this reason, Raman
spectroscopy is the most common sample characterization tool used by groups working with
carbon nanotubes and graphene.
436 R. Saito et al.

Figure 10. Qualitative comparative evaluation of the amount of information vs. the simplicity of performing
an optical experiment in SWNTs based on the experience of the authors. The position of each technique in the
plot is defined both by phyiscal limitations (e.g. photoluminescence is not available from metallic SWNTs)
and by aspects of practical implementation.

In Section 2, we review a number of optical and spectroscopic techniques that are used for
characterizing materials and especially focusing on sp2 carbon materials, carbon nanotubes and
graphene. We start this section with a brief introduction to the special properties of electrons and
phonons in graphene.

2.1. Electrons and phonons in graphene

In order to analyze optical phenomena and Raman spectra of sp2 carbons, we focus on the electronic
and vibrational structure of graphene, the mother material of sp2 carbons. In this section, we review
the geometrical structure, the electronic structure and the phonon structure of graphene.

2.1.1. The hexagonal crystal structure of graphene

Mono-layer graphene (1-LG) is a single atomic layer of graphite in which carbon atoms crystallize
into a hexagonal lattice (see Figure 2). The unit cell of the hexagonal lattice is a rhombus which
consists of two distinct carbon atoms A and B (see Figure 2(a)). The first BZ of graphene shown
in Figure 2(e) also has a hexagonal periodicity in k space.

2.1.2. The electronic structure and optical transitions

The carbon valence electrons of graphene have 2pz orbitals. This orbital is elongated in the direc-
tion perpendicular to the C–C bond and therefore is called a π orbital.8 A simple tight-binding
calculation for the two π orbitals for the A and B atoms in the unit cell gives two π energy bands.
The electron-occupied band is called the π band and the electron-unoccupied band is called the
π ∗ band. Of particular interest is the fact that the two π energy bands touch each other at the K
and K  points which are the two non-equivalent hexagonal corners of the BZ where the Fermi
energy for undoped graphene is located. What is special about the π bands in graphene is that the
energy dispersion of both π bands near the K and K  points have the same linear E(k) relation.
Thus, the corresponding electron and hole effective masses at the K and K  points become zero,
 Advances in Physics 437

and increase as we move away from the K and K  points. The anomalous and symmetric behavior
of the electrons and holes in graphene mainly originates from this unusual linear E(k) energy
dispersion relation of graphene near the Fermi energy, EF .
Electronic transitions occur from the electron-occupied π band to the electron-unoccupied
π ∗ energy band. Selection rules forbid intra-atomic transitions from 2p to 2p states. However,
inter-atomic transitions from 2p to 2p states are allowed both for the nearest-neighbor pairs of
carbon atoms and for further neighbors as well. Since the energy dispersion is linear near the K
and K  points, the electronic dispersion in graphene forms Dirac cones. Optical transitions for a
given laser excitation energy Elaser occur on the equi-energy lines around the K and K  points.
Because of the three-fold symmetry of E(k) around the K or K  points, the equi-energy lines are
distorted into a triangle with increasing energy [32,138]. This trigonal warping effect of the energy
dispersion and of the Fermi surface dominate the electronic properties.

2.1.3. Phonons and the el–ph interaction in graphene

The two carbon atoms in the graphene unit cell result in six phonon modes and six energy disper-
sions in the unit cell [1]. Three of the six phonon modes are acoustic (A) phonon modes whose
phonon energy is zero at the zone center of the BZ (the  point). The other three phonon energy
bands correspond to optical (O) phonon modes in which the vibration directions of the A and B
atoms are anti-symmetric with respect to each other. The three A (or O) phonon modes consist
of one longitudinal (LA or LO) mode and two tangential (TA or TO) phonon modes. The two
tangential phonon modes (TA or TO) consist of one in-plane (iTA and iTO) and one out-of-plane
(oTA and oTO) phonon mode. Since the L phonon modes of graphene are always in-plane phonon
modes, the i label for the LA and LO phonon modes is generally suppressed. Hexagonal symmetry
requires a degeneracy for the LO and iTO phonon modes at the  point and for the LA and LO
modes at the K points.
The electron motion of the A and B carbon atoms is perturbed by the atomic vibrations
which cause an electron to scatter from a k state to a k + q state by emitting a −q phonon.
In Raman spectroscopy, we generally consider the el–ph interaction for a photo-excited elec-
tron within a π ∗ energy band. There is thus a restriction imposed on the scattering event that
the final state must be unoccupied. Energy and momentum conservation applies to the scattering
of electrons by emitting a phonon, resulting in a maximum energy for E(k) − E(k + q) which
corresponds to the maximum phonon energy (0.2 eV for graphene). The scattering of electrons
only occurs within circles of equi-energy which specify the allowed optical transitions. Since
we have two Dirac cones which are near the K and K  points, this restriction does not mean
that the momentum transfer q is restricted to lie within the equi-energy circle. Electron scatter-
ing from the K to K  (or from the K  to K) regions is possible, for energy differences between
the incident and scattered photons up to 0.2 eV, and we call these processes inter-valley scat-
tering processes [139]. On the other hand, scattering within each of the K or the K  regions
can also occur and this type of scattering is called an intra-valley scattering event. Another
important concept for electron and phonon scattering is the distinction between forward and
backward scattering. The velocity of the electron can change its sign upon scattering, since
its group velocity is given by ∂E(k)/∂k. If the scattered states are on the same (another) side
of a Dirac cone, then the scattering event is classified as forward (backward) scattering. Con-
sidering the six phonon modes, the two types of valley scattering processes, and both forward
and backward scattering processes, there are 24 different possible types of scattering processes
that are relevant to the el–ph interaction and all of these 24 scattering processes are discussed
below.
438 R. Saito et al.

2.2. Electrons and phonons in 1D carbon nanostructures

2.2.1. 1D carbon structures
Carbon nanotubes [32] and graphene nanoribbons [140] are 1D sp2 carbon nanostructures based on
graphene and their physical properties can be discussed by zone-folding the 2D dispersion relations
of graphene onto the 1D BZs of carbon nanotubes and graphene nanoribbons both of which have
discrete wave vectors which are specified in the quantum confined direction. Both carbon nanotubes
and graphene nanoribbons have a 1D unit cell for which translational symmetry exists, respectively,
along the direction of the nanotube axis and the ribbon axis. The quantum confinement of electrons
and phonons in the circumferential direction of nanotubes or along the ribbon width direction of
nanoribbons applies to their discrete wavelengths and discrete wavevectors, which are expressed
by the cutting lines discussed in Section 1.5. The symmetries of the 1D unit cells are given by point
group theory which specifies the possible Raman-active modes and the possible dipole-allowed
optical transitions [141,142].

2.2.2. π bands and phonons in carbon nanotubes and nanoribbons

The electronic dispersion relations for the π (π ∗ ) bands of carbon nanotubes and graphene ribbons
consists of 2N energy subbands, where 2N denotes the number of carbon atoms in the 1D unit cell.
Each energy subband is obtained from the 2D energy bands of graphene by combining the pertinent
allowed 1D wave vectors on a cutting line in the 2D BZ of graphene. For zigzag graphene ribbons,
special electronic states, called edge states appear for which the amplitude of the wavefunction is
localized at one zigzag edge and on one sublattice (A or B) of graphene [140], and these zigzag edge
states have a very high density of electronic states at the Fermi level [143]. The density of states for
zigzag nanotubes including the spin–orbit interaction was calculated by Son and Louie [143,144].
The phonon dispersion of carbon nanotubes and graphene nanoribbons similarly contain 2N
energy subbands. Most phonon subbands are also obtained by combining the zone-folding of the
2D phonon dispersion relations and using appropriate cutting lines. However, in the case of carbon
nanotubes, special phonon modes which do not appear in graphene must be introduced to describe
carbon nanotubes. These additional phonon modes are the RBM and the twist modes (TM), in
which the carbon atoms are vibrating along the radial direction and along specific directions with
respect to the nanotube axis [32].

2.2.3. Optical transitions

Special to 1D systems is the quantum confinement of discrete k vectors associated with the van
Hove singularities in the 1D DOS for the top and bottom of each energy subband of the π bands
(see Figure 9). When the light is polarized along the nanotube axis or along the graphene ribbon
axis, optical transitions occur between π and π ∗ energy bands at the same cutting line for the
initial and final state and a strong absorption occurs from the top of the π bands to the bottom
of the π ∗ band at the wavevector kii . For 1D systems, the kii vector is special because the joint
density of states (JDOS) for the ith π band and the ith π ∗ band become singular at the energy
separation of Eii . For light perpendicular to the nanotube axis, optical transitions occur between
nearest-neighbor cutting lines with energies Ei,i+1 because the polarized light in this case has
an additional quantum unit of angular momentum. Thus, different selection rules apply to light
polarized parallel (Eii ) or perpendicular (Ei,i+1 , Ei,i−1 ) to the nanotube or graphene ribbon axes.

2.2.4. Type I and II semiconductor nanotubes

In discussing optical transitions, it is important to classify (n, m) semiconducting nanotubes into
two categories depending on whether mod(2n + m, 3) = 1 or 2, which we call Types I or II
 Advances in Physics 439

Figure 11. Cutting lines near the K point in the 2D BZ of graphene for (a) Type 0 (or Mod 0) metallic
SWNTs, (b) Type I (Mod 2) and (c) Type II (Mod 1) semiconducting SWNTs.

semiconducting nanotubes (S-SWNTs), respectively.9 In Figure 11, we show the geometry of the
cutting lines near the K point in the 2D BZ of graphene. In the case of type I (II) semiconducting
nanotubes, the K point lies at the one-third (two-thirds) position between two cutting lines, since
the distance of the K point from the cutting line at the  point is given by (2n + m)/3 times K1 ,
where K1 is a reciprocal lattice vector in the direction of the chiral vector [32]. In Figure 11(a),
we see that mod(2n + m, 3) = 0 corresponding to metallic nanotubes. Each cutting line results in
an optical transition for its Eii value. Further discussion of this classification is given in [107,135,
136,141,145,146]. Because of the anisotropy of the effective mass tensor around the K point, the
optical properties of S-SWNTs depend on their Type I or II classification (see Figure 11(b) and
(c)). We call this classification the semiconductor type-dependence.1

2.2.5. 1D exciton, exciton–photon and exciton–phonon interactions

For a pure 1D electron and hole pair, such as that occurs for a carbon nanotube or a graphene
ribbon, the exciton binding energy becomes singular (∞ energy) for the lowest energy state.
For a nanotube or a nano-ribbon, however, the binding energy of these nanostructures is also
finite, since the nanotube circumference or nanoribbon width is finite. Though finite, the bind-
ing energy is still very large for a 1 nm diameter nanotube or a nanoribbon of 1 nm width,
compared with the room temperature thermal energy of 25 meV. Thus, understanding excitonic
electron and hole binding is essential for understanding the optical properties of nanotubes at room
temperature.
Since the exciton is localized in real space, the exciton wavefunction is given by a linear
combination of many wave vector states near the kii points, whose mixing coefficients are given
by the Coulomb interaction between the photo-excited electron and the valence electrons (elec-
tron self-energy) or between a photo-excited electron and hole (exciton-binding energy). These
exciton wavefunctions for graphene ribbons or nanotubes are calculated by the Bethe–Salpeter
equation [120,147,148]. Because of the highly localized wavefunction of an exciton, the intensity
of the excitonic optical transition (absorption and emission) intensity is enhanced significantly
(10,000 times) relative to ordinary optical excitations between conventional band states. On the
other hand, since the el–ph interaction strongly depends on the wavevectors k near the Fermi
energy, the exciton–photon interaction energy (which is given by the integration of the el–ph
interaction over k weighted by the exciton wavefunctions) shows a strong (n, m) dependence.
In fact, the value of the exciton–phonon interaction is of the same order of magnitude as the
el–ph interaction in these 1D systems. The observation of excitons in carbon nanotubes was
first reported by the Heinz group at Columbia [44,121] and by the Reich–Thomsen group in
Berlin [122].
440 R. Saito et al.

2.3. The optical measurement techniques

In this section, we mention the various measurement techniques which have been used to study the
photophysics of carbon nanotubes, graphene and graphene nanoribbons. In general, the studies
on carbon nanotubes are more developed, while graphene and graphene nanoribbons are still in
an early stage. Some techniques are found to be more sensitive for certain systems and others
may be more sensitive for the specific applications that a particular system might be used for.
It is also important to know which photophysical techniques have been or could be used for
spectroscopic studies and for making related measurements on graphene, graphene nanoribbons,
carbon nanotubes and related sp2 nano-carbon systems. These various techniques are briefly
reviewed in this section.

2.3.1. Light absorption

Light absorption is one of the most fundamental photophysical measurement techniques. A typical
absorption spectrum is shown in Figure 12(a). This spectrum compares the optical absorption
spectra in 1D SWNTs to that in colloidal 3D graphite. Above 2.0 eV, the spectrum of the two
carbon forms is similar. At lower energies, however, the SWNT sample shows three broad peaks,
A, B and C in Figure 12(a), associated with the two lowest allowed transitions in semiconducting
S S
SWNTs (A, E11 and B, E22 shown in Figure 12(b)) and the first peak in the joint density of states
M
(JDOS) of the metallic nanotubes (C, E11 shown in Figure 12(b)). Measurements on isolated
SWNTs show sharper transitions than for SWNT bundles, and each sharp transition is related to
specific van Hove singularities, relevant to each SWNT. These transitions become merged and
broadened in samples of nanotube bundles.
Graphene, being a 2D system, has no such van Hove singularities10 and the optical absorption
of 2D graphene looks more similar to that of 3D graphite [42,149]. However, for a given optical
frequency, the optical absorption of graphene depends on the number of graphene layers. The
amount of light transmitted through an N-layer graphene sample only depends on N and on
fundamental constants. Experiments have confirmed the theoretically expected universal value
πe2 /hc for the optical conductivity of graphene [74] (or a 2.3% optical absorption per graphene
layer). Such experiments have also revealed departures of the quasi-particle dynamics from the

Figure 12. (a) The optical absorption spectra from SWNTs and from colloidal graphite shown comparatively.
(b) Illustration of interband processes giving rise to the optical absorption peaks in semiconducting and
metallic SWNTs [42].
 Advances in Physics 441

simple predictions for Dirac Fermions in idealized graphene as we move sufficiently far from the
Dirac point in k space.

2.3.2. Resonance Rayleigh scattering

Rayleigh scattering refers to the elastic scattering of light and the intensity of the Rayleigh scattered
light intensity in graphene depends on the number of graphene layers and this technique can be
used for determining the number of layers in a given few layer graphene sample. The Rayleigh
scattering intensity is stronger when the laser is in resonance with an optical transition energy in
which case the process is called resonance Rayleigh scattering. An important use of resonance
Raleigh scattering is to directly measure the Eii values of SWNTs when using the laser light source
whose frequency can be tuned. Figure 13(a) shows a schematic representation of the dark-field
configuration which is used for this type of Rayleigh scattering spectroscopy. This geometry is used
to efficiently reject the incident laser light so that the nearby Rayleigh signal can be clearly seen.
One main challenge of Rayleigh scattering experiments comes from sample preparation. In this
application, isolated SWNTs are prepared by chemical vapor deposition (CVD) and a substrate
with a slit etched in it is used, allowing exploitation of the dark-field configuration shown in
Figure 13(b). In the experiment of Figure 13(a), typical slit widths are tens of micrometers and
slit lengths are up to 1 mm [44,150].
Since the pioneering work of the Heinz group [150], Rayleigh scattering is becoming increas-
ingly popular as a method for the rapid identification of the (n, m) indices of individual SWNTs.
Figure 14 shows the Rayleigh spectrum of different (n, m) SWNTs for the type of sample shown in
Figure 13(b). The (n, m) of individual SWNTs are identified by comparing the observed resonance
peak frequencies with the expected theoretical values, anchored on some geometrical constraints
that are discussed in more detail from a theoretical standpoint in Section 3. For example, in
Figure 14(a) the (16,11) type I S-SWNT has a similar electronic structure to the (15,10) type I
SWNT, and the resonance Rayleigh spectra of both are displayed together in Figure 14(a). The
diameter of the (16,11) nanotube is 1.83 nm, which is 0.12 nm greater than the 1.71 nm diameter

Figure 13. (a) Schematic diagram for a Rayleigh scattering measurement. Microscope objectives focus the
incident light on a suspended nanotube and collect the radiation of the scattered light. Using a super-continuum
source, different wavelengths can be detected simultaneously using a spectrometer and a multichannel (CCD)
camera. (b) Electron micrograph of an individual suspended SWNT in a geometry used for Rayleigh scattering
spectroscopy. Adapted from Carbon Nanotubes: Advanced Topics in the Synthesis, Structure, Properties and
Applications, T.F. Heinz [44]. Copyright (2008) from Springer.
442 R. Saito et al.

Figure 14. The Rayleigh scattering spectra of two different (n, m) semiconducting SWNTs shown compar-
atively for type I(a), type II(b) semiconducting SWNTs. The peaks in both cases correspond to the ES33 and
ES44 interband transitions. The comparison in (a) corresponds to two nanotubes of the same (2n + m)mod3
type, but having different diameters; the comparison in (b) corresponds to two nanotubes of different type,
but with similar SWNT diameters. Adapted from M.Y. Sfeir et al. Science, 312, pp. 554–556, 2006, [150].
Adapted with permission from AAAS.

S S
of the (15,10) SWNT. As a result, a downshift of about 150 meV in the E33 and E44 transitions of
S S
the larger diameter nanotube is observed. The ratio of the E44 to E33 transition energies is similar
for these two nanotubes which are both of type I S-SWNT. However, a comparison of the (13,12)
type II S-SWNT with the (15,10) type I S-SWNT (Figure 14(b)) shows a different behavior for
the two different types of S-SWNTs. In this case, the average energies of the two transitions of
the (15,10) and (13,12) nanotubes are very similar due to their nearly identical diameters (1.71 nm
and 1.70 nm, respectively). However, the difference in behavior is manifested in the dissimilar
S S
intensity ratios for their E44 to E33 transitions. Rayleigh scattering measurements have been espe-
cially useful for advancing the theory related to optical transition energies in SWNTs and how
environmental effects influence these values [44]. Rayleigh scattering experiments have also been
carried out in graphene by Casiraghi [151].

2.3.3. Photoluminescence excitation spectra

The photoluminescence technique has become an especially popular method for characterizing
semiconducting SWNTs [152,153]. The photoluminescence excitation (PLE) spectra conveniently
characterize S-SWNTs by measuring their emission spectra and the technique is limited to semi-
conducting nanotubes. The PLE spectrum thus plots the light emission as a function of the laser
S
excitation energy of the light and most experiments focus on measuring E11 rather than higher
EiiS with PLE [152,153]. Here, we show a PLE spectrum for a semiconducting-(6,5)-enriched
SWNT sample, illustrated in Figure 15 [154], in order to show several light scattering peaks
(highlighted by circles in Figure 15(b)), which also include Raman scattering events in addition to
PLE. The vertical gray band in Figure 15(a) denotes photoluminescence emission at the band gap
EPL = E11 = 1.26 eV. The horizontal gray bands denote nearly continuous-luminescence emis-
sion bands associated with thermally excited processes involving different phonon branches for the
(6,5) SWNT. The cutoff energy at 1.06 eV is marked in Figure 15(b) by a vertical dotted line which
corresponds to the maximum phonon energy (1.26 − 1.06 = 0.20 eV) in the first-order phonon
spectra. Slanted dotted lines, which correspond to Eex − Eem = constant, denote emission from
resonance Raman spectra (RRS) for G-band, M-band and G -band phonons. Note that the strong
 Advances in Physics 443

Figure 15. (a) A 2D excitation vs. emission contour map for a dried (6, 5)-enriched DNA-CNT sample on a
sapphire substrate. The spectral intensity is plotted using the log scale shown on the right. (b) A schematic
view of the various observed light emissions plotted as the laser excitation energy vs. photon emission energy.
Reproduced figure with permission from S.G. Chou et al. Physical Review Letters 94, p. 127402, 2005
[154]. Copyright © (2005) by the American Physical Society.

emission spots, appearing in Figure 15(a) at E11 and denoted by circles, correspond to energies
where these bands cross the E11 transition energy. These intersection points are associated with a
mixture of PL and RRS processes involving one-phonon (VI) and two-phonon (I–V) processes,
following the labels in Figure 15(b). The various features in Figure 15 differ in linewidth. The
Raman peaks are much sharper (tens of cm−1 ) than the PL peaks which have linewidths of hun-
dreds of cm−1 . The various peaks also differ by the fact that, when changing the excitation laser
energy, the PL emission is fixed at E11 , while the Raman peaks change in frequency, keeping fixed
the energy shift from Elaser . With light emission occurring at the same energy, the PL and RRS
processes are sometimes confused in the literature, and the major reason is that RRS in solids often
has a much greater (typically 103 times larger) intensity than the non-resonance Raman spectra
[155]. To differentiate between the RRS and PL processes, one can just look at what happens to
the spectral output when changing Elaser .
The difference in linewidth between RRS and PL arises because in Raman scattering the
intermediate states that are excited between the initial state (incident photon plus the energy of
the system before light absorption) and the final state (emitted photon plus the energy of the
system after light emission) are “virtual” states.11 These virtual states do not have to correspond
to real states (and do not have to be eigenstates of the physical “system”) – any optical excitation
frequency will, in principle, suffice in RRS. In photoluminescence, on the other hand, the optically
excited state must be a real state of the system, and PL involves a real absorption of light at one
frequency, followed by a real emission of light at a different frequency.12 Photoluminescence is a
technique commonly used to study graphene samples engineered for an energy gap opening but
studies of energy gap opening are still at an early stage.

2.3.4. Electro-luminescence
Light emission by excited carriers can also be obtained by means other than photo-excitation, such
as electro-luminescence (EL). In EL the electrically induced excitation is followed by ambipolar
e–h recombination which comes from electrons and holes that are injected independently by using
doped semiconductor electrodes and impact excitation occurs by hot carriers. These effects have
444 R. Saito et al.

largely been studied in SWNTs [47,156,157], where both radiative decay of photo-excited and
electron-excited emission occur, as well as the non-radiative decay to create free carriers which can
then be studied by their photoconductivity spectra. EL processes could lead to the technological
use of carbon nanotubes as nanometer scale light sources and as photo-current or photo-voltage
detectors. EL has also been observed in graphene [157].

2.3.5. Infrared absorption spectroscopy

IR absorption spectroscopy has especially been useful for characterizing nanotubes [158–160]. In
the case of graphene the IR absorption per layer of graphene is 2.3% and is given by the simple
formula [74]
e2
πα = π = 2.293% (10)
c
and for this reason can be used to determine the number of graphene layers in a few-layer graphene
sample. IR spectroscopy typically is useful for studying the symmetry of the optical phonon modes.
When a particular phonon amplitude has the same symmetry as a vector, the corresponding
phonon can directly absorb (or emit) a photon. Such phonon modes are called IR-active phonon
modes. If the unit cell of a solid (or a molecule) has inversion symmetry, then the phonon modes
can further be classified as either even and odd functions of the inversion operator, corresponding
to Raman and IR-active phonon modes, respectively.13 Thus, by using polarized IR absorption
spectroscopy for such materials, we can get independent information on the symmetry of specific
phonons. For example, achiral nanotubes (armchair or zigzag nanotubes) have inversion symmetry
along the nanotube axis. Since carbon materials tend to absorb light readily, IR absorption is
typically measured by using reflected light in the back-scattered geometry. Complementary IR
and Raman studies on sp2 carbon materials have been performed [159,160] and used to study the
symmetry of specific phonons in these materials.

2.3.6. Coherent phonon spectroscopy

In this section, we first describe the generation and measurement of coherent phonons (CPs) in
general and then illustrate the use of this measurement technique in the case of carbon nanotubes
[45,161–165]. When the duration of a light pulse is of the order of 10 fs which is smaller than the
period of vibrational modes, then atomic vibrations start at the same time as the el–ph interaction
starts to act on all involved atoms. This means that the amplitude of each phonon vibration has
the same phase for all atoms, and we call this effect the coherent motion of a vibration. When
we now send a second light pulse, we can then observe the vibrations of the atoms by a time-
dependent transparency of the light, for which the mean frequency values correspond to the
experimentally determined phonon frequencies. When we make a Fourier transform of the time-
dependent transparency of materials, sharp peaks appear at the frequencies of the phonon modes
which have strong el–ph matrix elements. The observed results are then similar to those obtained in
Raman spectra. From the observation of the phase of the time-dependent transparency vibration,
we can obtain information on the coupling constant of the el–ph interaction. For example, in
the case of nanotubes, the nanotube can start a RBM phonon vibration by either increasing or
decreasing its diameter, depending on the mod(2n + m, 3) type of the nanotube. With ultra-fast
laser spectroscopy, it is possible to monitor the photo-induced electronic and vibrational dynamics
on a femto-second timescale. In general, time-domain observations of the phonon dynamics require
the generation of CPs, corresponding to in-phase lattice vibrations of an ensemble of the same
species that add up constructively to generate a detectable signal [45,161,164,165]. CPs have been
 Advances in Physics 445

Figure 16. (a) Pump–probe time-delay data taken on a nanotube sample at a central wavelength of 800 nm
in (b). The decay of the pump–probe signal within several picoseconds reflects the decay of the excited-state
population. The inset in (a) is a zoom-in of the data between 0.3 and 6 ps, highlighting the coherent phonon con-
tribution to the signal. (b) CP oscillations excited and measured at five different Elaser excitations (expressed
in terms of their wavelengths). The individual traces for each wavelength are offset for clarity. The slower
decay of the excited-state population has been subtracted in each case. (c) Phonon spectrum detected at a
center wavelength of 765 nm obtained from both resonant Raman scattering (RRS) and from CP measure-
ments in the frequency range of the RBM [45]. Adapted from carbon Nanotubes: Advanced Topics in the
Synthesis, Structure, Properties and Applications, A. Hartschuh [45]. Copyright (2008) from Springer.

observed in a variety of different systems including thin films, semiconducting nanostructures,

fullerenes and other carbon nanostructures.
Figure 16 shows results from coherent oscillations of the RBM coordinate, namely the diameter
of the nanotubes [45]. Upon short-pulse laser excitation, all nanotubes of a certain (n, m) species
within the sample will breathe in phase. By Fourier transforming this signal, the resulting response
is similar to what is obtained in the frequency domain of resonance Raman spectroscopy. Upon
close examination, however, noticeable differences are observed between CP and resonance Raman
spectroscopy (RRS) data that were obtained from the same sample. For example, the spectral width
of the phonon bands in the CP spectra is sharper, while the chirality dependencies of the CP and
RRS signal intensities are different. Namely, in Raman scattering, the signal strength decreases
as (2n + m) increases, whereas the CP signal increases with increasing (2n + m). These different
types of behaviors can be used to distinguish one type of mechanism from the other in fast
optics studies. Furthermore, pump–probe coherent phonon experiments on nanotubes (such as in
Figure 16) also are very useful for finding the lifetime of photo-excited electrons in excited states.

2.4. Raman spectroscopy of sp2 carbons

In this section, a brief introduction to Raman spectroscopy in sp2 carbons is presented, starting
with some historical background and then very briefly mentioning a number of different types of
Raman spectroscopy that are being used to study sp2 carbon materials.

2.4.1. Historical background

Raman spectroscopy has historically played an important role in the study and characterization
of sp2 carbon materials [19,166,167], which are being widely used in the last four decades to
characterize pyrolytic graphite, carbon fibers [19], glassy carbon, pitch-based graphitic foams
[101,168], nanographite ribbons [59], fullerenes [23], carbon nanotubes [32,167] and graphene
[5,97,112]. For sp2 nanocarbons, Raman spectroscopy can give information about crystalline
size, clustering of the sp2 phase within a given sample, the presence of sp3 hybridization and
chemical impurities, its mass density, optical energy gap, elastic constants, doping, defects and
446 R. Saito et al.

other crystal disorder, edge structure, strain, the number of graphene layers, nanotube diameter,
nanotube chirality and nanotube metallic vs. semiconductor behavior [169]. Another important
area where much work has been done is on disordered, amorphous and diamond-like carbons
[19,131,169], as well as graphite and graphene edges [151,170].

2.4.2. Raman spectra of graphite and graphene: G- and G -bands

Figure 4 shows the Raman spectra from different crystalline and disordered sp2 carbon nanostruc-
tures and emphasizes the difference in these spectra from one sp2 carbon material to another. The
first spectrum shown is that for mono-layer graphene – the building block of many sp2 nanocar-
bons. A clear message derived from Figure 4 is that every different sp2 carbon material in this
figure shows a distinct Raman spectrum, which can be used to understand the different properties
that accompany each of these different sp2 carbon structures. For example, 3D highly oriented
pyrolytic graphite (labeled HOPG in Figure 4) shows a distinctly different spectrum from that of
mono-layer graphene (1-LG), which in turn is distinct from the Raman spectra characteristic of
the various few layer-graphene materials, such as for 2-LG and 3-LG (see sect. 4.2) [112].

2.4.3. First-order RBM, G+ and G− Raman spectra of SWNTs

The Raman spectrum for SWNTs is also shown in Figure 4. These spectra show a variety of
features, such as the RBM and the splitting of the G-band into G+ and G− bands. These first-
order Raman features distinguish a SWNT from all other sp2 carbon nanostructures. Carbon
nanotubes are unique materials in many ways. For example, SWNTs exhibit transport properties
that are either metallic (where their valence band and conduction bands touch each other at the
K and K  points in the graphene BZ) or semiconducting (where a band gap, typically of several
hundred meV, separates their valence and conduction bands). The Raman G− spectral feature
differs in lineshape, linewidth and frequency, according to whether the resonant nanotubes are
semiconducting or metallic, as shown in Figure 17 [171] regarding their G− and G+ frequencies.

− +
Figure 17. ωG and ωG for semiconducting (filled circles) and metallic (open circles) SWNTs are plotted
+
as a function of 1/dt . The flat solid line shows ωG = 1591 cm−1 . The curves are given by the function
− −1
ωG = 1591 − C/dt , where C = CS = 47.7 cm nm for semiconducting SWNTs (long dashed curve) and
2 2

C = CM = 79.5 cm−1 nm2 for metallic SWNTs (short dashed curve). Also plotted (open squares) are the data
for the ∼ 1580 cm−1 Lorentzian peak sometimes observed in metallic SWNTs.Adapted with permission from
A. Jorio et al., Physical Review B 65, p. 75414, 2001 [171]. Copyright © (2001) by the American Physical
Society.
 Advances in Physics 447

2.4.4. The defect-induced Raman spectral features: D- and D -bands

The introduction of disorder breaks the crystal symmetry of graphene, 3D graphite and carbon
nanotubes. The presence of disorder activates certain vibrational modes that would otherwise be
silent, such as the D-band and the D -band features and the combination D + G mode, shown in
the spectrum-labeled damaged graphene in Figure [1,3,96,102,130,131,169,174]. The different
types of defects do in fact show their own characteristic Raman spectra, as illustrated in Figure 4
by comparing the spectra-labeled damaged graphene and SWNH (denoting single wall carbon
nanohorns, another nanostructured form of sp2 carbon which may include pentagons with a small
content of sp3 bonding [37]). The topic of distinguishing between the Raman spectra of one
and another type of defective sp2 carbon remains an area for future study. When the disorder
is so dominant that only nearest-neighbor structural correlations are present (labeled amorphous
carbon in Figure 4), broad one-phonon (<1600 cm−1 ) and two-phonon (1600–3200 cm−1 ) Raman
features are seen [3,130]. At larger disorder limits, both sp2 and sp3 bonding might be seen. Some
hydrogen uptake can also occur for such materials to satisfy their dangling bonds [131].

2.5. Laser Raman scattering measurements

In this section, we discuss the various types of Raman scattering studies that are generally carried
out and we briefly describe in more detail how the measurements are done and the information
that is provided by these measurements.

2.5.1. The Raman setup

In order to get a sharp and strong Raman signal, we generally use a laser light source. In the
back-scattering geometry, the scattered light is collected at the same side of the laser spectrum
relative to the position of the sample. Typical observation times to collect the scattered photons
are from 1 s to 10 min. The incident and Rayleigh scattered light is filtered from the scattered
light by a notch filter or some other method. The frequency shift of the Raman scattered light
is measured by a monochromator which splits the observed scattered light as a function of the
shift in the wavenumber of the scattered light relative to that of the incident light. The Raman
intensity of each spectral line is measured by counting the number of photons recorded for that
line and by considering the instrument function of the monochromator and the CCD detector to
get a properly calibrated Raman signal. A typical resolution of a high-quality Raman system is
1 cm−1 . Splittings of the Raman spectra caused by experimental perturbations (such as due to
external fields or applied stress) are directly measured by a CCD camera which counts photons,
measuring frequency shifts, intensities and lineshapes of each pertinent Raman line.

2.5.2. Polarized and micro Raman measurements

Polarization is used to get additional information about crystal symmetry and selection rules.
Symmetry information can be obtained by changing the polarization of the electric field of the
light through use of suitable polarizers. The polarized incident light beam is focused on the sample
and the scattered light is filtered by one polarizer and detected by a second polarizer. By selecting
the propagating direction of the incident and the scattering light relative to the aligned sample and
by selecting the polarization angles of the incident and scattering light beams, the optical dipole
selection rules can be studied. This is the most common method that is used to determine the
symmetry of the relevant phonons.
Micro Raman spectroscopy is a measurement technique in which the laser light of a given
wavelength (or photon energy) is focused on the sample by an optical microscope. Such a system
448 R. Saito et al.

is called a micro-Raman set up since the spatial resolution of the Raman signal is of the order of
one micron. Not only is the incident light focused on the sample, but also the scattered light is
collected by the same optical lens and is split by a half mirror [173]. An isolated graphene flake
or SWNT on a Si substrate is located by putting the substrate on a mobile stage that moves the
sample horizontally, and the mobile stage is controlled by stepping motors.

2.5.3. Confocal Raman spectroscopy and Raman imaging

Confocal Raman spectroscopy makes use of spatial filtering by means of pinholes in the optical
paths, and it can generate spatially resolved Raman emission with a resolution on the order of
λlaser /2 m, where λlaser is the laser wavelength. By scanning the substrate on which an isolated
nanotube or a graphene sample is located, we can get a Raman signal as a function of the position
on the sample, which is generally called Raman imaging (see Figure 18). In the case of nanotubes,
a micro-Raman imaging scan of a substrate can find the location of the nanotubes resonant with a
given laser line. If we instead use near-field spectroscopy (see Sections 2.6.3 and 2.6.4), we can
even get information on the Raman signal as a function of the position of an individual isolated
nanotube with a resolution on the order of 10 to 20 nm. In the case of graphene or graphite, the D-
band imaging technique is also important for determining the distribution and specific location of
defects over the sample, since the D-band feature is not symmetry allowed, but is instead induced
by a symmetry-breaking mechanism such as a defect or an edge [174].

2.5.4. Characterization of the sample edges and the imaging of defects

Since the D-band in sp2 carbons is a feature that is only observed when the crystal symmetry is
broken by a defect or a sample boundary, the D-band intensity can be used to characterize the dis-
order in graphene and carbon nanotubes. This effect is clearly observed in Figure 18 which shows
two confocal Raman images of a 6 μm highly oriented pyrolytic graphite crystallite deposited
on a glass substrate. Figure 18(a) shows a Raman image of the crystalline regions of the sample,
obtained by plotting the spatial dependence of the G-band intensity. In Figure 18(b) a map of the
intensity of the disorder-induced D-band is shown, while Figure 18(c) shows two Raman spectra,
one taken at the interior location of the crystalline region, and the other is taken at the edge. It
is clear from Figure 18(a–c) that the G-band intensity is uniform over the whole graphite sur-
face, while the D-band intensity is localized where the crystalline structure is not perfect, which

Figure 18. Raman spectra imaging of an HOPG micro-crystallite. In (a) the G-band intensity is plotted. In
(b) the D-band intensity is plotted. (c) Spectra 1 and 2 are the spectra at locations 1 and 2 in (b) [174]. This
experiment was performed in the laboratory of Prof. Achim Hartschuh and represent, to our knowledge, the
first Raman imaging of localized defect modes in graphitic materials. M.A. Pimenta et al., Physical Chemistry
Chemical Physics 9, pp.1276–1290, 2007 [174]. Reprinted by permission of the PCCP Owner Societies.
 Advances in Physics 449

occurs mainly at the edges of the crystalline region where symmetry-breaking occurs and D-band
intensity due to the edge discontinuity is also seen.
The intensity of the D-band can also be used to assign the atomic structure of the edge in
graphite and graphene [170], so that it can provide a useful tool to probe the edge chirality of
graphene. In particular, armchair edges have a large matrix element for D-band scattering while
for zigzag edges the matrix element for D-band scattering should vanish, and chiral edges show an
intermediate amount of D-band edge scattering. However, imperfect graphene edges produced by
the mechanical cleavage of graphite can produce ambiguous results that do not clearly discriminate
the armchair and zigzag edges from one another.

2.5.5. Resonance Raman spectroscopy

In the Raman measurement, the resonance Raman effect strongly enhances the Raman intensity
when the energy of either the incident or the scattered light matches the optical transition energy
of a carbon nanotube or of a graphene ribbon. In the case of carbon nanotubes, since the JDOS
becomes singular for an excitonic transition, the resonance Raman effect is especially strong. This
is a major reason why we can get a strong Raman signal even from one nanotube [111]. On the
other hand, if the resonance condition is not satisfied, we cannot get a measurable Raman signal
even if a nanotube intersects with the light beam. This situation makes the search for isolated
nanotubes in a sample very time consuming. Here, Rayleigh scattering is helpful for locating an
individual SWNT and in providing its (n, m) assignment because all the nanotubes in a sample
can thus be identified. However, with Raman spectroscopy we can always get a resonance Raman
signal for bundle samples, but in this case we see only those nanotubes that are in resonance with
Elaser . In the case of graphene which is a 2D system, the DOS is continuous near the Fermi energy
and is not singular, so that any laser energy satisfies the condition for observing Raman spectra.

2.5.6. The Raman excitation profile

Using a continuous laser excitation energy light source, the Raman excitation profile (or resonance
window) of a carbon nanotube is a plot of the Raman intensity measured at the same location on
a sample as a function of laser energy. To determine the resonance energies of a carbon nanotube,
denoted by Eii , it is necessary to find the frequency of the peak in the Raman spectra taken
as a function of the laser excitation frequency (Elaser ), as shown in Figure 19(a). The FWHM
intensity of the Raman excitation profile is defined as the resonance window width (typically
10 meV) which is inversely proportional to the lifetime of the photo-excited carriers (0.1 ps). The
lifetime is defined as the time for the photo-excited electron to emit a phonon when it is in the
excited state. In the case of the G-band, when we scan the laser excitation energy we can see
two peaks satisfying the incoming and outgoing resonance conditions for the incident and the
scattered light, while in the case of the RBM the two peaks for the incident and scattered light
normally overlap with each other, since the resonance window width is larger than the RBM
energy.

2.5.7. The Kataura plot

From the Eii energy measured by the Raman excitation profile, we can plot Eii for each (n, m)
SWNTs as a function of the SWNT diameter, which we call the Kataura plot (Figure 19(b))
[176,177]. From the Kataura plot, we can determine the resonance conditions for all (n, m) SWNTs
which is useful for assigning the (n, m) value to specific SWNTs from actual Raman measurements
[178]. Since the radial breathing mode frequency ωRBM depends on the SWNT diameter [111], we
can plot the Eii energy which is obtained by the resonance Raman excitation profile as a function of
450 R. Saito et al.

Figure 19. (a) RBM Raman measurements of HiPCO SWNTs dispersed in an SDS aqueous solution [152],
measured with 76 different laser lines Elaser [175]. The non-resonance Raman spectrum from a separated
CCl4 solution is acquired after each RBM measurement, and this spectrum is used to calibrate the spec-
tral intensities of each nanotube and to check its frequency calibration. (b) Filled circles are experimental
Eii vs. ωRBM data points obtained by Telg et al. [107] from analysis of an experiment very similar to
the one shown in (a). The label “Transition energy exp” actually indicates the excitation laser energy
(Elaser ) for each data point. Open circles come from third-neighbor tight-binding calculations, showing
that even the addition of interactions with more neighbors in the π-band based tight-binding model is
not enough to accurately describe the experimental results. Gray and black circles indicate the calculated
optical transition energies from semiconducting (E22 S and E S ) and from metallic (E M ) tubes, respec-
33 11
tively. (a) Adapted from M.J. O’Connell et al., Science, 297, p. 593, 2002 [152]. with permission from
AAAS. And adapted with permission from C. Fatini et al., Physical Review Letters 93, p. 147406, 2005
[175]. Copyright © (2005) by the American Physical Society. (b) Reprinted with permission from
H. Telg, et al., Physical Review Letters 93, p. 177401, 2004 [107]. Copyright © (2004) by the American
Physical Society.

ωRBM , and we call this plot an experimental Kataura plot. Figure 19(a) shows a 2D RBM map for
the HiPCO nanotube sample in aqueous solution wrapped by the SDS (sodium dodecyl sulfate)
surfactant [175]. For the construction of the plot in Figure 19(a), 76 different laser lines were
used. By fitting each of the spectra with Lorentzians, (n, m) indices were assigned to the different
SWNTs. Solid circles in Figure 19(b) denote the Eii values obtained experimentally by fitting the
resonance windows extracted from similar data to that shown in Figure 19(a), as compared with
the Eii obtained from tight-binding calculations (open circles) shown in Figure 19(b) [107].

2.6. Other measurement techniques related to Raman spectroscopy

In this section, we briefly describe several experimental techniques that have grown out of Raman
spectroscopy or are related to Raman spectroscopy and are sensitively employed for specific types
of optical studies used to gain unique and detailed information about nanostructures.

2.6.1. Surface-enhanced Raman spectroscopy

When light is incident on a small (nm in diameter) metallic particle, such as a gold or silver
particle, the electric field near the metallic nano-particle is enhanced by several orders of magni-
tude, due to the surface plasmon excitation of the particle. The enhancement of the electric field
 Advances in Physics 451

near the metallic nano-particle is given by the boundary condition for the transmitted electromag-
netic wave on the metallic surface. When the light beam on the sample is located at the metallic
nano-particle, the corresponding Raman signal becomes significantly enhanced (sometimes up to
1010 times that of the normal Raman signal). The surface-enhanced Raman spectroscopy (SERS)
technique was utilized initially to obtain a large enough signal to observe the spectra from an
isolated individual SWNT [128,179–181]. Changes in the symmetry selection rules are observed
due to local symmetry breaking [182]. SERS measurements were carried out on SWNTs before
researchers realized that there was a strong resonance effect occurring in the π -related states in
carbon nanotubes and that this resonant effect was large enough to generate a measurable signal,
without any SERS enhancement effect. In fact the observation of strong SERS spectra stimulated
the first effort to observe the spectrum from one individual carbon nanotube [111]. Recently, the
use of the SERS technique has been applied to graphene [183–186].

2.6.2. Surface and interference enhanced Raman spectroscopy

When we place a multi-layered sample on the top of a substrate, we can enhance the electric field
of the light at the substrate by the interference of the incident light and the multi-reflecting light.
This technique is called interference enhanced Raman spectroscopy and this technique can be
used to get a large Raman signal from a graphene or a graphene nano-ribbon sample. Interference-
enhanced Raman spectroscopy can also be combined with SERS and RRS, in which case the
effect is called surface and interference co-enhanced Raman spectroscopy (SICERS) [184,187].

2.6.3. Near-field enhanced Raman spectroscopy

In order to get higher spatial resolution, an apex with a small aperture and a sharp tip at the end of an
optical fiber is used to focus the light [45,188]. Since the diameter of the small aperture is smaller
than the wavelength of the light, the light cannot get out from the aperture as a electromagnetic
wave, but instead emerges as an exponentially decaying electric field which follows from a solution
to Maxwell’s equations. This decaying electric field of the light is called the near field while the
conventional electromagnetic wave is called the far field. The amplitude of the near field can be
much larger than that of the far field, and thus if the optical set up is sufficiently close to the sample,
then we can get a much stronger signal with high spatial resolution using the near-field Raman
technique, especially when tip enhancement techniques are used (Section 2.6.4). As discussed
above, confocal Raman spectroscopy exhibits a limited resolution due to the Rayleigh criterion
where the light cannot be spatially focused to better than ∼ λ/2. This limitation can be overcome
using near-field optical spectroscopy. We elaborate on this technique below in discussing the
enhancement produced by using a metal tip and then by applying a combination of near field
Raman and photoluminescence imaging to enhance atomic force microscopy imaging.

2.6.4. Tip enhanced Raman spectroscopy

Recently, laser-illuminated metal probes have largely been utilized to locally enhance electro-
magnetic fields, thereby generating a near-field effect [45]. So far, this so-called tip-enhanced
microscopy (TERS, in comparison with more usual surface-enhanced techniques) has been applied
to both Raman scattering and photoluminescence studies of SWNTs, and it is based on the local
field enhancement at a laser-illuminated metal probe (see Figure 20). The highest spatial res-
olution achieved up to now is about 10 nm, which is limited by the tip size [45]. Near-field
optical microscopy also provides significant signal enhancement by several orders of magni-
tude, which is crucial for efficient nanotube and graphene ribbon detection. The combination of
nanoscale resolution and signal amplification makes the near-field optical techniques ideally suited
452 R. Saito et al.

Figure 20. Imaging of DNA-wrapped SWNTs at different magnifications: (a) A confocal Raman image of a
DNA-wrapped SWNT using an excitation wavelength of 632.8 nm. Topographic images in (b) and (c) indicate
a periodic height modulation expected for wrapping with short DNA segments. Near-field Raman images
(d) and (e) show the G-band intensity around 700 nm, and in (f) and (g) the intensity of photoluminescence
images at around 950 nm corresponding to the emission wavelength of an (8,3) nanotube are shown. PL
occurs only in the lower section of the nanotube where the Raman intensity is significantly weaker. The
abrupt transition from strong to weak Raman scattering combined with the appearance of PL is interpreted as
a local change in the nanotube (n,m) chirality [45]. Here, we see the power of near-field spectroscopy to show
images with spectroscopic information at high spatial resolution. Adapted from carbon Nanotubes: Advanced
Topics in the Synthesis, Structure, Properties and Applications, A. Hartschuh [45]. Copyright (2008) from
Springer.

for the investigation of nano-materials or localized perturbations caused by defects or nonuniform

environments. The use of the TERS technique is time consuming and requires great care in its
execution.

2.6.5. Simultaneous atomic force microscopy, near-field Raman and PL imaging

Figure 20 shows simultaneous near-field Raman and photoluminescence (PL) imaging for SWNTs
grown on glass by the arc discharge method [45], where highly confined PL imaging from short
20 nm long segments of about 20 nm in length has been observed from SWNTs. The PL from
micelle-encapsulated SWNTs on mica typically comes from much longer (more than 100 nm
long) SWNT segments of up to several hundreds of nanometers. Figure 20 shows PL emission
in images coming from a DNA-wrapped nanotube at different magnifications, starting with about
75 nm in the lower part of the nanotube (Figure 20g). The intensity of the Raman signal is seen to
decrease sharply at the position where PL starts to occur and the results obtained in this particular
image indicate a change in nanotube chirality (Figure 20e).

2.6.6. Coherent anti-Stokes Raman spectroscopy (CARS)

The CARS technique [189,190] consists of a pump–probe experiment using laser light. The elec-
trons in the ground state are excited by the pump laser light and the electron excitation is transferred
to a phonon via a Stokes Raman process. Then by the second probe light pulse, the phonon exci-
tation is transferred to another electron by an anti-Stokes Raman process. In this subsequent
process, we can observe the anti-Stokes Raman signal. An important concept in CARS is the
coherence of the Raman processes. Since the amplitude of the anti-Stokes Raman scattered light
has the same phase as the pump laser light, the CARS amplitude (intensity) is proportional to the
square of the number of molecules or atoms in the sample. In order to obtain this coherence, a
non-resonance condition is selected for both the first Stokes and the second anti-Stokes processes.
 Advances in Physics 453

Since CARS observes a blue-shifted signal, the signal is well separated from the related red-shifted
photoluminescence signal.

2.7. Kohn anomaly in graphene and carbon nanotubes

The use of the gate voltage to vary the electrochemical potential of a nanostructure has provided
a very sensitive tool to probe many different phenomena that are sensitive to the carrier density,
such as the Kohn anomaly (KA) [127,191–194]. We discuss the dependence of the Kohn anomaly
on the frequency and linewidth of the G-band phonon for mono-layer graphene in Section 2.7.1,
and for bi-layer graphene in Section 2.7.2, and for metallic SWNTs in Section 2.7.3. A theoretical
approach to the KA of carbon nanotubes is later discussed in Section 3.6.5 and 5.4.4.

2.7.1. Kohn-anomaly of the G-band of graphene

In Figure 21, the G-band Raman spectra of graphene is shown as a function of applying a gate
voltage to the substrate on which the graphene sample is positioned [195]. Here, the G-band is
observed to upshift in frequency with both negative and positive applied potentials (see Figure 21(a)
and (b)) and this frequency upshift is accompanied by a decrease in its linewidth (see Figure 21(c))
which is observed for both electron or hole doping. These effects are known as the Kohn anomaly
[194,196–199]. These observations are explained as follows.
In the presence of the el–ph interaction of free electrons near the Fermi energy EF , the phonon
frequency and spectra become, respectively, down-shifted and broad, which is given by second-
order, time-independent perturbation theory for the phonon energy and the phonon self-energy
[194,196–199]. When the Fermi energy shifts from the Dirac point by more than 0.1 eV (±ωG /2),
the virtual absorption of an electron by a G-band phonon with an energy of 0.2 eV is suppressed
and thus a phonon hardening occurs. Furthermore, since the el–ph interaction is suppressed, the
lifetime of a phonon becomes large, which corresponds to the decrease in linewidth shown in
Figure 21(c).

2.7.2. Kohn anomaly of bi-layer graphene

In bi-layer graphene, the unit cell has four C atoms rather than two, and as a result there are
two π bands and two π ∗ bands at the K point. In this case, there will be more than two Kohn
anomalies in the G-band gate-dependent frequency renormalization [200]. When the Fermi energy
reaches ±ωG /2, the π–π ∗ transition from the valence band to the lower conduction band shown
in Figure 22(I) is no longer allowed, as it is in mono-layer graphene. However, the transition from
the now filled lowest energy π ∗ band to the higher energy π ∗ band, shown by the dashed arrow
in Figure 22(II), is possible. Therefore, when the gate voltage rises further and the Fermi energy
reaches the second band, then the π ∗ –π ∗ transitions shown in Figure 22(III) are suppressed. These
effects are seen in the G-band frequency and linewidth of bi-layer graphene (see Figure 22)[191,
200], where a distinctly different behavior with respect to the mono-layer case (see Figure 21) is
clearly observed for both the G-band frequency and linewidth.
Furthermore the interlayer interaction between phonons in bi-layer graphene gives rise to even
and odd functions for the G-band vibrational modes in which only the even function mode is
Raman active. However, if an interaction with the substrate exists, this symmetry-imposed effect
is relaxed and both even and odd types of G-band spectra can be observed [5].
454 R. Saito et al.

2.7.3. Kohn anomalies of SWNTs

Similar Kohn anomalies of the G-band for SWNTs when compared with mono-layer graphene
are observed [193], but for SWNTs there is also a dependence on both the diameter and chiral
angle [196,199,201–203]. In this case, the RBM mode also shows a weak phonon softening effect
[170,204,205], which is further discussed in Section 5.
The Raman spectra of the G-band of sp2 carbons have an asymmetric lineshape if there are
free electrons at the Fermi energy. The asymmetric lineshape is expressed by Breit–Wigner–Fano
lineshapes (BWF), whose origin is generally understood by an interference of discrete energy levels
with a continuum energy spectrum (Fano resonance [129]). The second-order perturbation theory
expression for the el–ph interaction for the KA only gives a symmetric broadening of the phonon
spectra and thus the el–ph interaction by itself is not sufficient to account for the observed BWF
lineshapes. There is, however, another continuum spectra for the electron excitations involving
the Coulomb interaction in which there is an electron–electron interaction between photo-excited
electrons and the free electrons associated with the SWNTs that is relevant to accounting for the
BWF lineshapes. Recent measurements of the electronic Raman spectra of SWNTs clearly show
a BWF asymmetric shape depending on the energy position of Eii [202,206].

Figure 21. The dependence of the Raman G peak frequency of mono-layer graphene on doping using a gate
voltage to provide positive and negative potentials. (A) The G-band spectra at 295 K for many values of the
gate voltage Vg are shown. The darker line spectrum is at V = 0 but the spectrum at V = 0.6 V corresponds to
the undoped case, which occurs at V  = 0 due to the natural doping of graphene by the environment. (B) The
G peak position (frequency) and (C) the G peak FWHM linewidth as a function of electron concentration
as deduced from the applied gate voltage data are shown. Black circles show the measurements and the
solid lines show results from a finite-temperature non-adiabatic calculation. Adapted with permission from
Macmillan Publishes Ltd. Nature Nanotechnology [195], Copyright © (2008).
 Advances in Physics 455

2.8. Classification of Raman processes

In this section, we classify the Raman spectra of sp2 carbon materials from the point of view of
Raman processes.

2.8.1. First-order Raman process

After photo-absorption, if one phonon is emitted by the el–ph interaction, we can observe the
inelastically scattered light coming from the electron, whereby a phonon is emitted through the
el–ph interaction in the first-order Raman spectra. Through the recombination of a photo-excited
electron with the hole that was left behind, the scattered electron should have the same k value
as the hole, which requires that the phonon q vector should be zero. The phonon thus produced
is called a zone-centered (or a  point) phonon [104,207,208]. In the case of graphene, the
degenerate LO and iTO phonon modes at the  point are Raman-active and are known as the E2g
symmetry phonon modes. These modes occur around 1585 cm−1 , while the oTO phonon mode
occurs around 850 cm−1 and is an IR-active phonon mode, which does not contribute to the Raman
spectra for either defect-free graphene or SWNTs. The remaining three modes are acoustic modes
and have zero phonon energy at the  point. In the case of SWNTs, the RBM and G-band phonons
correspond to first-order Raman processes.

2.8.2. Two-phonon second-order Raman process

After photo-absorption, if two-phonons are emitted by the el–ph interaction, then we can observe
inelastically scattered light as a two-phonon second-order Raman process. In this case, there are
two possibilities for the actual Raman processes. One process involves the harmonic formed by two

Figure 22. On the left, the peak frequency (Pos(G)) and linewidth (FWHM(G)) for the Raman G-band
feature of doped bi-layer graphene vs. Fermi energy are shown. The black circles show the measurements
and the solid line shows the finite-temperature non-adiabatic calculation. On the right, schematics of the
el–ph coupling at three different doping levels, as indicated by the thicker lines on the electronic bands, are
shown. Adapted figure with permission from A. Das et al., Physical Review B 79, p. 155417, 2009 [191].
Copyright © (2009) by the American Physical Society.
456 R. Saito et al.

 point phonons, which is an over-tone phonon mode and the overtone mode of the G-band appears
at around 3170 cm−1 (twice the G-band mode frequency at 1585 cm−1 ). A feature associated with
the harmonic of the D -band also appears at 3240 cm−1 .
The other process is given by emitting two K point phonons with q and −q wavevectors by
an inter-valley scattering process. By emitting two phonons with opposite q and −q vectors, the
photo-excited electron can go back to its original position and recombine with a hole. Because
of the many possible q = 0 vectors, the spectral width for two-phonon Raman processes is broad
compared with the linewidths typically found in the first-order Raman spectra. The G -band around
2700 cm−1 arises from a two-phonon Raman process occurring near the K point iTO phonon mode.
For both cases of two-phonon scattering processes, there is no restriction on q = 0 for either of
these processes. Furthermore, the two phonons are not always the same types of phonon modes
and can have the same wavevector but different frequencies. If two different phonon modes are
involved in the two-phonon process, the Raman shift becomes the sum of the two phonon mode
frequencies, which we call a combination mode. It is important to emphasize that combination
modes which combine via an intra-valley scattering process together with an inter-valley scattering
process are not possible since the corresponding q vectors would be completely different from
each other14 and conservation of momentum would be violated.

2.8.3. One-phonon and one-elastic second-order Raman process

In a real crystal lattice, there are many defect structures by which a photo-excited electron can be
scattered by defects, such as the edges of graphene or point defects in both graphene and carbon
nanotubes. As far as there is no elementary excitation creation or annihilation at the defect site,
the scattering is elastic and only the momentum of the electron k can be changed to k + q by the
q component of the defect potential V (q). There is also the possibility of a second-order Raman
scattering process in which one of the two phonon scattering processes can be changed from
an inelastic scattering process to an elastic scattering process. Here, we still call this scattering
process a second-order Raman process, since the number of the order is defined by the number
of scattering events that occur. For this process, the corresponding Raman shift involves only one
phonon energy. What is different here from the first-order process is that the restriction on q = 0
is relaxed and many different phonon modes with q  = 0 can contribute to the Raman spectra. The
D-band around 1350 cm−1 is near the K point iTO phonon mode and the D-band intensity come
from a Raman process involving one-phonon and one-elastic inter-valley scattering process. The
D -band around 1620 cm−1 is near the  point LO phonon mode which consists of one-phonon
and one-elastic intra-valley scattering process. Since many q vectors are possible in this Raman
process, defect-related Raman peaks are generally broad compared with typical first-order Raman
features. The optical process which consists of only elastic scattering processes is the Rayleigh
scattering process.
In Figure 23, we show the schematics for (a1,a2) first-order and (b1-b4) one-phonon second-
order and finally (c1,c2) two-phonon second-order, resonance Raman spectral processes [209].
In Figure 23, we also show the incident and the scattered resonance conditions. For second-
order Raman processes, a DR Raman condition is satisfied [104,209,210], which is clarified in
Section 2.8.4. Although not shown in Figure 23, the Raman process for inter-valley phonons which
connects the K and K  points in the BZ is a second-order process.

2.8.4. Double resonance Raman spectra

Referring to Figure 23 [104,208], there are three intermediate electronic states in second-order
Raman processes: (1) excited states at k, (2) the intermediate state at k + q, and (3) the second
 Advances in Physics 457

Figure 23. Schematic diagrams for (a1,a2) first-order and (b1–b4) one-phonon second-order, (c1,c2)
two-phonon second-order, resonance Raman spectral processes for which the top diagrams refer to inci-
dent photon resonance conditions and the bottom diagrams refer to the scattered resonance conditions. For
one-phonon, second-order transitions, one of the two scattering events is an elastic scattering event (dashed
line). Resonance points are shown as solid circles [104,207–209]. Adapted with permission from R. Saito et
al., New Journal of Physics 5, p. 157, 2003 [209]. Copyright © (2003) by the Institute of Physics.

intermediate state back to k (see Figure 23(b) and (c)) [209]. This two-scattering amplitude process
is expressed by perturbation theory in which the numerator of the resulting term consist of four
scattering matrix elements (two for photon absorption and emission, and two for phonon emission
or absorption), while the denominator of this term consists of three energy difference factors. If
two of the three energy difference factors becomes zero, the scattering intensity becomes strongly
enhanced by each of these factors. We call a process containing two resonance denominators
double resonance (DR) Raman scattering [104,174,207,210,211]. The G -band of mono-layer
graphene represents a resonance Raman spectral feature for an iTO phonon mode near the K point
which is resonant for each of the three scattering processes.

2.8.5. Dispersive behavior of the phonon energy in DR Raman processes

An important aspect of the DR Raman spectra is that the observed phonon energy changes by
changing the laser excitation energy, which gives rise to a dispersive behavior of the scattered
phonon. Since the electronic energy dispersion of the π and π ∗ bands is linear in k as measured
from the K (K  ) point in the 2D BZ of graphene, a special k value is selected on an equi-energy
line for a given laser energy (Elaser ). In order to satisfy energy–momentum conservation for the
DR scattering from k to k − q and from k − q to k, the phonon momentum q is selected by the
circles near the K point shown in Fig. 23(c). When we calculate the phonon density of states for
this DR process, the DR wave vector qDR ≈ 2k for a phonon frequency ω(q) is selected [104,207].
Thus by changing the photon energy, we can probe the phonon energy along the phonon energy
dispersion [174,210,212]. The D- (G -) band frequency is 1350 (2700) cm−1 for Elaser = 2.41 eV
and the D-(G’)-band frequency increases by 53 (106) cm−1 by changing Elaser by 1 eV [213]. In the
case of SWNTs, the selection of qDR becomes more selective because of the 1D character of qDR ,
which is the reason why a sharp D-band spectral feature occurs in the case of SWNTs [210,211].
The pioneering work on the variation of the G -band peak frequency with laser excitation energy
was first studied by Vidano et al. [214,215] and by Baranov et al. [207].
458 R. Saito et al.

Figure 24. (a) The schematic diagram shows the light-induced e–h formation and the one electron–one
phonon scattering event taking place in the DR process with two different excitation laser energies (associated
with phonon wave vectors q1 and q2 , respectively), which are indicated by the gray and black arrows,
respectively. The two events in the DR process can occur in either order in time. (b) The phonon dispersion in
graphene is shown where the phonon wavevector q that fulfills the DR requirements for each Elaser value in
(a) is also indicated in terms of the phonon wavevectors q1 and q2 (see text). Reprinted figure with permission
from A. Jorio et al. Spectroscopy in Graphene Related Systems, 2010 [1]. Copyright © Wiley–VCH Verlag
GombH & Co. KGaA.

2.8.6. The inter-valley double resonance Raman scattering processes

When a photon with a given energy is incident on mono-layer graphene, it will excite an electron
from the valence band to the conduction band vertically in momentum space (vertical arrows in
Figure 24(a)). Since the graphene energy band does not have an energy gap, we always have
an electron with wavevector k for any Elaser value which satisfies energy conservation Elaser =
E c (k) − E v (k). In the single-resonance Raman process, a zone center (q = 0) phonon is created
or destroyed by coupling with the excited electron or hole, and the e–h recombination generates
the Raman shifted photon.
In contrast, the photo-excited electron at k can be scattered by emitting a phonon with wave
vector q to a state at k − q, as shown by the quasi-horizontal arrows in Figure 24(a). The phonon
emission in Figure 24(a) corresponds to an inter-valley scattering process in which the phonon
q vector connects two energy bands at the K and K  points of the BZ. If there is a phonon in
the vibrational structure of graphene with the wavevector q and phonon energy Eq so that this
photon can connect the two conduction electronic states, then this phonon scattering process will
be resonant. The DR process (involving both electron–photon and el–ph scattering, shown in
Figure 24(a)) will then take place.

2.8.7. Forward and backward scattering

The slope of the energy dispersion ∂E/∂k is called the group velocity. When we consider only the
direction of the group velocity for the initial k, there are two possibilities for the scattered k − q
states, as shown in Figure 25, where each of the inter-valley (a,b) and intra-valley (c,d) scattering
processes correspond to backward (a,c) and forward (b,d) scattering. Here the backward (forward)
scattering means that the direction of the group velocity does (does not) change after scattering.
The corresponding q vectors for inter-valley scattering are given by (see Figure 26)

q = K + qDR = K + k + k  ≈ K + 2k (backward scattering), (11)

q = K + qDR = K + k − k  ≈ K (forward scattering), (12)

where K is the magnitude of the reciprocal lattice vector that connects K and K  , and k (k  ) here
is measured from the K (K  ) point, which means that the double resonance wave vector qDR is the
 Advances in Physics 459

Figure 25. The full DR Stokes Raman processes for inter-valley (a,b) and intra-valley (c,d) scattering. Here
(a,c) relates to the backward scattering process with qDR = k + k  and (b,d) relates to the forward scattering
process with qDR = k − k  , with k and k  measured from the K point. The reciprocal lattice vector K is the
distance between the K and K  points, k (k  ) is the distance of the resonant states from K (K  ), as defined in
(a). Reprinted figure with permission from A. Jorio et al. Spectroscopy in Graphene Related Systems, 2010
[1]. Copyright © Wiley–VCH Verlag GombH & Co. KGaA.

Figure 26. One of the possible DR Stokes Raman processes involving the emission of a phonon with wavevec-
tor −q. The set of all phonon wavevectors q which are related to transitions from points on the two circles
around K and K  gives rise to the collection of small circles around the K  point obeying the vector sum rule
q = K − k + k (here we neglect the trigonal warping effect). Note that this collection of circles is confined
to a region between the two circles with radii qDR = k + k  ≈ 2k and qDR = k − k  ≈ 0. The differences
between the radii of the circles around K and K  and thus the radius of the inner circle around K  are actually
small in magnitude and are here artificially enlarged for clarity in presenting the concepts of the double res-
onance process. Adapted with permission from L.G. Cançado et al., Physical Review B 66, p. 35415, 2002
[216]. Copyright © (2002) by the American Physical Society.
460 R. Saito et al.

phonon wavevector distance linking the K and K  points. In the case of intra-valley scattering, we
just put K = 0 into Equations (11) and (12). Since the phonon energy is usually small compared
to the excited electronic energy levels, k ≈ k  , these two double resonance conditions approach
qDR = 2k and qDR = 0 (as is commonly used in the literature [114,172,210]).

2.8.8. DR q circles in 2D graphene

The picture discussed up to now is not the full story because graphene is a 2D material. For a given
laser energy, not only is the e–h excitation process shown in Figure 24 resonant, but any similar
process on a circle in these cones defined by Elaser (see Figure 26) is also resonant. Furthermore,
the mechanism of DR is actually satisfied by any phonon whose wavevector connects two points
on two circles around the K and K  points, as shown in Figure 26 [216]. (In constructing this
figure, we neglect the trigonal warping effect of the constant energy surface of graphene for
simplicity). A phonon with wavevector q connects two points along the circles with radii k and
k  around the K and K  points, respectively, where the difference between k and k  (for k  = k  )
comes from the energy loss from the electron to the phonon.15 By translating the vector q to
the  point, and considering all possible initial and final states around the K and K  points, the
doughnut-like figure shown in Figure 26 is generated [216]. Therefore, there is a large set of q
vectors fulfilling the DR condition. However, there is also a high density of phonon wavevectors
q satisfying the DR mechanism for which the end points of the wavevectors measured from the 
point are on the inner and outer circles of the “doughnut” in Figure 26. Therefore, the radii of the
inner and outer circles around K  (see Figure 26) are, respectively, k − k  and k + k  . Exactly as
given by the 1D model (Equations (11) and (12)), these wavevectors correspond to the phonons
associated with the singularities in the density of q vectors that fulfill the DR requirements, and
these special wavevectors in Figure 26 are expected to make a significantly larger contribution to
the second-order Raman scattering process. However, for a full description and lineshape analysis,
it is important to consider the 2D model, which seems to generate the asymmetric lineshape that
is observed for the 2450 cm−1 G -band feature.
Finally, Figure 24 shows that, if Elaser is changed, then the specific wavevector q and phonon
energy Eq that will fulfill the DR conditions will also change. This effect gives rise to the dispersive
nature of the G -band, which comes from an inter-valley DR Raman process involving an electron
with wave vector k in the vicinity of the K point and two iTO phonons with wave vectors qDR ≈ 2k,
where both k and qDR are measured from the K point.
To analyze experimental data for graphene in Figure 27 one has to consider the electron
and phonon dispersion of a graphene mono-layer. Near the K point, the electron and phonon
dispersions can be approximated by the linear relations E(k) = vF k and E(qDR ) = vph qDR ,
respectively, where vF = ∂E(k)/∂k and vph = ∂E(q)/∂q are the electron and phonon velocities
near the K point, respectively (in which vF is usually called the Fermi velocity, vF ∼ 106 m/s) for
graphene. We denote the electron (phonon) wave vector by k (qDR ) which is measured with respect
to the K point, so that the conditions for the DR Raman effect are given by [1]

Elaser = 2vF k,
Eph = vph qDR , (13)

qDR = k ± k ,

where Elaser and Eph are, respectively, the laser and phonon energies, and k  is the scattered electron
wave vector near the K  point in the graphene BZ. It is important to remember that we are dealing
here with combination modes, so that the observed Eph has to reflect this combination. For example,
for the G -band, the observed G -band energy is given by EG = 2Eph , where Eph is the energy for
 Advances in Physics 461

the iTO phonon mode at qDR .16 Making another commonly used approximation in Equation (13),
i.e., qDR = k + k  ≈ 2k, then EG can be written as [1]
vph
EG  = 2 Elaser . (14)
vF
A drawback in using the DR Raman features to define the electron and phonon dispersion relations
is that the measured values depend on both vph and vF , and one has to be known in order to obtain
the other. In addition to this problem, the physics of the phonon dispersion for graphene near
the K point is rather complex due to the Kohn anomaly, and the KA also occurs for phonons at
q → K (see Section 2.7). The high frequency of the iTO phonon when combined with the KA
near the K point are together responsible for the strong dispersive behavior observed for ωG . The
exact values for vph and vF are still under debate since they also depend on the complex physics
of many-body effects [112,194,217–220]. This is one area where more work for future research
is needed.

2.8.9. Dispersive behavior of the G - and G∗ -band

Since both G∗ and G features in Figure 27 are due to double resonance processes, both Raman
features show dispersion with Elaser , but with quite different characteristics. So far, we explained
that there are two possibilities for selecting q while satisfying the double resonance condition:
qDR ≈ 2k and 0. The condition for qDR ≈ 0 will or no select a phonon frequency at q = 0 measured
from the K point, and these qDR show weak dispersion even though the signal is due to a DR Raman
spectra. Figure 27(a) shows the Raman spectra in the region of both the G∗ (∼ 2450 cm−1 ) and
G (∼ 2700 cm−1 ) bands for different laser excitation energies. Also the G∗ -band can either be
explained by the q ≈ 0 DR relation [221] or by the q ≈ 2k relation applied to an inter-valley
process [213], but involving two different phonons in each case. Figure 27(b) shows the G and G∗

Figure 27. (a) Raman spectra of the G and the G∗ -bands of mono-layer graphene for 1.92, 2.18, 2.41, 2.54
and 2.71 eV laser excitation energy. (b) Dependence of ωG and ωG∗ on Elaser . The circles correspond to
the graphene data and the lozenges correspond to data for turbostratic graphite. Adapted with permission
from D. Mafra et al., Physical Review B, 76, p. 233407, 2007 [213]. Copyright © (2007) by the American
Physical Society.
462 R. Saito et al.

frequencies ωG and ωG∗ as a function of Elaser for graphene and turbostratic graphite (for which
the stacking between graphene layers is random). The G -band in Figure 27 exhibits a highly
dispersive behavior with (∂ωG /∂Elaser ) 88 cm−1 /eV for mono-layer graphene, 95 cm−1 /eV for
turbostratic graphite [213] and 106 cm−1 /eV for carbon nanotubes (see Figure 27 and [222]). The
G∗ -band feature exhibits a much less pronounced dispersion than the G -band, and of opposite
sign, with (∂ωG∗ /∂Elaser ) −10 to −20 cm−1 /eV for both mono-layer and turbostratic graphite
[208,213], and no dispersion is reported for carbon nanotubes [221]. It should be noted that a
different interpretation to the origin of the G∗ -band is given in [131,223], which together with
[221] identified the origin of the 2450 cm−1 peak with the overtone of the 1225 cm−1 feature
which has a peak in the phonon density of states for two phonons [223], while [224,225] assigned
this feature to the combination modes iTA + iTO. As already stated, the qDR ≈ 2k wavevector
gives rise to the G -band, while the qDR ≈ 0 wavevector gives rise to a DR feature coming from
the iTO phonon very close to the K point. The qDR ≈ 0 processes are expected to be less intense
than the qDR ≈ 2k processes because the destructive interference condition is exactly satisfied for
qDR = 0 [208].

2.8.10. Double resonance, overtone and combination modes

The sp2 carbons exhibit several combination modes and overtones, which are shown in Figure 28
for graphite whiskers as a function of frequency up to 7000 cm−1 [226]. Basically all the branches
in the phonon dispersion can be seen to have a combination and overtone Raman features which
obey the DR condition [104,210]. Many of the peaks observed in the spectra of Figure 28 below
1650 cm−1 are actually one-phonon bands activated by defects (see Section 4.3).Above 1650 cm−1 ,
the observed Raman features are all multiple-order combination modes and overtones, and here
too some of the features in Figure 28 are also activated by defects.
As shown in Figure 28, the DR peaks change frequency with changing Elaser , and they can be
fitted onto the phonon dispersion diagram shown in Figure 29 using DR theory. The data points
displayed in Figure 29 all stand for the qDR ≈ 2k DR backward resonance condition, those near
 and K coming from intra-valley and inter-valley scattering processes, respectively. Actually,
in the Raman spectra there are no characteristic features distinguishing peaks associated with
the intra-valley scattering process from those associated with inter-valley scattering processes,
or even features that distinguish between the qDR ≈ 2k or qDR ≈ 0 resonance conditions. All we
have in hand in analyzing actual Raman spectra is the Elaser dependence of each peak. Here, each
peak has to fulfill one of the DR processes and to fit the predicted phonon dispersion relations.
For example, the data points near the K point in Figure 29 that are assigned as the iTO+LA
combination mode (TO+LA) could alternatively be assigned to a qDR ≈ 0 process, since this
combination mode is weakly dispersive [221]. Supporting this assignment is the asymmetric (DR
phonon-density-of-states-like shape) observed for this peak, and against this identification is the
destructive interference working towards DR Raman processes at exactly q = K [208]. The debate
about the iTO + LA combination mode assignment near the K point remains for future clarification.
Near the  point, the dispersive behavior is more clear and the assignment of the observed Raman
features is on a more solid foundation [227].

2.9. Summary
The power of Raman spectroscopy for studying carbon nanotubes is in particular revealed through
exploitation of the resonance Raman effect, which is greatly enhanced by the singular density
of electronic states of SWNTs and the resonant effect comes from the 1D confinement of the
electronic states due to the small diameters of carbon nanotubes. Soon after the discovery of the
 Advances in Physics 463

Figure 28. Raman spectra of graphite whiskers obtained at three different laser wavelengths (excitation
energies) [226]. Note that some phonon frequencies vary with Elaser and some do not. Above 1650 cm−1 the
observed Raman features are all multiple-order combination modes and overtones (see Figure 29), though
some of the peaks observed below 1650 cm−1 are actually one-phonon bands activated by defects. The inset to
(c) shows details of the peaks labeled by L1 and L2 . The L1 and L2 peaks, which are dispersive, are explained
theoretically by the defect activation of double-resonance one-phonon processes (see Section 2.8.10) involv-
ing the acoustic iTA and LA branches, respectively, as discussed in Ref. [210]. Adapted figure with permission
from P.H. Tan et al., Physical Review B 64, p. 214301, 2001 [226]. Copyright © (2001) by the American
Physical Society.

resonance Raman effect in SWNTs [103], it was found that the resonance lineshape could be used
to identify the nanotube structure, i.e., the chiral indices (n, m) [111], and to distinguish metallic
from semiconducting SWNTs [128,228]. It is clear that most of the results achieved up to now
have been developed for SWNTs, while the optics of graphene and nanoribbons is still at an early
stage. This is the way it happened historically, and the knowledge developed in carbon nanotube
science is now fostering an amazingly fast development of graphene photo-physics. It is expected
that graphene photo-physics will follow a similar path of development that will reveal much new
physics as this very fundamental field develops such as the understanding of the KA which helped
to elucidate the phonon dispersion relation of graphite and all related sp2 carbons.
464 R. Saito et al.

Figure 29. Two-phonon dispersion of graphite based on second-order double-resonance peaks in the Raman
spectra (dark circles). Solid lines are dispersion curves from ab initio calculations considering combination
modes and overtones associated with totally symmetric irreducible representations. Adapted figure with
permission from J. Maultzsch et al., Physical Review B 70, p. 155403, 2004 [208]. Copyright © (2004) by
the American Physical Society.

3. Calculation method of resonance Raman spectra

Within the tight-binding approximation methods, we can calculate the Raman spectra and the
Raman excitation profile of an (n, m) nanotube by considering the electron–photon, el–ph,
exciton–photon and exciton–phonon interactions. In this section, the (n, m) dependence of the
RBM and G-band intensities, spectral width and resonance window are directly compared with
experiment. The exciton energy calculations can reproduce the Eii energy within high accuracy.
In Section 3, however, we do not mention the environmental effects (see Section 5.2.1) which are
here comparatively considered here by experiment [229,230] and theory [231,232]. The electron–
electron interaction and elastic scattering matrix elements [233] need to be developed further for
obtaining the asymmetric shape of the BWF lines and the D- (G -) band spectra, respectively.
Using the established computer library, we can extend these calculations to consideration of the
coherent phonon response of a nanotube [165]. In this section, we do not mention the polariza-
tion dependence of the Raman spectra, in which the screening effect (the so-called depolarization
effect [234]) is important.

3.1. Overview of calculations reviewed in this section

In Section 3, we review the theoretical calculation of the resonance Raman spectra and exciton
energy states for carbon nanotubes. The method used here for carbon nanotube calculations can
also be applied to graphene Raman spectra, though we do not need to consider exciton states for
graphene. Most of the quantitative comparisons are made in Section 3 with regard to resonance
energy, Raman frequencies and spectral linewidths. In order to obtain the Raman intensity, we
review the calculations of the excitonic (electronic) interaction matrix element for nanotubes
(graphene). Since the derivations consist of many topics, each topic is first briefly mentioned in
this subsection and is then described in more detail in the following subsections.
 Advances in Physics 465

3.1.1. Raman scattering and phonon energy dispersion

In Raman spectroscopy studies of solids, we generally observe the phonon frequency at the center
of the BZ (zone-center phonon). Other inelastic scattering techniques, such as inelastic neutron
scattering [235,236], or inelastic X-ray scattering [208] or electron energy loss spectroscopy
[237,238] provide measurements of the phonon dispersion inside the BZ, which we can reproduce
theoretically [235,239] either by fitting force-constant calculations to experimental data or by first-
principles calculations [32,240,241]. Graphene-related systems have a special electronic structure
which allows the observation of phonons in the interior of the BZ by DR Raman spectroscopy. In
Section 3.2, we mention how to obtain the phonon dispersion relations by force-constant models.

3.1.2. Electronic energy bands

Optical processes can be studied by Raman spectroscopy as well as by other techniques. If there is
either a photo-absorption or a photo-emission process that couples the ground state to an excited
state of an electron, then the amplitude of the phonon scattering process is greatly enhanced if the
excited state is a real electronic state. This resonant process is known as resonance Raman spec-
troscopy. In order to obtain the resonance condition by which a given laser energy Elaser matches
the transition energy of the actual electronic states, the electronic energy bands are calculated
by a simple tight-binding method in Section 3.3 or by an extension of this method as given in
Section 3.3.1 using the so-called extended tight-binding model. Angle-resolved photo-electron
spectroscopy (ARPES) experiments [123–126] are especially relevant for providing information
about the occupied electronic energy bands in graphene.

3.1.3. The double resonance process

While the most usual first-order Raman processes measure only zone center phonon modes,
excited electron scattering processes may also take place involving phonons in the interior of the
BZ. Such processes can become Raman allowed either by two phonon scattering processes, thus
conserving momentum, or in the presence of a lattice defect, where the momentum conservation
requirement can be broken. However, these are generally low probability processes. In graphene-
related systems, however, such DR scattering processes become highly probable because of the
so-called DR phenomena in graphene [104,207,210]. One resonant phenomenon is light absorption
or emission, and the other resonant phenomenon is the scattering of the excited electron (or hole)
by phonons. Here “resonant” means that the phonon brings the electron from one real state to
another real state, which matches the energy and momentum transfer required for momentum and
energy conservation.

3.1.4. Electron–photon and electron–phonon interactions

To obtain good agreement with experiment, it is necessary to include el–ph and electron–photon
interactions in such calculations of the Raman intensity as a function of Elaser . In this connection,
we here discuss the calculation of the electron–photon matrix elements (Sections 3.4 and 3.4.1)
and of the el–ph matrix elements (Section 3.4.3), which are the matrix elements that appear in the
numerator of the perturbation calculation of the Raman scattering amplitude (Section 3.6). Cal-
culation of the Raman excitation profile (Section 3.6.3) in which the Raman intensity is observed
as a function of Elaser is of great interest for obtaining the Raman intensity at resonance which is
the quantity of greatest interest for experimental studies. Many experimental Raman studies that
are found in the literature are actually not carried out under fully resonance conditions, but only
466 R. Saito et al.

within the resonance window, so care must be exercised in making proper comparisons between
experiment and theory.

3.1.5. Excitons
In the case of carbon nanotubes, the exciton binding energy is much larger (up to 1 eV)
[120–122,147,242] than that for Si (which is in the meV range). The exciton, which is
formed from a photo-excited electron and the hole that is left behind, is especially impor-
tant and dominates the observed optical processes in carbon nanotubes which are 1D systems
where excitonic effects are exceedingly strong. Even at room temperature, the excitoni-
cally mixed electronic states are specified by a wavevector k so as to form a spatially
localized state. In order to obtain excitonic states and their corresponding wavefunctions,
the Bethe–Salpeter equation for π electrons is used here within the tight-binding method
(Section 3.5). Using excitonic wavefunctions, we can calculate the relevant exciton–photon
(Section 3.6) and exciton–phonon (Section 3.6.3) matrix elements. Two-photon absorption or
time-dependent Raman spectroscopy have also been used to observe many specific exciton-related
phenomena.

3.1.6. Resonance window and the Kohn Anomaly

Since the photo-excited electron (or hole) has a finite lifetime (less than 1 ps), the transition
energy of an exciton has an energy uncertainty which is observed as an energy width in the
measurement of the Raman excitation profile, which we also call the resonance window. A typical
value for the experimentally reported resonance windows is 100 meV and the origin of the finite
lifetime of the photo-excited carriers lies in the exciton–phonon scattering process, which depends
on the metallicity, chiral angle and diameter of the SWNT. In the case of metallic nanotubes,
phonons typically couple to an e–h pair excitation by the el–ph (or more precisely the exciton–
phonon) interaction. Second-order perturbation theory for the phonon energy gives values for the
energy shifts and spectral broadening that arise through the el–ph interaction, and we call these
energy shifts and broadening effects the KA [127,194,197–199,243]. This topic is discussed in
Section 3.6.5. The KA is observed experimentally in the G-band of graphene (Section 2.7.1) and
in both the G-band and the RBM features for metallic nanotubes in gating or electro-chemical
doping experiments Section 2.7.3 [196,203–205].

3.2. Tight-binding calculation for phonons

The phonon energy dispersion can be calculated by using a force constant tensor which connects the
relevant motion of nearest-neighbor atoms through these theoretical calculations. The equations
of motion are given by

Mi üi = K (ij) (uj − ui ), (i = 1, . . . , N), (15)
j

where Mi and ui are, respectively, the mass and the vibrational amplitude of the ith atom and K (ij)
represents a 3 × 3 force constant tensor which connects ith and jth atoms. The summation on j is
taken over the jth nearest neighbor atoms so as to reproduce the phonon energy dispersion relation
(see Figure 30). The K (ij) terms are obtained by fitting to experimentally obtained phonon disper-
sion relations, such as are determined from neutron or X-ray inelastic scattering measurements
[208,235,239]. The fitting procedure to the experimental phonon dispersion is possible even if the
KA effect is included. However, the broadening of the phonon dispersion due to the finite lifetime
 Advances in Physics 467

Figure 30. Phonon dispersion of graphene in the 2D BZ. The symbols are experimental data obtained by
inelastic X-ray scattering [244] and the lines are fitted to the experimental phonon data using up to 14th
nearest-neighbor interactions [245].

of phonon cannot be expressed by the present method and the self-energy for the phonon should
then be calculated as discussed below [196,205,219,220,246].
Since the lattice is periodic, each displacement ui in the unit cell can be expressed by a wave
with a phonon wavevector k and frequency ω as follows:
1  i(k·Ri −ωt)
uk(i) = √ e ui , (16)
Nu R
i

where the sum is taken over all Nu lattice vectors Ri in the crystal for the ith atom in the unit cell.
The equation for uk(i) (i = 1, . . . , N), where N is the number of atoms in the unit cell, is given by
[32]
⎡ ⎤
 
⎣ K (ij) − Mi ω2 (k)I ⎦ uk(i) −
(j)
K (ij) eik·Rij uk = 0 (17)
j j

in which I is a 3 × 3 unit matrix and Rij = Ri − Rj denotes the relative coordinate of the ith
atom with respect to the jth atom. The simultaneous equations implied by Equation (17) with 3N
unknown variables uk ≡ t (uk(1) , uk(2) , . . . , u(N)
k ), for a given k vector, can be solved by a diagonal-
ization of the 3N × 3N matrix in brackets, which we call the dynamical matrix. By diagonalizing
the dynamical matrix for each k, we get the phonon frequencies and corresponding amplitudes
as a function of k, ω(k) and uk , respectively, which are the eigenvalues and eigenfunctions of the
dynamical matrix.
In Figure 30, the phonon dispersion relations of graphene are plotted in the 2D BZ. Lines
are fitted for the calculated phonon energy dispersion to the experimental data for inelastic X-ray
scattering (symbols) by a set of force constants that includes force constants up to 14th nearest
neighbors [245]. This force constant set is obtained by minimizing the square of the difference
between experiment and theory for each experimental data point. In order to get good convergence
for the nonlinear fitting, we must start with a small number of nearest neighbors and we then
increase the number of neighbors one by one. Further, in order to get the required zero value for
the acoustic phonon modes at the  point, we should consider the relationships between the force
constant set, known as the force constant set sum rule [217]. Degenerate in-plane optical phonon
modes around 1600 cm−1 at the  point are known by symmetry requirements to correspond to the
Raman-active phonon mode (G-band), while the out-of-plane optical (oTO) phonon mode around
860 cm−1 at the  point is an infrared-active phonon mode. The acoustic modes are discussed
in [247].
468 R. Saito et al.

The phonon modes near the K point and  point can be observed by defect-induced or
two-phonon derived features in the Raman spectra. The phonon modes along the phonon dis-
persion relation can be observed by studying phonon modes arising from DR Raman processes
(see Section 2.8.6) [210]. The LO phonon mode (the highest frequency mode) of graphene has
a local minimum at the  point and the phonon energy increases with increasing k by a pro-
cess which we call “over-bending”. The “over-bending” can be reproduced using force constants
going beyond the fifth nearest neighbor. Both the in-plane optic phonon modes near the  point
and the iTO phonon mode near the K point show phonon softening phenomena for graphene and
metallic SWNTs and the resulting phonon frequency down-shifts are known as the KA effect
[127,194,196–199,205,219,220,243,246] (see Section 3.6.5). When we calculate phonon disper-
sion by first principles [248–250], the effect of the Kohn anomalies should be taken into account
when calculating a force constant set, while in the simple tight-binding method, we just obtain
a force constant set either by fitting to the experimental results [208,244,251,252] or by first-
principles calculation in which the Kohn anomalies are taken into account. When we obtain a
force constant set by the atomic potential as a function of the C–C bond distance, such as by the
extended tight-binding method [253,254], we should consider the additional effect of the KA in
the calculation.
Interlayer force constants of multi-layer graphene are much weaker than the intralayer force
constant set. Each phonon mode of a graphene sheet is split into symmetric and anti-symmetric
vibrational modes with respect to the inversion or mirror symmetry of multi-layers, depending on
whether the multilayer graphene consists of an even or odd number of graphene layers, respectively.
It is important that some phonon modes (oTO, oTA, LO) become Raman active (or inactive) by
the interlayer interaction [4].

3.3. Simple tight-binding calculation for the electronic structure

Tight-binding calculations of the electronic energy bands for sp2 carbons are useful for understand-
ing the physics of sp2 carbons and for saving computational time. The tight-binding wavefunction

j (k), where j denotes the energy band index, is given by a linear combination of a small number
of tight-binding Bloch wave functions j


N

j (k, 
r) = 
Cjj (k) 
j (k, 
r) (j = 1, . . . , N), (18)
j =1

 are coefficients to be determined and N the number of atomic orbitals in the unit
where Cjj (k)
cell. When we consider π orbitals for n-layer graphene, then N = 2n. Here j denotes the Bloch
function for an atomic orbital ϕj which is given by

1 
Nu
 
j ( k, 
r ) = √ )
eik·R ϕj (r − R (j = 1, . . . , N), (19)
Nu 
R

where the summation takes place over Nu lattice vectors R  in the crystals. When we put
Equation (18) into the Schrödinger equation H j = E j for a Hamiltonian H, we get


N 
N
 ij = Ei (k)
Hjj (k)C   ij
Sjj (k)C (i = 1, . . . , N). (20)
j =1 j =1
 Advances in Physics 469

 and Sjj (k)

Here Hjj (k)  are the Hamiltonian, and the overlap matrices are defined by

 =
Hjj (k) j |H| j ,
 =
Sjj (k) j| j  (j, j = 1, . . . , N). (21)

By defining a column vector Ci as

⎛ ⎞
Ci1
⎜ ⎟
Ci = ⎝ ... ⎠ , (22)
CiN
then Equation (20) is expressed by the eigenvalue equation


HCi = Ei (k)SCi. (23)

By diagonalization of a given H and S for each k vector, we get the energy eigenvalues Ei (k)
 and

eigenfunctions Ci (k).
The ij matrix element of H is given by

 = 1  ik(R−R )
Hij (k) e ϕi (r − R )|H|ϕj (r − R)
Nu R,R
 (24)
= eik(R) ϕi (r − R)|H|ϕj (r),
R

where R ≡ R − R and in the second of line of Equation (24), we use the fact that the tight-
binding parameter ϕi (r − R )|H|ϕj (r − R) only depends on R. Similarly, the matrix elements
of S are given by

 =
Sij (k) eik(R) ϕi (r − R)|ϕj (r). (25)
R

The tight-binding parameters ϕi (r − R)|H|ϕj (r) and ϕi (r − R)|ϕj (r) are obtained by:
(1) integrating the matrix elements using the atomic orbitals ϕj (r) [247] or (2) fitting them so as
to reproduce experimentally obtained energy dispersion measurements. Values for a typical fitted
parameter set (TBP) are listed in Table 1.
As seen in Table 1, tight-binding parameters are listed for up to the third nearest neighbor
(3NN) within a layer (upper half) and for interlayer interactions between graphene layers (lower
half) [255]. In Figure 31, we show a definition of the tight-binding parameters listed in Table 1 for
the Hamiltonian for pairs of carbon atoms separated by their corresponding distances R [255].
j
The notation used for the parameters γi follows that of Slonczewski and Weiss [257], while γ0 and
sj (j = 1, 2, 3) denote the in-plane parameters with the jth nearest neighbors up to the 3rd nearest
neighbor (3NN). As far as we consider transport properties near the K point of the first BZ, the
in-plane nearest-neighbor parameter γ01 is sufficient. However, when we consider optical transition
phenomena around the K point, it is necessary to include the parameters γ02 and γ03 which are
indicated explicitly in Figure 31 [256]. The parameters γ1 , γ3 and γ4 denote interactions between
carbon atoms in the nearest-neighbor layers, while the parameters γ2 and γ5 couple carbon atoms
in next nearest-neighbor layers. The parameters γ3 and γ4 introduce a k-dependent interlayer
interaction and γ2 sensitively determines a small energy dispersion along the KH direction in the
3D BZ for energy bands near the Fermi energy of graphite (see [5, Fig. 1f]) which gives rise to
the semi-metallic nature of 3D graphite.
The overlap tight-binding parameters, s0 , s1 and s2 , are essential for describing the asymmetry
between the valence and conduction energy bands relative to the Fermi energy. The energy band
470 R. Saito et al.

Table 1. Third nearest-neighbor tight-binding (3NN TB) parameters for few-layer

graphene and graphite.

TBP 3NN TB-GWa 3NN TB-LDAa EXPb 3NN TB-LDAc R, paird
√
γ01 −3.4416 −3.0121 −5.13 −2.79 a/ 3, AB
γ02 −0.7544 −0.6346 1.70 −0.68 a, AA and BB
√
γ03 −0.4246 −0.3628 −0.418 −0.30 2a/ 3, AB
√
s0 0.2671 0.2499 −0.148 0.30 a/ 3, AB
s1 0.0494 0.0390 −0.0948 0.046 a, AA and BB
√
s2 0.0345 0.0322 0.0743 0.039 2a/ 3, AB
γ1 0.3513 0.3077 – – c, AA
γ2 −0.0105 −0.0077 – – 2c, BB
√
γ3 0.2973 0.2583 – – (a/ 3,c), BB
√
γ4 0.1954 0.1735 – – (a/ 3,c), AA
γ5 0.0187 0.0147 – – 2c, AA
E0e −2.2624 −1.9037 – −2.03
f 0.0540g 0.0214 – –
a
Fits to LDA and GW calculations [255].
b
Fit to ARPES experiments by Rotenberg et al. [124].
c
Fit to LDA calculations by Reich et al. [256].
d
In-plane and out-of-plane distances between a pair of A and B atoms.
e
The energy position of π orbitals relative to the vacuum level.
f
Difference of the diagonal term between A and B atoms for multi-layer graphene.
g
The impurity doping level is adjusted in order to reproduce the experimental value of  in graphite.
All values are in eV except the dimensionless overlap parameters of s0 -s2 . The parameters of fits to LDA and
GW calculations are shown. The 3NN Hamiltonian is valid over the whole 2-D (3-D) BZ of graphite (graphene
layers) [255]. Mopac93 and Gaussian 9 software packages were used for implementing Gaussian and other
software applications.

Figure 31. Identification of the various Slonczewski–Weiss parameters for the tight-binding parameters for
a pair of carbon atoms separated by a distance R. Adapted figure with permission from A. Grüneis et al.,
Physical Review B 78, p. 205425, 2008 [255]. Copyright © (2008) by the American Physical Society.

width of the conduction band is larger than that of the valence band when using this set of tight-
binding parameters [32], thereby inducing asymmetry between the electrons and holes in few layer
graphene. Further, depending on whether the number of graphene layers is odd (even), the linear
(quadratic) k energy dispersion behavior appears near the K point. Koshino and Ando [258] have
explained analytically the reason for the odd-even dependence of the electronic structure of few
layer graphene on the number of graphene layers by the tight-binding method.
 Advances in Physics 471

3.3.1. Extended tight-binding calculation for the graphene electronic structure

The simple tight-binding parameters obtained in Section 3.3 are only for π orbitals in graphene.
When we consider SWNTs, the curvature of the cylindrical tube surface should be considered.
The curvature effect mixes π orbitals with σ orbitals (2px , 2py and 2s). Furthermore, when we
consider the geometrical optimization of the lattice, we need to calculate tight-binding parameters
as a function of the actual C–C bonds.
The extended tight-binding (ETB) calculation is a tight-binding calculation for π, σ and 2s
orbitals, in which the tight-binding parameters for a pair of orbitals are given as a function of the
C–C bond lengths and bond angles. The basic treatment of the mixing between π and σ orbitals
uses a formalism known as the Slater–Koster method in which p orbitals can be projected on to
a chemical bond [259]. Values for the ETB parameters as a function of the C–C bond length are
given by first-principles calculations for several sp2 molecules or solids. For carbon systems, the
tight-binding parameters as a function of the C–C bond length have been calculated by Porezag
[253], and the optimized structure of SWNTs using ETB parameters reproduced well the transition
energy separation Eii especially for SWNTs with diameters smaller than 1 nm [254].
In quantum chemistry calculations, great effort has been given to ETB-like calculations for
reproducing the energy levels for many different molecules, which are known as semi-empirical
methods. MNDO, MINDO, AM3 and PM5 are names of the parameter sets for popular semi-
empirical methods, which are used in many chemistry molecular level calculations, such as
MOPAC [260] and Gaussian [261], etc. An advantage of the ETB or the semi-empirical methods
is that the calculation is fast and small in size. This calculational approach is suitable for the
calculation of SWNTs since a SWNT has a large number of carbon atoms in the unit cell.

3.4. Calculations of matrix elements

Using the electronic and phonon eigenfunctions, we can calculate the matrix elements for the
electron–photon and el–ph interactions.

3.4.1. The electron–photon matrix element

Using the simple tight-binding wavefunction, the electron–photon matrix element is calculated
within the dipole approximation. The perturbation Hamiltonian of the optical dipole transition is
given by
ie
Hopt = A(t) · ∇, (26)
m
where A is the vector potential. When we adopt the Coulomb gauge ∇ · A(t) = 0, the electric field
of the light is given by E = iωA. Hereafter, we consider only a linear polarization of the light, and
thus the vector potential A is given by

−i I
A= exp(±iωt)P, (27)
ω c0

where P is the unit vector (polarization vector) which specifies the direction of E, I the intensity
of the light in W/m2 and 0 the dielectric constant for the vacuum in SI units. The “±” sign
corresponds to the emission (“+”) or absorption (“−”) of a photon with frequency ω. Here, we
can assume that the wavevector k of an electron does not change during the transition (vertical
transition). Then the matrix element for optical transitions from an initial state ı (k) to a final
472 R. Saito et al.

f
state (k) at k is defined by

fı
Mopt (k) =  f
(k)|Hopt | ı
(k). (28)

The electron–photon matrix element between initial and final states in Equation (28) is calculated
by

fı e Iρ i(ωF −ωı ±ω)t f ı
Mopt (k) = e D (k) · P (29)
mωρ c0

where the weak spatial dependence of the vector potential A is neglected and Df ı (k) is the dipole
vector defined by the matrix element

Df ı (k) =  f
(k)|∇| ı
(k). (30)

For a given polarization, P, the optical absorption (or stimulated emission) becomes large (absent)
when D is parallel (perpendicular) to P.

3.4.2. Electric dipole vector for graphene

When we expand in Equation (30) into atomic orbitals ϕj (r − R  ) (Equation (19)), the dipole
  ). The optical dipole
vector D can be expressed by the atomic dipole vector ϕj (r − R )|∇|ϕj (r − R
transition of an electron from a π (2pz ) band to an unoccupied π band within an atom (R = R
∗ )
is forbidden in the case of graphene, which is understood by the mirror symmetry occurring at
z = 0. However, the optical transition between a π and a π ∗ energy band is possible when the
optical transition between nearest-neighbor interaction is allowed, as shown below.
Here, we consider the electric dipole vector for graphene [262]. The wavefunction in
Equation (18) with N = 2 is given by (k) = CA A (k, r) + CB B (k, r), in which is the
Bloch wavefunction for 2pz atomic orbitals for the A and B sites of graphene. If we neglect the
next nearest-neighbor interaction between the A and A atoms (or the B and B atoms), the electric
dipole vector Df ı (kF , kı ) for graphene is given by,

f∗
Df ı (kF , kı ) = CB (kF )CAı (kı ) B (kF , r)|∇| A (kı , r)
(31)
f∗
+ CA (kF )CBı (kı ) A (kF , r)|∇| B (kı , r).

Since both the 2pz orbital and the ∂/∂z component of ∇ have odd symmetry with respect to the z
mirror plane, the z component of D becomes zero. Thus, we conclude that the dipole vector lies
in the xy plane.
When we expand into atomic orbitals, the leading term of  A (kF , r)|∇| B (kı , r) is the
atomic matrix element mopt between nearest neighbor atoms given by
   
∂ 
mopt  
= φ(r − Rnn )   φ(r) , (32)
∂x

where Rnn is the lattice vector between nearest-neighbor C atoms along the x-axis.
When we use a linear approximation for the coefficients CA and CB for a k point around the
corner point of the 2D BZ K = (0, −4π/(3a)) for the valence (v) and conduction (c) bands, we
 Advances in Physics 473

Figure 32. (a) The normalized dipole vector Dcv (k) is plotted as a function of k over the 2D BZ. (b)
The oscillator strength in units of the atomic matrix element mopt is plotted as a function of k over
the 2D BZ. The separation between two adjacent contour lines is 0.4 mopt . The darker areas have a
larger value for the oscillator strength. Reprinted figure with permission from A. Grüneis et al., Physical
Review B 67, pp. 165402–165407, 2003 [262]. Copyright © (2003) by the American Physical Society.

write [32]
1 ky − ikx
CAv (K + k) = √ , CBv (K + k) = √ .
2 2k
(33)
1 −ky + ikx
CAc (K + k) = √ , CBc (K + k) = √ ,
2 2k
The electric dipole vector coupling the valence and conduction bands is then given by
3mopt
Dcv (K + k) ≡  c 
(K + k)|∇| v
(K + k) = (ky , −kx , 0). (34)
2k
In Figure 32(a) we plot the normalized directions of the normalized dipole vector Dcv (k) as arrows
over the 2D BZ of graphene [262]. Around the K points, the arrows show a vortex behavior. Note
also that the rotational directions of Dcv (k) around the K and K  points are opposite to each other
in Figure 32(a). In Figure 32(b) we plot the values of the magnitude of the oscillator strength O(k)
in units of mopt on a contour plot. Here Ocv (k) is defined by

Ocv (k) = Dcv∗ (k) · Dcv (k). (35)

As shown in Figure 32(b), the oscillator strength Ocv (k) has a maximum at the M points and a
minimum at the  point in the 2D BZ. The k dependent Ocv (k) will be relevant to the calculation for
the type-dependent photoluminescence (PL) intensity of a single wall S-SWNT [263] in which the
PL of type I (mod(2n + m, 3) = 1) is stronger than for type II (mod(2n + m, 3) = 2) S-SWNTs,
though we need to consider the electric dipole vector for each carbon nanotube individually in
terms of its diameter and chiral angle [262,264,265].
The optical absorption intensity is given by the inner product Dcv (k) · P up to linear terms in
kx and ky for a given polarization vector P = (px , py , pz )

3mopt
P · Dcv (K + k) = ± (py kx − px ky ). (36)
2k
Equation (36) shows that the line py kx − px ky = 0 in the 2D BZ denotes the conditions for the
occurrence of a node in the optical absorption for a given polarization vector P = (px , py ). In
the case of graphene, however, the optical transition events take place along equi-energy contours
474 R. Saito et al.

around the K points, and we cannot see the nodes. This phenomena might be observed in graphene
nanoribbons in which a 1D k value is specially selected. The polarization dependence of the optical
absorption relative to the edge of graphene nanoribbon is now an interesting problem. In the case
of a normal semiconductor, since the dipole vector is not a linear function of k, we cannot get a
node in such cases.

3.4.3. Calculation of the electron–phonon interaction

The el–ph interaction which is the focus of this section is expressed by a modification to the tight-
binding parameters that are pertinent to describing the lattice vibrations. In most theoretical works
on the el–ph interaction, modification to the electron transfer energy as a function of the C–C
bond length is considered for only the nearest neighbor C–C bonds as a parameter [219,220,266].
Here we calculated the el–ph interaction not only for long distance C–C bonds, but also for the
so-called on-site el–ph interaction in which the site energy is modified by the vibration [267,268].
Their values are obtained by using the wavefunction and atomic potential as a function of the C–C
distance [253].
Here, we rewrite the wavefunctions appearing in Equations (18) and (19) using a different
notation, which is suitable for calculating the el–ph matrix elements [268]. We then write the
Bloch functions
1  
a,k (r) = √ Cs,o (a, k) eik·Rt φt,o (r − Rt ), (37)
Nu s,o R t

where s = A and B is an index denoting the two carbon atoms in the unit cell, and Rt denotes the
equilibrium atom positions relative to the origin. φt,o denotes the atomic wave functions for the
orbitals o = 2s, 2px , 2py and 2pz at Rt , which are real functions (with no imaginary components).
The potential energy of the lattice V can be expressed by the atomic potentials v(r − Rt ) at
Rt ,

V= v(r − Rt ), (38)
Rt

where v in Equation (38) is given by the first-principles calculation for the Kohn–Sham potential
of a neutral pseudo-atom [253]. The matrix element for the potential energy between the two
different states i = a,k and f = a ,k is then written as

1  ∗
 a ,k (r)|V | a,k (r) = C   (a , k )Cs,o (a, k)
Nu s ,o s,o s ,o
 
× ei(−k ·Ru ,s +k·Ru,s ) m(t  , o , t, o), (39)
u u

with the matrix element m for the atomic potential given by

⎧ ⎫
 ⎨ ⎬
m= φs ,o (r − Rt  ) v(r − Rt  ) φs,o (r − Rt ) dr. (40)
⎩ ⎭
Rt 

The atomic matrix element m thus comes from an integration over three centers of atoms, Rt ,
Rt  and Rt  . We neglect m for the cases for which the three centers t, t  and t  are different from
one another. When we consider only two center integrals, m consists, respectively, of off-site and
 Advances in Physics 475

on-site matrix elements mα and mλ as follows:


mα = φs ,o (r − Rt  ){v(r − Rt  ) + v(r − Rt )}φs,o (r − Rt ) dr,
⎧ ⎫
 ⎨ ⎬ (41)
mλ = φs ,o (r − Rt ) v(r − Rt  ) φs ,o (r − Rt ) dr.
⎩ ⎭
Rt   =Rt

The potential v(r − Ri ) is vibrating within an adiabatic approximation with a phonon amplitude
S(Rt ). Then the potential modification δV due to a lattice vibration is given by

δV = v[r − Rt − S(Rt )] − v(r − Rt )
Rt
 (42)
≈− ∇v(r − Rt ) · S(Rt ).
Rt

Since the potential modification δV breaks the periodicity of the lattice, the wavevector for an
electron is no longer a good quantum number and thus the scattering of an electron by the el–ph
interaction occurs. If we consider δV as a perturbation, then the el–ph matrix element is defined
on the basis of perturbation theory: as [139,267–270]
Ma,k→a ,k ≡  a ,k (r)|δV |
a,k (r)

1     (43)
=− Cs∗ ,o (a , k )Cs,o (a, k) ei(−k ·Ru ,s +k·Ru,s ) δm(t  , o , t, o),
Nu s ,o s,o u ,u

where δm(t  , o , t, o) is the atomic deformation potential which consists of the off-site and on-site
deformation potentials δmα and δmλ given by

δmα = φs ,o (r − Rt  ){∇v(r − Rt  ) · S(Rt  ) + ∇v(r − Rt ) · S(Rt )} × φs,o (r − Rt ) dr,
⎧ ⎫
 ⎨  ⎬
δmλ = δRt ,Rt φs ,o (r − Rt  ) ∇v(r − Rt  ) · S(Rt  ) φs ,o (r − Rt  ) dr. (44)
⎩ ⎭
Rt   =Rt 

It is noted that both terms δmα and δmλ are of the same order of magnitude and that they work in
a different way for each phonon mode [271].
The atomic deformation potential for any orbitals and for any vibration can be expressed by a
small number of terms which are defined by the bonding or force constants between atoms along
or perpendicular to the two atoms by using the Slater–Koster scheme [32,253]. The atomic el–ph
matrix elements φ|∇v|φ are thus defined for four fundamental hopping and overlap integrals
denoted by (ss), (sσ ), (σ σ ) and (ππ), which are defined as a function of the C–C distance [253,
268] as follows:

α p (τ ) = φμ (r)∇v(r)φν (r − τ) dr = αp (τ )Î(αp ),
 (45)
λ p (τ ) = φμ (r)∇v(r − τ)φν (r) dr = λp (τ )Î(λp ),

which are, respectively, denoted by α p (τ ) and λ  p (τ ) for the off-site and on-site deformation
potentials. Here Î(αp ) and Î(λp ) are unit vectors describing the direction of the off-site and on-site
476 R. Saito et al.

Figure 33. (a) The nine non-zero off-site deformation potential vectors α p . The dashed curves represent
the atomic potentials. (b) The six non-zero on-site deformation potential vectors λ  p . The dashed curves
represent the atomic potentials. For λss , λσ σ and λπ π , the two same orbitals are illustrated by shifting
them with respect to each other. Reprinted figure with permission from J. Jiang et al., Physical Review B
72, pp. 235408–235411, 2005 [268]. Copyright © (2005) by the American Physical Society.

Figure 34. (a) The off-site deformation potential α p and (b) the on-site deformation potential λ  p as
a function of inter-atomic distance. The vertical line corresponds to 1.42 Å which is the C–C dis-
tance in graphite Reprinted figure with permission from J. Jiang et al., Physical Review B 72, pp. 235408–
235411, 2005 [268]. Copyright © (2005) by the American Physical Society.

deformation potential vectors α p and λ  p , respectively [271], and p = μν as given in Equation (45).
The 2p orbital φμ (φν ) is along or perpendicular to the bond connecting the two carbon atoms and
τ is the distance between the two atoms.17 In Figure 33, we show the non-zero matrix elements for
 p atomic deformation potentials for 2s, σ and π atomic orbitals.
the (a) off-site α p and (b) on-site λ
In Figure 34, the calculated values of α p and λ  p are plotted as a function of inter-atomic
distance between two carbon atoms [268]. At r = 1.42 Å, which is the bond length between a
carbon atom and one of its nearest neighbors, we have απ π ≈ 3.2 eV/Å, and |λπ π | ≈ 7.8 eV/Å, and
|απ σ | ≈ 24.9 eV/Å. In order to calculate the el–ph matrix element of Equation (43) for each phonon
mode, the amplitude of the atomic vibration S(Rt ) for the phonon mode (ν, q) is calculated by


S(Rt ) = Aν (q) n̄ν (q)eν (Rt )e±iων (q)t . (46)
 Advances in Physics 477

Here ± is for phonon emission (+) and absorption (−), respectively, and A, n̄, e and ω are the
zero-point phonon amplitude, number, eigenvector, and frequency, respectively. At equilibrium,
the phonon number in Equation (46) is determined by the Bose–Einstein distribution function
nν (q) for phonon ν:
1
nν (q) = ω/k T . (47)
e B −1
Here, T = 300 K is the lattice temperature at room temperature and kB is the Boltzmann constant.
For phonon emission, the phonon number is n̄ = n + 1, while for phonon absorption, n̄ = n. The
amplitude of the zero-point phonon vibration is


Aν (q) = , (48)
Nu MC ων (q)

where MC is the mass of a carbon atom and the phonon eigenvector eν (Rt ) is given by diagonalizing
the dynamical matrix Eq. (17)18 .

3.5. Calculation of excitonic states

In order to calculate the excitonic states, we first introduce in section 3.5.1 the Bethe–Salpeter
equation Equation (49) which makes a localized wavefunction in real space. Using the exciton
wavefunction, we show how to obtain the exciton–photon matrix element in Section 3.5.2 and the
exciton–phonon matrix element in Section 3.5.3 within the tight-binding method.

3.5.1. The Bethe–Salpeter equation for exciton states

The exciton is a photo-excited electron and hole pair that is bonded by an attractive Coulomb
interaction. In a SWNT, because of its 1D properties, the e–h binding energy becomes as large as
1 eV, so that exciton effects can be observed even at room temperature. Thus excitons are essential
for explaining optical processes in SWNTs, such as optical absorption, photoluminescence and
resonance Raman spectroscopy. The localization of the wavefunction can be obtained by mixing
different k states with one another. The equation for making localized wavefunctions is called the
Bethe–Salpeter equation.
Here, we show the Bethe–Salpeter equation for the tight-binding method in order to calculate
the exciton energy n and the corresponding wavefunction n [120,147,148,272,273]. Since
the exciton wavefunction is localized in real space by a Coulomb interaction, the wavevector
of an electron (kc ) or a hole (kv ) is not a good quantum number any more, and thus the exciton
wavefunction n for the nth exciton energy n is given by a linear combination of Bloch functions
at many kc and kv wavevectors. In the case of carbon nanotubes, since the range of the Coulomb
interaction is larger than the nanotube diameter, the mixed k’s are selected near the kii point on one
cutting line of the 1D BZ [148]. The mixing of different wavevectors by the Coulomb interaction
is obtained by the Bethe–Salpeter equation [147]

{[E(kc ) − E(kv )]δkc kc δkv kv + K(kc kv , kc kv )} n
(kc kv ) = n n
(kc kv ), (49)
kc kv

where E(kc ) and E(kv ) are the quasi-electron and quasi-hole energies, respectively (see
Equation (52)). Here “quasi-particle” means that the particle has a finite lifetime in an excited
state because of the Coulomb interaction. Equation (49) is solved by a matrix that includes many
478 R. Saito et al.

kc and kv points. The mixing term of Equation (49) which we call the kernel, K(kc kv , kc kv ), is
given by
K(kc kv , kc kv ) = −K d (kc kv , kc kv ) + 2δS K x (kc kv , kc kv ) (50)

with δS = 1 for spin singlet states and 0 for spin triplet states [167]. The direct and exchange
interaction kernels K d and K x are, respectively, given by [274]

K d (kc kv , kc kv ) ≡ W (kc kc , kv kv )


= dr drψk∗ (r )ψkc (r )w(r , r)ψkv (r)ψk∗v (r), (51)
c

K x (kc kv , kc kv ) = dr drψk∗ (r )ψkv (r )v(r , r)ψkc (r)ψk∗v (r),
c

where the functions w and v are the screened and bare Coulomb potentials, respectively, and ψ
denotes the quasi-particle wavefunction. The quasi-particle energies are the sum of the single
particle energy ((k)) and the self-energy ((k)),

E(ki ) = (ki ) + (ki ) (i = c, v), (52)

where the self-energy (k) is expressed by


(kc ) = − W (kc (k + q)v , (k + q)v kc ),
q
 (53)
(kv ) = − W (kv (k + q)v , (k + q)v kv ).
q

In order to determine the kernel and self-energy, the single particle Bloch wavefunction ψk (r)
and the screening potential W are evaluated by either a first-principles calculation [147] or by using
an extended tight-binding wavefunction within a random phase approximation (RPA) calculation
[148]. In the RPA, the static screened Coulomb interaction for π electrons is expressed by

V
W= , (54)
κ(q)

with a static dielectric constant κ and a dielectric function (q) = 1 + v(q)(q). For describing
the exciton energy and exciton wavefunction it is essential to select a reasonable function for the
unscreened Coulomb potential v(q) [120,148]. For 1D materials, the Ohno potential is commonly
used for the unscreened Coulomb potential v(q) for π orbitals [275] at two sites, Ru s and R0s , (u:
unit cell, s: atom position) with

U
v(|Ru s − R0s |) =  , (55)
((4π0 /e2 )U|Rus − R0s |)2 + 1

where U is the energy cost to place two electrons on a single site (|Rus − Ros | = 0) and this energy
cost is taken as U ≡ Uπa πa πa πa = 11.3 eV for π orbitals [275]. The Ohno potential works well in
reproducing the ground state and low-energy electronic excitations [276].
 Advances in Physics 479

3.5.2. Exciton–photon matrix element

The exciton–photon matrix element Mex-op is given by a linear combination of the electron–photon
n∗
matrix element Dk at k, weighted by Zkc,kv


Mex-op =  0 |Hel-op |0
n
= n∗
Dk Zkc,kv , (56)
k

where 0n is the exciton wavefunction with a q = 0 center of mass momentum. Since the center
of mass momentum is conserved before and after an optical transition, only q = 0 excitons can
be excited.
In the case of a SWNT, since the lattice structure is symmetric under a C2 rotation around an
axis which is perpendicular to the nanotube axis and goes through the center of a C–C bond, the
C2 exchange operation between A and B carbon atoms in the hexagonal lattice is equivalent to
the exchange of k and −k states. Since the exciton wavefunction of a carbon nanotube should
transform as an irreducible representation of the C2 symmetry operation, we can get A1 , A2 , E
and E ∗ symmetry excitons [142]. For example, the A1 and A2 exciton wavefunctions which are,
respectively, symmetric and antisymmetric under a C2 rotation, are given by

1  n + +
| n
0 (A1,2 ) =√ Zkc,kv (ckc ckv ∓ c−kc c−kv )|0, (57)
2 k

where k and −k are located around the K and K  points, respectively, and −(+) in ∓ corresponds
to an A1 (A2 ) exciton.19 When we use the relation Dk = D−k , the excitonic-optical (ex-op) matrix
elements for the A1 and A2 excitons are given by

Mex-op (An1 ) = 0,
√  (58)
Mex-op (An2 ) = 2 n∗
Dk Zkc,kv .
k

Equation (58) directly indicates that A1 excitons are dark and only A2 excitons are bright, which is
consistent with the predictions by group theory [277]. Because of the spatially localized exciton
wavefunction, the exciton–photon matrix elements are greatly enhanced (on the order of 100
times) compared with the corresponding electron–photon matrix elements [273].

3.5.3. The exciton–phonon interaction

The exciton–phonon interaction is given by a linear combination of el–ph interactions weighted
by the exciton wavefunction. Using creation and annihilation operators, the el–ph interaction for
a phonon mode (q, ν) is given by

+ + +
Hel-ph = [Mk,k+q
ν
(c)c(k+q)c ckc − Mk,k+q
ν
(v)c(k+q)v ckv ](bqν + bqν ), (59)
kqν

+
where M(c) (M(v)) denotes the el–ph matrix element for the conduction (valence) band, and bqν
(bqν ) is a phonon creation (annihilation) operator for the νth phonon mode at q. From Equation (59),
we obtain the exciton–phonon matrix element between the initial state | q1 n1
 and a final state | q2
n2
,
480 R. Saito et al.

by writing
Mex−ph =  q2
n2
|Hel-ph | q1
n1


= [Mk,k+q
ν n2∗
(c)Zk+q,k−q1 n1
Zk,k−q1 − Mk,k+q
ν n2∗
(v)Zk+q2,k n1
Zk+q2,k+q ], (60)
k

with q = q2 − q1 accounting for momentum conservation. Since the electron–phonon interaction

smoothly changes as a function of k for the region of the exciton wavefunction, the value of the
exciton–phonon interaction is similar to the electron–phonon interaction [273].

3.6. The resonance Raman process

3.6.1. Matrix elements for the resonance Raman process
Combining all the matrix elements discussed above, we can formulate the first-order Stokes Raman
intensity of graphene by time dependent perturbation theory as
 
 1  D2 [M (k → k, c) − M (k → k, v)] 2
 el-ph el-ph 
Iel =  k
 , (61)
L [E − Ecv (k) + iγ ][E − Ecv (k) − Eph + iγ ] 
k

where γ is the width of the resonance Raman window (Section 3.6.3) [278]. The γ value is
essential in determining Iel as a function of laser excitation energy (Raman excitation profile).
When we use the exciton–photon and exciton–phonon interactions, we apply the formula to the
Raman intensity of SWNTs as follows:
 2
1  M 
 ex-op (a)Mex-ph (a → b)Mex-op (b) 
Iex =  
L (E − Ea + iγ )(E − Ea − Eph + iγ ) 
a
 2 (62)
1  Mex-op (a)2 Mex-ph (a → a) 
 
=  .
L (E − Ea + iγ )(E − Ea − Eph + iγ ) 
a

In the second line of Equation (62), we assume that the virtual state b can be approximated by
the real state a.20 In the case of a first-order Raman process, since q = 0, the matrix element of
Equation (60) is simplified as

Mex-ph = [Mk,k
ν
(c) − Mk,k
ν
(v)]|Zk,k |2 . (63)
k

When we consider the second-order Raman intensity, we should consider q  = 0 phonon scat-
tering. In this case, the exciton–phonon interaction between an A2 exciton state and an E exciton
state is important, in particular, for the case where the E exciton state consists of an electron near
the K point and a hole near the K  point and vice versa. Here the inter-valley exciton–phonon
interaction is generally large.

3.6.2. Matrix elements for double resonance Raman scattering

The two-phonon process is described in quantum mechanical terms by using fourth-order, time-
dependent perturbation theory and the scattering intensity can be calculated using:
 2
   

I(ω, Elaser ) ∝  Jm ,m (ω1 , ω2 ) , (64)
   
i m ,m ,ω1 ,ω2
 Advances in Physics 481

in which the summation is taken over two intermediate electronic states m and m and the cor-
responding phonon frequencies ω1 and ω2 with phonon wavevectors −q1 and −q2 , respectively,
and for the initial states i, after taking the square of the scattering amplitude, Jm,m that is given by

Mex-op (im )Mex-ph (m m )Mex-ph (m m)Mex-op (mi)

Jm ,m (ω1 , ω2 ) = , (65)
(Em i − ω1 − ω2 − iγ )(Em i − ω1 − iγ )(Emi − iγ )

where Em i ≡ Elaser − (Em − Ei ) and Mex-op (mi) denote the optical transition from i to m states,
etc. In general, energy and momentum conservation for the incident (i) and scattered (s) electrons
requires:

Es = Ei ± Eq1 ± Eq2 , (66)

ks = ki ∓ q1 ∓ q2 , (67)

where + (−) in Equation (66) and − (+) in Equation (67) correspond to phonon absorption and
emission with the wavevectors q1 and q2 . By considering ks ≈ ki (see Section 1.5), momen-
tum conservation requires q2 ≈ −q1 for satisfying the DR condition for two of the three energy
denominators in Equation (65).

3.6.3. Resonance window width

The resonance width, or γ in eV, of the Raman excitation profile is related by the uncertainty
relation in quantum mechanics and to the lifetime of the photo-excited carriers. Usually, the
dominant contribution to the lifetime of the carriers in the Raman spectra is in an inelastic scattering
process by the emission or absorption of phonons. The Raman spectral width,  in cm−1 , in the
Raman spectra, on the other hand, which has a different physical value from γ , is related to
the phonon lifetime.  is determined by the elastic (or inelastic) scattering of a phonon due to
defects, anharmonicity or by the electron–phonon interaction. The carrier transition rate τ (=/γ ),
[139,263,267,278] is estimated by the Fermi Golden rule for the electron–phonon matrix elements
[268,269]. For metallic systems (graphene and M-SWNTs), an additional interaction of phonons
with free electrons can shorten the lifetime (broaden the γ values) significantly, and this additional
interaction is known as the Kohn anomaly (KA) (Section 3.6.5).
The transition rate 1/τ or the scattering rate per unit time of an excited electron from an initial
state k to all possible final states k by the νth phonon mode is given by [263]

1
= Wkν
τν
 
S  |Dν (k, k )|2 dE(μ , k  ) −1
= (68)
8π Mdt μ ,k  ων (k − k) dk 

δ(ω(k ) − ω(k) − ων (k − k)) δ(ω(k ) − ω(k) + ων (k − k))
×  +  ,
eβων (k −k) − 1 1 − e−βων (k −k)

where S, M, dt , β and μ , respectively, denote the area of the graphene unit cell, the mass of a
carbon atom, the diameter of a SWNT, 1/kB T (where kB is the Boltzmann constant), and the
cutting line indices of the final state. Here Dν (k, k ) denotes a matrix for scattering an electron
from k to k by the νth phonon mode. The relaxation process is restricted to 24 possible phonon
scattering processes satisfying energy–momentum conservation [267]. The two terms in braces
in Equation (68), respectively, represent the absorption and emission processes of the νth phonon
482 R. Saito et al.

mode with energy ων (k − k). The calculation of the transition rates as in Equation (68) have
been considered by the Ferrari group using another approach [151,219].
For the result, in the case of S-SWNTs, we can obtain calculated γ values in agreement with
experiments by just considering the electron–phonon coupling model [278]. The calculated γ value
shows a strong dependence on chirality and diameter for S-SWNTs. However, the calculated γ
value for M-SWNTs is much smaller than the experimental γ value which shows the presence of

Figure 35. (a) The calculated G-band spectra for S-SWNTs with the same family number p = 2n + m = 28.
(b) The calculated electron–phonon matrix elements vs. chiral angle θ for the LO and TO phonons and
for two different 2n + m family numbers (22 and 28). (c) Plot of γ vs. θ for members of p = 28. (d)
Plot of R vs. θ for three 2n + m families of M-SWNTs. (e) The angle R between the circumferential
vector K1 and the cutting line for the polar coordinate of a k vector at the van Hove singular point. (f)
The angle φ between the tube axis and the phonon eigenvector direction for a (12,6) SWNT. The calcu-
lated angles φ vs. θ for the TO phonons (g) and the LO phonons. (h) For the results for the LO phonons
as a function of θ (fitted by the function of Equation (70) (see text)). Reprinted with permission from
J.S. Park et al., Physical Review B 80, p. 81402, 2009 [86]. Copyright © (2009) by the American Physical
Society.
 Advances in Physics 483

an additional scattering path associated with the charge carriers in M-SWNTs. Such a scattering
path might come from the electron–electron interaction, but this theory is not yet well described.

3.6.4. G-band intensity for semiconducting SWNTs

Next, we consider the G-band Raman intensity as a function of (n, m) [86]. Figure 35(a) shows the
calculated resonance Raman spectra for the G-band for type I S-SWNTs [86] with family number
p = 2n + m = 28. The (n, m) SWNTs with the same family number p have a similar diameter
and Eii value to one another. In this figure, values for EL and γ (see Figure 35(c)) are taken from
S
E22 for each (n, m) SWNT. The chiral angle can vary from θ = 0◦ to θ ∼ 30◦ . The intensity of
the G− peak (TO) is always smaller than that of the G+ peak (LO), because MR,LO ep > MR,TO
ep for
the electron–phonon matrix elements, as shown in Figure 35(b), in which the above notation R
indicates Raman scattering. In particular, the intensity of the G− peak vanishes for a (14,0) SWNT,
since MR,TO
ep for zigzag SWNTs is zero, as shown in Figure 35(b). Here MRep is calculated for
the phonon amplitude at 300 K. These calculated G-band Raman spectra can be compared with
previous experimental results which show only one peak in the G-band spectra of (n, m) SWNTs
with smaller chiral angles [193,279].
In order to explain the chiral angle dependence of the el–ph matrix elements for the LO and
TO phonons, we obtain the analytical formulae for the el–ph matrix element within the effective
mass approximation [196]

MR,LO
ep ≡ e(k), ωLO |Hep |e(k) = gu cos R (k),
(69)
MR,TO
ep ≡ e(k), ωTO |Hep |e(k) = −gu sin R (k),

respectively, where g is the el–ph coupling constant, u the phonon amplitude and R (k) is defined
by an angle between the k vector from the K point of the 2D BZ to the van Hove singular point, kii ,
and the circumferential direction vector, K1 , [32,136,177] as shown in Figure 35(d). The values of
g are consistent with the work by Basko et al. [151,219]. Since R (k) is zero for all zigzag SWNTs
(k  K1 ), we obtain MR,TO
ep = 0, while MR,LO
ep has a maximum value [196]. The meaning of R
vs. θ for SWNTs with the same family number p is shown in Figure 35(e). For the TO phonon
mode, the magnitude of the matrix element MRep for SWNTs with a similar θ value increases with
decreasing dt because of the diameter dependence in the circumferential direction [86] as shown
in Figure 35(b). The angle φ between the SWNT- axis and the phonon eigenvector for the LO
and TO phonons [280] is essential for determining the value of the el–ph matrix element. In fact,
when we consider φ, then Equation (69) is modified and becomes

e(k), ωLO |Hep |e(k) = gu cos(R (k) + φ),

(70)
e(k), ωTO |Hep |e(k) = −gu sin(R (k) + φ).

Figure 35(g) and (h) show that the calculated angle φ here changes smoothly as a function of θ
[86]. The sum φLO + φTO for a general chiral angle θ always becomes π/2, because of symmetry.
The angle φ vs. θ for the LO and TO phonons can be fitted by the chiral angle dependence
(A + Bθ + Cθ 2 ) sin(6θ ), where A, B and C are fitting parameters and θ is the chiral angle in units
of degrees (◦ ). Values obtained for A, B and C for φLO , are A = 26.9, B = −76.3 and C = 84.5,
respectively, and for φTO , the corresponding values are A = −26.7, B = 75.4, and C = −83.2.
The units for the fitting parameters are degree (◦ ). This φ dependence of φLO and φTO should be
taken into account when carrying out Raman spectral calculations.
484 R. Saito et al.

3.6.5. G-band intensity for metallic SWNTs: The Kohn Anomaly

In the case of metallic SWNTs, the G-band spectra become soft and broad and they are represented
by the Breit–Wigner–Fano (BWF) lineshape [128]. The BWF lineshape has been widely observed
for graphite intercalation compounds as a function of doping concentration and also as a function
of carrier density [281]. The phonon softening phenomena for metallic SWNTs is understood
by the electron–phonon interaction between phonons and free electrons at the Fermi energy,
(EF ), and these phenomena are known as the KA effect [127] which has been widely discussed
[151,193,194,197–199,243]
The phonon frequency ω of the LO and TO phonon modes at the  point for M-SWNTs is
modified by the KA effect, which we understand by second-order perturbation theory. The phonon
energy ω becomes ω = ω(0) + ω(2) , where ω(0) is the original phonon frequency without the
el–ph interaction, and ω(2) is the quantum correction to ω(0) which is given by [196]
 |eh(k)|Hep |ω(0) |2
ω(2) = 2
k
ω(0) − [Ee (k) − Eh (k)] + i (71)
× {f [Ee (k) − EF ] − f [Eh (k) − EF ]},

in which the factor 2 comes from spin degeneracy, and Ee (k) [Eh (k)] denotes the electron (hole)
energy as a function of wave vector k, while eh(k)|Hep |ω(0)  represents the el–ph matrix element
for creating an e–h pair with wave number k by the el–ph interaction Hep and f (E) is the Fermi
distribution function. The G-band spectral width is given by the decay width  in Equation (71),
which is calculated self-consistently by evaluating  = −Im(ω(2) ) [196,205]. The electron–
phonon interaction is used, too, for defining the ω(0) and thus we should be careful about not
double counting the constituents of this interaction [198].
Figure 36(a) shows the calculated Raman spectra for the G-band of M-SWNTs with family
number p = 30 and EF =0. The EL and γ values (see Figure 36(c)) are taken from E11 M
for each
(n, m) SWNT. The G peak intensity is larger than that of the G peak, because the G (G+ ) peak
− + −

corresponds to the LO (TO) phonon due to the LO phonon softening, in which MR,LO ep > MR,TO
ep
for any θ value, as shown in Figure 36(b). The relative intensities of the two peaks, G and G− ,
+

are affected by the Raman spectral width which relates to the phonon lifetime, . For the (10,10)
armchair SWNT, the G+ (TO) peak width is significantly smaller than those for the G− (LO) peak

Figure 36. (a) The calculated G-band spectra of M-SWNTs with the same family number p = 30 and EF =0.
(b) el–ph matrix elements vs. θ for the LO and TO phonons and for two different 2n + m family numbers.
Open-circles indicate the Mep values for the family number p = 30. (c) γ vs. θ for members of p = 30.
Reprinted with permission from J.S. Park et al., Physical Review B 80, p. 81402, 2009 [86]. Copyright ©
(2009) by the American Physical Society.
 Advances in Physics 485

Figure 37. The calculated G-band spectra for three M-SWNTs with different chiral angles taken by changing
the Fermi energy from EF = −0.2 eV to 0.2 eV. (a) (10,10). (b) (11,8) and (c) (15,0). Reprinted with permis-
sion from J.S. Park et al., Physical Review B 80, p. 81402, 2009 [86]. Copyright © (2009) by the American
Physical Society.

and of the G+ peaks for the other chiral tubes. Therefore, the G+ peak intensity of the (10,10) tube
becomes large compared with the other chiral SWNTs, even though the MR,TO ep for the armchair
tube has a smaller value than that for the other chiral tubes. Since the Raman peak intensity is large
for large Mep and small  values, the chiral angle dependence of these values gives an irregular
behavior to the G+ /G− spectra as a function of (n, m), as seen in Figure 36.
For a zigzag SWNT ((15, 0)), only the G+ peak appears, because MR,TO ep vanishes for zigzag
nanotubes as seen in Equation (69). The other chiral tubes in this p = 2n + m > 30 family, (11,8),
(12,6), (13,4) and (14,2), show various intermediate intensity ratios. In Figure 36(c), we show that
γ decreases monotonically with increasing θ. Because of the small difference between the γ and
the el–ph coupling for the LO phonon as compared to that for the TO phonon as a function of θ,
the G− peak intensity does not show a large change for the different chiral SWNTs. These results
show that the G-band intensity for both the G+ and G− components depends on θ, but the Raman
intensity is more sensitive to the EF position, especially for M-SWNTs.
This effect is shown more clearly by varying the Fermi Level, as shown in Figure 37(a), where
the calculated G-band spectra is plotted vs. EF at 300 K for a (10,10) armchair SWNT. Here
neither are the changes in the C–C bond nor the changes in the Eii transition energy by doping
with electrons or holes considered [86].
In Figure 37, the positive (negative) Fermi energy +EF (−EF ) corresponds to electron (hole)
doping. When EF is changed from EF = 0, the G− peak shows a frequency shift and a sharpening
of the spectral width, while the G+ peak does not show any change in intensity or width. The el–ph
interaction for the photo-excited electron does not couple to the TO phonon for armchair SWNTs
[196]. For the chiral M-SWNT (11,8) as shown in Figure 37(b), both the LO and TO phonons
couple to the intermediate e–h pair state, which is excited by a lower energy phonon. The TO
phonon becomes harder for EF = 0 eV, since the intermediate state of an e–h pair for E < ωTO
contributes to a TO phonon hardening [196]. In the case of the (15,0) SWNT, the G+ peak always
vanishes because of a vanishing MR,TO ep (See Figure 36(b)).
The matrix element Mep for the KA effect in Equation (71) is given by [196]
KA

MKA,LO
ep ≡ eh(k)|Hep |ωLO  = igu sin KA (k),
(72)
MKA,TO
ep ≡ eh(k)|Hep |ωTO  = −igu cos KA (k),
486 R. Saito et al.

Figure 38. (a,c,e) Experimental G-band Raman spectra which are given by the electro-chemical doping effect.
(a) Vg = 1.5 to −1.5 V. (c) Vg = 1.9 to −1.3 V. (e) Vg = 1.3 to −1.3 V with the traces taken at uniform changes
in Vg . (b,d,f) Calculated G-band Raman spectra taken by changing the Fermi energy EF in equal steps (b) 0.45
to −0.45 eV, (d) 0.60 to −0.42 eV and (f) 0.39 to −0.39 eV. The tube chiralities are: (a,b) (11,11), (c,d) (24,4)
and (e,f) (12,0). Reprinted with permission from J.S. Park et al., Physical Review B 80, p. 81402, 2009 [86].
Copyright © (2009) by the American Physical Society.

where KA (k) is defined as the angle between the k point taken on a cutting line21 for two-linear
metallic sub-bands and the nanotube circumferential direction of a unit vector, K1 . For the armchair
nanotube, the cutting line for the two-linear metallic bands lies on the nanotube axis direction unit
vector, and then KA is π/2 (−π/2), which gives a vanishing MKA,TO ep . For a chiral nanotube,
KA is not zero, since the cutting line for the two-linear metallic bands deviates from the K point
due to the curvature effect, and then the KA effect appears in both the LO and TO modes. For the
zigzag M-SWNT (15,0), only the G− peak that is related to the LO phonon appears, since the el–ph
matrix element for the Raman scattering process for iTO phonon has a zero value for a zigzag
tube, as shown in Figure 36(b). Thus, only an LO phonon softening is measured experimentally,
even though a TO phonon hardening was expected theoretically.
The calculated G-band Raman spectra vs. EF can be directly compared with the experimental G-
band Raman spectra which are obtained for electro-chemically doped individual SWNTs, as seen
in Figure 38 [86]. Here, we assume EF =0.3Vg according to Sasaki et al. [196]. The experimental
Raman spectra are shown in Figure 38(a,c,e), and the corresponding calculated Raman spectra
are shown in Figure 38(b,d,f). In Figure 38(a), the experimental Raman spectra show only a LO
phonon softening, and a TO phonon frequency shift does not occur. As mentioned above, for the
armchair SWNT, the TO phonon frequency shift does not appear and only LO phonon softening
appears. Therefore, we can predict that Figure 38(a) shows an armchair-type behavior by changing
the gate voltage. The RBM peak for these experimental Raman spectra appears at 161 cm−1 with
EL = 1.72 eV. Then we can select possible (n, m) values for a tube by using a simple tight-binding
(STB) model with γ0 = 2.9 eV for simplicity and by using the relation between the RBM frequency
and diameter, ωRBM (cm−1 ) = 248/dt (nm)22 , the possible for identifying the possible (n, m) values
for SWNTs we obtain these (n, m) values as (19,1), (18,3), (14,8) and (11,11). If our prediction
is correct, Figure 38(a) can be assigned as an (11,11) armchair SWNT. Figure 38(c) and (e) are
assigned as chiral (24,4) and zigzag (12,0) SWNTs, respectively, from the possible (n, m) values,
 Advances in Physics 487

{(21, 6), (22, 4), (23, 2)} and {(10, 4), (11, 2), (12, 0)}. For the chiral M-SWNTs, not only is there
a LO phonon softening, but there is also a TO phonon hardening that appears in the calculation
of the G-band Raman spectra vs. EF . However, in Figure 38, the TO peak is too small to see on
the intensity scale of the figure. Figure 38(e) shows that the zigzag SWNT has only a G− peak
and thus only the LO phonon softening appears by changing EF , experimentally. Brown et al.
[128] and others [282,283] pointed out that asymmetric line shapes appear in the G− band Raman
spectrum for metallic tubes, which is related to the Fano resonance (Breit–Wigner–Fano, BWF line
shape) lines. Recently, Farhat et al. showed that this asymmetry is sensitive to the relative position
of the scattered light energy relative to Eii(M) , suggesting that the electron–electron interaction is
important for understanding BWF lineshapes [206].

4. Raman spectra of graphene

In the family of sp2 carbon systems, mono-layer graphene is the simplest crystal structure (see
Figure 2), having the highest symmetry and, consequently, the simplest Raman spectra (see
Figure 39). The big rush into graphene research started in 2004 [7,55,61]. The large research
community that had become knowledgeable about the Raman spectroscopy associated with other
nano-carbon systems was ready for a quick appreciation of the Raman spectra in mono-layer
graphene as a perfect prototype spectra for the study and characterization of sp2 carbons more
generally [112,114,250]. For example, the detailed study of effects of inter-layer coupling on
the electronic structure was carried out using the dispersion of the G -band when changing the
excitation laser energy in bi-layer graphene [284]. Strain[285–287], charge transfer and disorder
effects due to doping, top gates and different types of substrates were addressed using the G and
G -bands of graphene [151,191,192,195,284–291]. Actually, graphene provides an ideal system
to study defects using light as a probe because there are no aspects related to the penetration
depth [3,65,292]. Interestingly, most of these findings in graphene are helping our basic under-
standing of long standing experimental results on carbon nanotubes and other nanocarbon systems
[151,191,192,195,288,289,293]. Another important aspect that is peculiar to graphene is the fact
that the G-band phonons (energy of 0.2 eV) can promote electrons from the valence band to the
conduction band. This happens because graphene is a zero gap semiconductor, and the linear E(k)
dispersion relation centered around the Dirac cones for the valence and conduction bands (see
Figure 9(a)) near the Fermi level makes the effect especially interesting, and it gives rise to a

Figure 39. Raman spectrum of single-layer graphene in comparison to graphite measured with a
Elaser = 2.41 eV (514 nm) laser. The two most intense features are named the G and G -bands. The Raman
spectrum of pristine mono-layer graphene is unique among sp2 carbons, i.e., the second-order G feature is
very intense when compared to the first-order G-band feature (see discussion in Section 4.1) [112,250].
488 R. Saito et al.

renormalization of the electronic and phonon energies, including a sensitive dependence of the
electronic structure on electron or hole doping [195]. Raman imaging can be used to define the
number of layers in different locations of a given graphene flake by measuring the dependence
of the Raman spectra (e.g., for the G-band intensity) on the number of scattering graphene layers
[294]. It is true that such information has to be analyzed with care since doping and other phys-
ical phenomena perturb the graphene Raman spectra. The effect of environmental interactions
on few-layer graphene samples have also been studied using Raman spectroscopy, including the
epitaxial growth of graphene on a substrate [295]. In this brief survey a number of important topics
are reviewed, including the spectra of mono-layer graphene, the layer number dependence in few
layer graphene, disorder-related phenomena, edge phonon phenomena, polarization effects and
the effects of doping.

4.1. The G-band and G -band intensity ratio

The first intriguing result that was observed when the Raman spectrum from single layers of
graphene was measured was the unusual G to G intensity ratio IG /IG . While the second-order
G Raman band in 3D graphite has a much smaller intensity than the first-order Raman-allowed
G-band, in single-layer graphene the G -band intensity is much stronger, reaching 4 times the
G-band intensity. In principle, the G to G intensity ratio can be used to determine the number of
layers in a few layer graphene sample, since IG /IG is reduced by increasing the number of layers.
However, it is also true that this ratio is sensitive to doping [220,291] and disorder [130], and
the intermixing of information (doping vs. number of layers) makes it complicated to use Raman
spectroscopy to determine the number of layers in few-layer graphene samples accurately, unless
the parameters for the environmental effects are clearly delineated.
Basko [219,296] argues that the special Raman spectrum of mono-layer graphene (1-LG) is an
indication that the very strong G -band comes from a fully resonant scattering process where both
the el–ph absorption and emission are resonant, as well as the el–ph and hole–phonon scattering
processes, so that the absorption and recombination occur at different Dirac cones. The very strong
G -band intensity could also be related to different el–ph matrix elements near the K point (for the
G -band) and near the  point (for the G-band) phonons [218–220]. The fully resonant process
should, in principle, be much more probable than the other processes which exhibit a virtual
(non-resonant) state. However, this can only happen if the electron and hole electronic dispersion
relations are symmetric within the phonon uncertainty and if the electron and hole scattering by
phonons is equally probable. Since the electron wave function overlap in graphene results in a
different normalization for the valence and conduction bands, an e–h dispersion asymmetry is
introduced, and for this reason the two processes could select double resonance phonons with
somewhat different q vectors. This asymmetry is relatively small and is generally neglected in
common descriptions of the electronic structure of graphene in terms of mirror band cones. More
theoretical and experimental work is required to fully understand the differences in the el–ph vs.
hole–phonon scattering, including the differences in the matrix elements for these two processes.

4.2. Layer number dependence of G -band

4.2.1. The number of graphene layers with AB stacking
Because of its dependence on the layer number, the Raman G -band has been used to characterize
the number of layers in few layer graphene samples and the stacking order between these layers,
as shown in Figure 40. The Raman spectra for highly ordered pyrolytic graphite (HOPG) and
for turbostratic graphite (which has random inter-layer stacking) are also shown in Figure 40 for
comparison. To explain this observed behavior, we first remind the reader about the dispersive
 Advances in Physics 489

Figure 40. The differences in the G Raman band for (a) 1-LG, (b) 2-LG, (c) 3-LG, (d) 4-LG, (e) HOPG
and (f) turbostratic graphite. All spectra are measured with Elaser = 2.41 eV. The original work was done by
Ferrari et al. [112] and is summarized in the review article of Malard et al. [5].

behavior discussed in connection with Figures 24 and 27. Then we turn to a discussion of the elec-
tronic properties of bi-layer graphene with AB Bernal layer stacking (as also occurs in graphite),
since the band structure change from mono-layer to bi-layer graphene is the most striking and both
mono-layer and bi-layer graphene have been probed by Raman scattering [284]. The change in the
electronic structure of graphene due to layer stacking can be probed in some detail by the DR Raman
features (see Section 3.1.3), and most sensitively by the detailed lineshape of the G -band [112].
Bi-layer graphene has a 4-peak G -band spectrum (Figure 40(b)) while mono-layer has a 1-peak
G -band (Figure 40(a)), and this fact is explained by the special electronic structure of bi-layer
graphene, which consists of two conduction and two valence bands [112], as discussed below.
Figure 41(a) shows the dispersion (peak frequency as a function of Elaser ) of each one of the four
peaks in Figure 40(b), which comprises the G -band for bi-layer graphene with AB stacking. The
double Raman resonance processes for bi-layer graphene are shown in Figure 41(b)–(e), where the
diagrams show the possible DR Raman processes that give rise to the four G peaks in Figure 41(a).
The processes are labeled by Pij (i, j = 1, 2) [284], where the states with energy Ei in the valence
band and Ej in the conduction band are connected in the photon absorption process using laser
energy Elaser . The highest frequency G peak for a given Elaser energy is associated with the P11
process, since the P11 process has the largest wave vector (q11 ) and the iTO phonon along the
KM direction in the BZ increases its frequency with increasing wave vector q (see Figure 40(b)
and Figure 41). The lowest frequency G -band peak is associated with the process P22 , which
gives rise to the smallest phonon wave vector q22 . Processes P12 and P21 shown in Figure 41(b)
give rise to the two intermediate frequency peaks of the G -band [112,284,297]. This DR Raman
490 R. Saito et al.

model has been used to relate the electronic and phonon dispersion of bi-layer graphene with the
experimental dependence of ωG on Elaser [284].
Tri-layer graphene has 15 possible DR processes [1,4], but the frequency spacings between
these peaks are not large enough to allow identification of each of the 15 scattering events
(Figure 40(c)). Increasing the number of layers increases the number of possibilities for the
G -band DR scattering processes, and an in depth analysis would get more and more complicated
for N-layer graphene (N > 3). However, experimentally the G -band spectra at a typical Elaser
energy (e.g., 2.41 eV) actually gets simpler in appearance when the number of layers increases
(see, for example Figure 40(d) for 4-LG). The spectra of increasing N converge to the two-peak
structure observed in HOPG, where N → ∞, as shown in Figure 40(e). The two-peak structure of
HOPG (Figure 40(e)) is the result of a 3D electron and phonon dispersion, as discussed in [298],
which can be seen as a convolution of an infinite number of allowed DR processes along the third
dimension of N → ∞ graphite.

4.2.2. Characterization of the graphene stacking order by the G spectra

The use of G -band Raman spectroscopy to assign the number of layers has to be a cautious
procedure, because the G -band lineshape is related not only to the number of layers, but also to
the stacking order of these layers. The G -band has actually been used to quantify the structural
ordering along the c- axis in graphite [299–301] much before the rise of the graphene field in 2004.
The change from one peak to two peaks in the G -band profile observed in the Raman spectra from
polycrystalline to crystalline graphite was shown in the late 1970s [223,302]. Raman spectroscopy
studies of carbon materials heat treated at different temperatures Thtt 23 show that, by increasing
Thtt , the G -band changes from a one-peak to a two-peak structure (see Figure 40(e,f)) [299,300].
Lespade et al. [299] associated this evolution with the degree of graphitization of the samples,
suggesting that the origin of the two-peak structure of the G -band in crystalline graphite was
related to the stacking order occurring along the c- axis. Finally, the evolution from the 2D to 3D
aspect (from one to two peaks [303]) has been quantitatively systematized (see also Section 4.2.1).
Barros et al. have used the G -band to identify three G -band peaks due to the coexistence of 2D

Figure 41. (a) Plot of the frequency of the four Raman G -band peaks vs. Elaser observed in
bi-layer graphene. These four peaks arise from the four processes shown in (b)–(e) which com-
prise the G -band scattering processes that are expected for the phonon frequencies in bi-layer
(2-LG) graphene plotted in (a) as a function of laser energy Elaser . Reprinted with permission from
L.M. Malard et al., Physical Review B 76, p. 201401, 2007 [284]. Copyright © (2007) by the American
Physical Society.
 Advances in Physics 491

and 3D graphite phases in more complicated carbon-based materials, such as pitch-based graphitic
foams [101].
The differences between stacked and non-stacked graphene layers became even more clear
when the G-band Raman spectra of AB stacked and misoriented folded bi-layer graphene were
compared [4,295,304]. While the AB stacked bi-layer graphene shows a four-peak structure, as
illustrated in Figure 40(b), the G-band spectra of misoriented bi-layer graphene shows a one-peak
profile, with an upshift of ∼14 cm−1 . This result is generally consistent with the observations
for turbostratic graphite shown in Figure 40 and it was explained as due to changes in the Fermi
velocity of graphene due to interlayer interactions in AB-stacked samples [295]. These aspects
also explain why a broadened single G peak is observed for regions of a sample that contains
domains of mono-layer or bi-layer graphene. For example, CVD-grown graphene often shows
such domains of mono-layer and bi-layer graphene and furthermore the stacking of the layers is
often not AB Bernal stacking [85,305,306]. Using a similar path of reasoning as was followed
to understand the difference in the G -band lineshape between mono-layer graphene and bi-layer
graphene with AB stacking, it is easy to understand that the G lineshape will be different for
ABC and ABA trilayer graphene stacking. Recently, Liu et al. [307] showed that the G -band can
indeed be used to distinguish between ABC versus ABA stacking in trilayer graphene samples.
The G -band for ABC stacked samples is generally broader than that for ABA, and by mapping
the G width, Liu et al. showed that these two types of stacking order coexist in trilayer graphene
samples. By mapping several samples, they showed that about 15% of the samples generated by
the mechanical exfoliation of HOPG are ABC-stacking-like, and this value for the mixed stacking
order is in very good agreement with X-ray studies on HOPG [9].

4.3. D-band and G-band intensity ratio and other disorder effects
Graphene provides an ideal structure to study the effect of disorder on a Raman spectrum, because
in a mono-layer 2D structure one does not have to worry about cascade effects and the penetration
depth of the light [3,65,292]. Here we discuss the effect of disorder caused by low energy Ar+
bombardment [3].

4.3.1. Ar+ ion bombardment on graphene

Raman spectroscopy is one of the most sensitive techniques that can be used to characterize disorder
in the sp2 network of carbon materials [308]. It is widely used to identify disorder in diamond-
like carbon, amorphous carbon, nanostructured carbon, carbon nanofibers, carbon nanotubes and
carbon nanohorns [2,174]. Just as Raman spectroscopy has been used for the characterization of
defects through the observation of symmetry-breaking features in the Raman spectra, point defects
have been used as a characteristic defect that can be readily reproduced [308] and the uses of ion
implantation to create these point defects has been widely adopted.
The first-order Raman spectra of crystalline graphene is shown in Figure 42(a), where the
presence of the Raman-allowed G-band is observed. When graphene is bombarded by Ar+ ions
(starting with a low dose, 1011 Ar+ /cm2 in Figure 42(b)), point defects are formed and the Raman
spectra of the disordered graphene exhibit two new sharp features, named by D and D , appear-
ing at 1345 cm−1 and 1626 cm−1 , respectively, for Elaser = 2.41 eV (Figure 42(b)). The D and
D labels indicate that these Raman bands are induced by disorder [102]. Both of these bands
are dispersive and change frequency when changing Elaser = 2.41 eV. Actually, the DR process
discussed for the symmetry-allowed G -band was developed to explain the dispersive behavior of
the D-band [104]. Instead of electron scattering by two phonons with momentum q and −q, the
breaking of the translational symmetry of crystals can be activated by introducing defects into the
492 R. Saito et al.

lattice. Introducing disorder breaks down the momentum conservation requirement, and phonons
at interior k points of the BZ can contribute to the Raman scattering process. Similar to the case
of the G -band, those scattering processes due to point defects which fulfill the DR process are
privileged in the disorder-induced Raman scattering process discussed in this section.
Finally, if the periodic structure of graphene is largely disordered, for example from a high
defect density caused by applying a large ion dose bombardment, such as 1015 Ar+ /cm2 , the
Raman spectrum evolves into a phonon DOS-like profile, where most of the higher-energy optical
phonon branch would be contributing to the spectra, rather than solely the special phonons fulfilling
the DR process (see Figure 42(c)) [3]. Of course in a fully disordered material, not only are all
phonons activated, but also changes in the structure, bonding and strain fields change the vibrational
frequencies and lineshapes.

4.3.2. The D to G intensity ratio and the LD dependence

Mono-layer graphene samples were bombarded with Ar+ ions and consecutive Raman spectra
were performed to study the evolution of the disorder-induced Raman peaks [3]. Figure 43 shows
the Raman spectra of a graphene mono-layer sample subjected to the ion bombardment procedure
that is described in the beginning of Section 4.3.5. From the pristine sample (bottom spectrum
in Figure 43) to the lowest bombardment dose in Figure 43 (1011 Ar+ /cm2 ), the D-band process
is activated, showing a very small D-band intensity relative to the G peak which is symmetry-
allowed. Within the bombardment dose range 1011 -1013 Ar+ /cm2 , the intensities of the disorder
induced peaks increase. A second disorder-induced peak around ∼1620 cm−1 (the D -band) also
becomes clearly evident at a dose of 1012 Ar+ /cm2 , but we do not focus on this feature here.
The Raman spectra start to broaden significantly above 1013 Ar+ /cm2 , and the spectra end up
exhibiting the graphene phonon density of states-like spectrum, corresponding to Figure 42(c)).
From 1014 Ar+ /cm2 (top spectrum in Figure 43) and above, the Raman scattering shows a lineshape
broadening with no measurable change in the peak frequencies of these broad features.

Figure 42. The Raman spectrum of (a) crystalline graphene, (b) defective graphene, (c) and fully disordered
single-layer graphene. These spectra were obtained with Elaser = 2.41 eV and the graphenes are deposited
on a SiO2 substrate using the mechanical exfoliation method (scotch-tape). Reprinted with permission from
A. Jorio et al., Journal of Physics: Condensed Matter 22, p. 334204, 2010 [309]. Copyright © (2010) by the
Institute of Physics.
 Advances in Physics 493

The development of disorder in sp2 carbon nano-crystallites is conveniently described by

plotting the ID /IG ratio as a function of crystallite size [102]. Here we perform a similar analysis,
but plotting the ID /IG ratio as a function of the average distance LD between defects, as shown
in Figure 44. Note that the ID /IG ratio has a non-monotonic dependence on LD . The ID /IG ratio
is here seen to increase initially with increasing LD , up to LD ∼ 3.5 nm, and then it decreases
for LD > 3.5 nm, consistent with the proposed amorphization trajectory for sp2 carbon nano-
crystallites [131]. Such a behavior indicates the existence of two competing mechanisms that
contribute to the Raman D-band, as described below.

4.3.3. The D to G intensity ratio: the “local activation” model

The impact of a single ion on a graphene sheet causes modifications on two length scales, which
we denote here by rA and rS . These two length scales represent, respectively, the radii of two
circular areas measured from the impact point, as shown in Figure 45. Structural disorder from the
impact position occurs within the shorter radius rS , where the subscript S stands for structurally
disordered. For distances larger than rS but shorter than rA , the lattice structure is preserved but
a break-down of the selection rule is caused by the proximity to the structurally disordered area
(S-region), thus leading to a local enhancement of the Raman D-band. We call this the A-region,
where A stands for activated. When the Raman scattering process occurs at distances larger than
φ = rA − rS from the defective region, only the G-band is active. The ID /IG ratio is then given
as a function of the average distance between two defects, LD , by [3]:
ID
(LD ) ∝ ID (LD ) = CA fA (LD ) + CS fS (LD ). (73)
IG
The intensity IG remains constant, independent of LD , while fA and fS are the fractions of the A
and S areas in the sheet, respectively, with respect to the total area. Both the A and S regions
break momentum conservation and give rise to a D-band. However, the A-regions will contribute

Figure 43. Evolution of the first-order Raman spectra using a λ = 514 nm laser (Elaser = 2.41 eV) to investi-
gate a graphene mono-layer sample deposited on an SiO2 substrate, and subjected to Ar+ ion bombardment.
The Ar+ ion doses from the bottom trace to the top trace are: zero (pristine), 1011 , 1012 , 1013 and 1014
Ar+ /cm2 for 90 eV ions. The spectra in this figure are also displaced vertically for clarity. Reprinted From
Carbon 48(5), M.M. Lucchese et al., pp. 1592–1597 [3]. Copyright © (2010) Elsevier.
494 R. Saito et al.

Figure 44. The ID /IG data points from three different mono-layer graphene samples as a function of the aver-
age distance LD between defects, induced by the Ar+ ion bombardment procedure described in Section 4.3.2.
The solid line is a modeling of the experimental data with Equation (73). The inset shows a plot of ID /IG vs.
LD on a log scale for two samples: (i) a ∼50-layer graphene sample; (ii) a 2 mm-thick HOPG sample, whose
measured values are here scaled by (ID /IG ) ×3.5. Reprinted From Carbon 48(5), M.M. Lucchese et al., pp.
1592–1597 [3]. Copyright © (2010) Elsevier.

Figure 45. (a) Definition of the “activated” A-region (darkest gray) and “structurally disordered” S-region
(dark gray). The radii rS and rA are measured from the impact point which is chosen randomly in this
simulation. (b–e) show 55 nm×55 nm portions of the graphene simulation cell, with snapshots of the structural
evolution of the graphene sheet for different defect concentrations: (b) 1011 Ar+ /cm2 ; (c) 1012 Ar+ /cm2 ;
(d) 1013 Ar+ /cm2 and (e) 1014 Ar+ /cm2 , as in Figure 43. Reprinted From Carbon 48(5), M.M. Lucchese et
al., pp. 1592–1597 [3]. Copyright © (2010) Elsevier.

most strongly to the D-band, while the S-regions will make less contribution to the D-band due
to the break-down of the lattice structure itself. These two different scattering cross sections for
the disorder-induced processes will give rise to the non-monotonic behavior observed in the LD
dependence of the ID /IG ratio, as shown in Figure 44.
The structurally disordered (S) region and the activated (A) region are shown in Figure 45(a) by
light and dark gray regions, respectively. The evolution of the S and A regions for a graphene sheet
under ion bombardment was simulated by randomly choosing a sequence of impact positions on
a graphene sheet. As the number of impacts increase, the activated A-region increases, leading to
a decrease in LD and an increase of the D-band intensity ID . When the graphene is fully covered
with A-regions, an increase in ion bombardment fluence causes the structurally disordered S-
regions to take over from the A-regions, thus leading to a decrease of the D-band intensity ID
(see Figure 45(b–e)). This model is the basis for the evolution of ID /IG based on Equation (73)
which, with the parameters CA = 4.56, CS = 0.86, rA = 3 nm and rS = 1 nm, give the line curve
 Advances in Physics 495

in Figure 44 that fully describes the experimental evolution of ID /IG , shown by the black bullets
in Figure 44 [3].
For low defect concentrations (large LD values), ID /IG = (102 ± 2)/LD2 , which means the total
area contributing to scattering is proportional to the number of defects. This regime is valid for
LD > 2rA , while below this limit for LD , the activated regions start to overlap (see Figure 45(e)),
thus changing the simple ID /IG ∝ LD−2 dependence. The D-band intensity then reaches a maximum
and a further increase in the defect concentration decreases the D-band intensity because the
graphene sheet starts to be dominated by the structurally disordered areas (S-region).
The rS = 1 nm value is in agreement with the average size of the disordered structures seen
in the STM images [3,130]. This is not a universal parameter, but is a parameter that is actually
specific to the ion bombardment process. The φ = rA − rS = 2 nm value represents the Raman
relaxation length for the defect-induced resonant Raman scattering in graphene. This value is valid
for the laser excitation energy 2.41 eV and room temperature, and may change with changing Elaser
and temperature. Be aware that this is the relaxation length for the excited electrons, which should
not be confused with the relaxation length for the phonons. The value CA = 4.56 is in rough
agreement with the ratio between the electron–phonon coupling for the iTO phonons evaluated
between the  and the K points in the BZ [218–220], which is consistent with the expectation that
the CA parameter should be related to the electron–phonon matrix elements. The CS parameter is
related to the size of the highly disordered area, and there is no theoretical work yet available on
this matter.
It is important to have an equation relating ID /IG to LD that can be used by researchers looking
for a Raman characterization of the defect density present in a specific graphene sample. The
entire regime (0 → LD → ∞) can be fitted using [3]:
   
ID r 2 − rS2 −π rS2 −π(rA2 − rS2 ) −π rS2
= CA 2A exp − exp + C S 1 − , (74)
IG rA − 2rS2 LD2 LD2 LD2
which comes from solving rate equations for the bombardment process. Fitting the data in Figure 44
with Equation (74) gives CA = (4.2 ± 0.1), CS = (0.87 ± 0.05), rA = (3.00 ± 0.03) nm and rS =
(1.00 ± 0.04) nm. This equation represent the results very well, since the fitting obtained with
Equation (74) is also in very good agreement with experiment and the fitting parameters are fully
consistent with the parameters obtained by computational modeling using Equation (73) [3].

4.3.4. The Local Activation Model and the Raman Integrated Areas
The dependence of the intensity ratio ID /IG on LD was found to accurately follow an analytical
formula (Equation (74)), as described above, and this result is useful for practical applications and
for inter-laboratory comparisons. However, the physics behind this effect has to take into account
that both ID and IG vary when LD is changed. As discussed in Section 1.4.5, the evolution of the
Raman profile can be discussed as related to the peak intensity or to the integrated peak area.
In this section, we choose to use the same model as was used to derive Equation (74) when we
analyze the evolution of the intensity and integrated area of the many Raman peaks that vary with
increasing structural disorder, by normalizing each of them to the G-band integrated area (see
Figure 46). As shown in the inset to Figure 46 (top-right panel), the integrated area of the G-band
does not show any simple evolution with disorder [130].
The lower-left panels of Figure 46 show that this analytical expression fits the quantities
AD /AG and AD /AG nearly perfectly, where A refers to the integrated area. For the D-band, the
fitting parameters are rS = 2.6 nm, rA = 4.1 nm, CS = 2.4 and CA = 3.6, whereas for the D -
band the fitting parameters are rS = 2.6 nm, rA = 3.8 nm, CS = 0.28 and CA = 0.19. Note that
we obtain close to the same value of rS for both the D and D modes, indicating that indeed rS
496 R. Saito et al.

Figure 46. Normalized intensities (upper panel) and areas (lower panel) of the Raman D-, D -, G-
and G -bands as a function of LD . All quantities are normalized by the area of the G-band (see
the as-measured AG in the inset to the upper-right panel). The solid lines in the lower panel are
theoretical results based on the model described in Section 4.3.4. Reprinted figure with permission
from E.H. Martins Ferreira et al., Physical Review B 82, p. 125429, 2010 [130]. Copyright © (2010) by the
American Physical Society.

is a geometrical, structure-related length. Also, we find 1.5 and 1.3 nm for the spatial extent of
the Raman processes rA − rS , which is of the same order of magnitude as the rough estimates
vF /ωD = 4.3 nm and vF /ωD = 3.6 nm. We remind the reader that the distance rA − rS is a rough
measure of the length traveled over the lifetime of the e–h pair, vF /ωX , where vF is the graphene
Fermi velocity of the electron and hole carriers and ωX is the frequency of any X phonon mode
[130]. More interestingly, the ratio between rA − rS for the D- and D -bands matches very closely
to the ratio of the inverse frequencies ωD /ωD ≈ 1.2.
Similar ideas can be applied to a discussion of the AG /AG ratio, but in this case, since the G -
band is already active for pristine graphene, the intensity ratio is only affected by the disruption
of the hexagonal network, leading to a decrease in the AG /AG ratio as a function of increasing
 Advances in Physics 497

disorder described by the simple formula [130]

 
AG AG π rS2
(LD ) = (∞) − B 1 − exp − 2 , (75)
AG AG LD

where AG /AG (∞) is the area ratio for pristine graphene while AG /AG (LD ) is the area ratio for
an actual sample characterized by its LD value. The fitting of the experimental data, shown in
the lower-right panel of Figure 46, gives in this case rS = 2.5 nm, which is also similar to the
structural damage length obtained for the D- and D -band spectra. This result is in accordance
with the typical defect-size estimates found independently from the STM analysis [3,130]. In
Section 4.3.5, we describe what happens to the frequency and linewidth of the Raman peaks as a
result of ion implantation-induced structural damage.

4.3.5. Modeling disorder effects in the Raman linewidths and frequency shifts: the spatial
correlation model for defects
Disorder introduced by a random distribution of defects causes a broadening and a shifting of
the Raman mode frequencies and increases in the asymmetry of both the Raman-allowed and
the newly disorder-activated Raman bands discussed in Section 4.3.4. Here, we use the so-called
“spatial-correlation model” introduced by Capaz and Moutinho in [130] to describe these effects
in graphene. Other work on this topic that should also be referred to is in Refs. [65,292],
As described in Section 1.4.6, a random distribution of point defects will scatter phonons and it
will also add a contribution to the FWHM by an el–ph coupling of phonons with wave vectors q0
and q0 + δq. In the limit of low levels of disorder, the Raman intensity for the disordered graphene
I(ω) can be calculated by Equation (7). With this model, we can calculate the full lineshape of I(ω)
and from that we can extract the disorder-induced peak shifts ωq0 (Figure 47, lower panel) and
the increases in the FWHM q0 (Figure 48, lower panel). Since we use experimentally available
dispersion relations ω(q), the only fitting elements in this model are: (1) the relationship between
the coherence length L and the typical inter-defect distance LD , and (2) the weighting function
W (q) in Equation (7).
We now describe in more detail the application of the above model to the different Raman bands
considered in graphene, including the G-band, the D -band, the D-band and the G -band [130].
A. G-band – The G-band in perfect graphene is associated with phonons at the -point, i.e., q0 =
0 phonons. We consider that disorder mixes equally the -point phonons with nearby phonons in
both the LO and iTO phonon branches. We find that the best agreement with experiment is obtained
by using a constant weighting function (which is equivalent to not use a weighting function at all).
For the LO and TO phonon dispersions, we take

ωLO (q) = ωG + 181q − 230.29q2 ,

ωiTO (q) = ωG − 135.42q, (76)

where ωn (q) is in cm−1 (n = LO or iTO) and ωG = 1580 cm−1 is the experimental G-band frequency
for pristine graphene used in this work. Here q is measured from the -point in units of Å−1 . These
dispersions are taken from the work of Maultzsch et al. [208] by interpolating the frequencies at
high-symmetry points and by averaging the dispersions between the –K and –M directions.
Also, since the main contribution to the integral in Equation (7) will come from q vectors near the
 point, the BZ can be safely approximated by a circular disk and the integral will be considered
explicitly in the radial coordinate only. Taking all these considerations into account, Equation (7)
498 R. Saito et al.

Figure 47. The upper panel shows peak frequencies of the D, G, D and G -bands as a function of LD denoting
a typical distance between defects. The inset compares the frequency of the D-band and the G -band divided
by two, showing that we always have ωG /2 < ωD , in agreement wtih Ref. [216]. The lower panel shows
frequency shifts with respect to the zero-disorder limit. Dots are experimental points and solid lines are theo-
retical results based on the model described in the text. Experimental error bars are 2 cm−1 . Reprinted figure
with permission from E.H. Martins Ferreira et al., Physical Review B 82, p. 125429, 2010 [130]. Copyright
© (2010) by the American Physical Society.

becomes [130]
 exp[−q2 L 2 /4]
IG (ω) ∝ 2π qdq (77)
n
[ω − ωn (q)]2 + [0 /2]2

in which the sum is over the two (LO and iTO) phonon branches.
B. D -band – The D -band arises from intra-valley phonons with a linear wavevector intensity
dependence with respect to the laser energy. Since the D -band has been assigned to LO phonons,
only this branch is considered in calculations of the D -band intensity using Equation (7). We
 Advances in Physics 499

Figure 48. (a) FWHM intensity of the D-, G-, D - and G -bands as a function of LD , denoting the typical
distance between defects. (b) Disorder contribution to the peak widths, , for the D, G, D and G -bands.
Points denote are experiments and solid lines are theoretical results based on the model described in the text.
Reprinted figure with permission from E.H. Martins Ferreira et al., Physical Review B 82, p. 125429, 2010
[130]. Copyright © (2010) by the American Physical Society.

average over all possible directions θ of the wavevector q0 and, similarly to the case of the G-
band, there is no need to introduce a q-dependent weighing function W (q). Then, the D -band
intensity becomes [130]:

exp[−(q − q0 )2 L 2 /4]
ID (ω) ∝ qdq dθ . (78)
[ω − ωLO (q)]2 + [0 /2]2

For the laser energy of 2.41 eV, the value for |q0 | in Equation (78) is found to be |q0 | = 0.42 Å−1
measured from the  point.
C. D-band – The D-band arises from inter-valley phonons which also show a linear wavevector
dependence with respect to the laser energy. In fact, for the laser energy of 2.41 eV, we also find
|q0 | = 0.42 Å−1 for the D-band, but now q0 is measured from the K point. Since the D-band
has been assigned to iTO phonons along the K–M direction in the BZ, we choose q0 along this
500 R. Saito et al.

direction and the weighting function W (q) is also restricted to be non-zero only along the same
direction. Mathematically, W (q) = δ(θ − θK−M )f (q), where θK−M indicates the K–M direction
and f (q) = 1 + a(q0 − q) is a function that linearizes the radial dependence of the electron–
phonon coupling along the K–M direction near q0 . With these conditions, the D-band intensity
becomes [130]

f (q) exp[−(q − q0 )2 L 2 /4]
ID (ω) ∝ dq . (79)
[ω − ωiTO (q)]2 + [0 /2]2
For the iTO phonon dispersion along the K–M direction, we use [244]:

ωiTO (q) = ωK + 589.35q − 485.46q2 , (80)

where ωiTO is in cm−1 and q is measured from the K point in units of Å−1 .
D. G -band – The G -band is related to a DR process associated with the same inter-valley
phonons as the D-band. For this reason, the expression for the intensity becomes more complicated
and it involves a double integral over the forward (q) and backward (q ) phonon wavevectors. Using
the same considerations for the el–ph matrix elements, which essentially select phonons in the
K–M direction, we have
 
! 
"
 f (q)f (q ) exp (−[(q − q0 ) + (q − q0 ) ]L )/4
2 2 2
IG (ω) ∝ dq dq (81)
[ω − ωiTO (q) − ωiTO (q )]2 + [0 /2]2

where f (q) is the same linear function as in the D-band case and q0 is also the same. We also
impose the condition that the same relation between L (the disorder-induced phonon coherence
length) and LD (the average distance between defects) must be valid for the D and G -bands.
In Figures 47 and 48, we see the results for the frequency and linewidth as a function of
the typical distance LD between defects for the data fitting of the frequency shifts and widths,
respectively, as described above. Note that the general agreement is good, especially for large
values of LD . Indeed, this spatial correlation model, because of its perturbation character, is not
expected to be valid in the highly disordered regime. In Figures 47 and 48, the best relationships
between L and LD in each case are shown (as obtained by the fits between the model and the
experimental data). It is physically reasonable to see that L and LD are similar to each other. This
condition was not imposed, but it comes automatically from the fitting procedure. This means that
the disordered-induced phonon coherence length L is of the same order of magnitude as the typical
inter-defect distance LD , which is physically reasonable. There is no reason to expect that the same
relation between L and LD should be found for the different phonon modes, since different modes
should have different defect scattering cross-sections. From the results shown here, it seems that
the D modes are the most affected by point defect disorder, showing a smaller coherence length
than the other modes for the same amount of disorder. Finally, the model allows us to explain the
greater increase in the FWHM for each of the modes near the K point relative to the modes near
the  point as being simply a consequence of the larger magnitude of the phonon dispersions near
the K point.

4.3.6. Evolution of overtone and combination modes

In Figure 49, we present the spectral evolution of the G -band and other second-order processes in
mono-layer graphene for three different ion dose levels. The G and G -band intensities decrease
as the line widths increase for increasing ion dose. The defect-related combination modes D +
G at 2930 cm−1 and G + D at 3190 cm−1 can be observed at higher ion bombardment doses
(1013 Ar + /cm2 ), but the G -band is too weak to be seen in these measurements. At a dose of
 Advances in Physics 501

Figure 49. Evolution of the G -band (at 2670 cm−1 ) and other second-order peaks, the (D + G) at 2930 cm−1 ,
the (D + G) at 3190 cm−1 , and the G (2D in the figure) at 3220 cm−1 with increasing ion doses. The
intensities of the two lower graphs are multiplied by a factor of 10 for the sake of readability [130]. Here the
notation 2D is used instead of G , as has also been used in the literature by other authors. Reprinted figure
with permission from E.H. Martins Ferreira et al., Physical Review B 82, p. 125429, 2010 [130]. Copyright
© (2010) by the American Physical Society.

1014 Ar + /cm2 the results show a frequency downshift for all DR features, in agreement with the
results of Section 4.3.5.

4.3.7. Disorder and the number of layers

The ID /IG results for ion bombarded graphene depend on the number of graphene layers N [130]
in the case of low energy ions (90 eV). Because of the low ion energy, the ion bombardment
process is limited generally to one defect per bombarding ion, so that the ID /IG scales with N.
For many-layer graphene (∼ 50 and higher), a monotonic evolution of ID /IG with increasing ion
fluence is seen because in this case there are always more unperturbed graphene layers available
to be bombarded.

4.4. Edge phonon Raman spectroscopy

The disorder-induced Raman bands should also depend on the type of defect structure, and not
only on the number of defects. This dependence has been demonstrated for graphene and graphite
edges, where the orientation of the carbon hexagons with respect to the edge axis was determined
502 R. Saito et al.

experimentally, thereby distinguishing the so-called zigzag edge from the armchair or random
atomic edge structures [170]. The armchair/random vs. zigzag edge structure can be identified
spectroscopically by the presence vs. absence of the D-band, and this effect results from the
momentum requirements of the DR model, as discussed below.
The defect associated with a step edge has a 1D character, which means that it is able to transfer
momentum solely in the direction perpendicular to the edge. In this sense, the wave vectors of
the defects associated with zigzag and armchair edges are represented in Figure 50(a) by d a (a
for armchair) and d z (z for zigzag) edges. When we translate these vectors into reciprocal space,
we see that different selection rules apply for the electron scattering by phonons for each of these
edge types. This is illustrated in Figure 50(b), where the first BZ of 2D graphite (graphene) is
shown, oriented in accordance with the real space directions shown in Figure 50(a).
Light-induced e–h pairs will be created on an equi-energy circle around points K  and K (here
neglecting the trigonal warping effect for simplicity), which has a radius that is defined by Elaser , as
shown in Figure 50(b). Note that for inter-valley electron-defect scattering, which connects K to
K  points, only the d a vector for armchair edges can connect points belonging to circles centered at
two inequivalent K and K  points. In contrast the zigzag d z vector to connect inequivalent points,
which means that inter-valley scattering is not allowed for zigzag edges. This therefore means that
the inter-valley DR process, which is the process responsible for the observation of the D-band in
graphitic materials, is not allowed for a perfect zigzag edge [170]. The D-band phonon connects
two inequivalent K and K  points, and along the zigzag edge there will be no defect able to connect
those points to achieve momentum conservation in the final process.
On the other hand, intra-valley electron-defect scattering can occur for both zigzag and armchair
edges (see Figure 50(b)). Therefore, intra-valley scattering processes induced by phonons can
achieve final momentum conservation using both d  a and d
 z vectors. Another well-known defect-
induced band is the so-called the D -band, which appears at around 1620 cm−1 , and it is related


to intra-valley el–ph processes. For this reason, the D -band observation should be independent of
the zigzag vs. armchair structure of the edges, in agreement with experimental observation.
Another selection rule aspect refers to the D-band intensity dependence on the polarization
direction of the light with respect to the edge orientation. The D-band intensity has a maximum
value when the light is polarized along the edge, and should give a null value when the light is
polarized perpendicular to the edge. The physics behind this selection rule is the optical absorption
(emission) anisotropy around the K(K  ) point in 2D graphite, which can be represented by [262]

Wabs,ems ∝ |P  2.
 × k| (82)

Here the polarization of the incident (scattered) light for the absorption (emission) process is
represented by P 
 , while the wave vector of the electron measured from the K point is given by k.
These selection rules were first observed for graphite edges, as reported in [170], and similar
results have been observed later in mono-layer graphene [151,311]. However, only edge-dependent
variations in the D-band intensity consistent with the selection rules have been reported. Raman-
based indications for the high crystallinity of zigzag edges have indeed been observed by Krauss
et al. [312], although the complete absence of the D-band together with the observation of the
D -band, which is expected for a zigzag edge structure, has never been reported, which might
imply that, up to now, no perfect zigzag structure has been measured by Raman spectroscopy. In
general, the polarization direction dependence for the D-band intensity, as given by Equation (82),
together with the zigzag vs. armchair dependence, can be used for an identification of the edge
orientation and structure. Raman spectroscopy is, therefore, a valuable tool for the development
of our understanding of edge structures, important for the science of graphene ribbons, and more.
The results reported here represent an effort to improve our understanding of the influence of
 Advances in Physics 503

Figure 50. (a) Schematic illustration of the atomic structure of edges with the zigzag and armchair ori-
entations. The boundaries can scatter electrons with momentum transfer along d  z for the zigzag edge,

and along da for the armchair edge. (b) First BZ of 2D graphite (graphene), showing defect-induced
 z is too short to connect the K and K  points,
inter-valley and intra-valley scattering processes. Since d
the defect-induced DR inter-valley process is forbidden at zigzag edges. Reprinted figure with permission
from L.G. Cancado et al., Physical Review B 93, p. 47403, 2004 [59]. Copyright © (2004) by the American
Physical Society.

the specific defect structure on the Raman spectra of sp2 carbon systems. Other defect-dependent
effects are expected, which may be very useful to characterize defects in nanographite-based
devices, but both theory and experiment have to be developed along these lines.

4.5. Polarization effects in graphene nanoribbons

Polarization effects have also been observed in the G-band of graphene nanoribbons, as shown
in Figure 51 [59]. Again, this development came before the graphene rush, and the experimental
results were actually obtained on a one-layer thick ribbon grown by CVD on top of HOPG. The
lower frequency G1 band in Figure 51 comes from the nano-ribbon, while the higher frequency
G2 band comes from the HOPG substrate. The reason why the frequencies are distinct is actu-
ally related to the different heat dissipation, whereby the nano-ribbon and the substrate get into
equilibrium at different temperatures when heated independently by the laser (see Figure 51).
Thus the nanoribbon is heated to a higher temperature by the laser heating than the substrate, and
therefore ωG1 for the nanoribbon decreases more than ωG2 for the substrate. This happens because
the thermal conductivity of the substrate is much higher than that of the graphene ribbon, as shown
in Figure 51(c).
Finally, the nanoribbon G1 band shows a clear dependence on the excitation laser light polar-
ization with respect to its axis, as illustrated in Figure 51(b). The Raman signal for the ribbon
disappears when the light polarization direction is perpendicular to the ribbon axis. This result is
related to both the anisotropy of the optical absorption (emission), according to Equation (82),
and the quantum confinement perpendicular to the ribbon axis.
504 R. Saito et al.

5. Raman spectra of carbon nanotubes

Since the Raman spectra of carbon nanotubes have additional unique spectral features that are
not found in other carbon nano-structured materials, Raman spectroscopy has provided an espe-
cially important tool for developing characterization techniques for nanocarbon materials [313].
In particular the RBM, which is unique to carbon nanotubes (see Figure 4), provides a highly
sensitive tool for determining the presence of carbon nanotubes in a particular carbon-based sam-
ple and for characterizing the (n, m) chirality of the carbon nanotubes present in the sample. The
KA effect provides a mechanism for study of the el–ph interaction through its effect on both the
G-band and the RBM mode features in the Raman spectra. The DR effect provides a mechanism
for studying aspects of the 1D quantum confined electronic energy band structure through the
el–ph interaction. Since the electronic transitions in carbon nanotubes are dominated by excitonic
effects, study of the Raman spectra of carbon nanotubes provides important insights into exci-
tonic phenomena in 1D systems. Near-field Raman spectroscopy has been especially sensitive for
revealing new and important spatial information about specific defects in carbon nanotubes. These
topics are discussed in this Section.

5.1. The radial breathing mode and the Kataura plot

The RBM is the nanotube normal mode vibration where all the C atoms vibrate in phase in the
radial direction, as if the tube is breathing. Since the atomic vibrational motion does not break
the tube symmetry, the RBM is a totally symmetric mode according to group theory, belonging
to the A1 symmetry irreducible representation (IR) [1,141,142]. The RBM only occurs in carbon

Figure 51. (a) The G-band Raman spectra from a graphene nano-ribbon (G1 ) and from the HOPG substrate
(G2 ) on which the nano-ribbon was grown. (b) The dependence of the G1 frequency on the light polarization
direction, with respect to the ribbon axis. Points are experimental results and the dashed curve is the theoretical
expectation. (c) Frequency of the G1 and G2 peaks as a function of the incident laser power density. Reprinted
figure with permission from L.G. Cancado et al., Physical Review B 93, p. 47403, 2004 [59]. Copyright ©
(2004) by the American Physical Society.
 Advances in Physics 505

nanotubes and, therefore, it can be used to distinguish SWNTs from other sp2 carbon structures in
the samples. In general the intensity of the RBM is unusually strong when compared with other
non-resonant spectral features coming from other carbonaceous materials or from the substrate
on which the tubes are sitting [111]. Furthermore, the RBM frequency ωRBM depends on the tube
diameter, following the proportionality relation ωRBM ∝ 1/dt . This dependence was predicted
initially using force constant calculation models (e.g. [100]), but a rather simple and instructive
analytical derivation can be made using elasticity theory, and we present this approach in sequence.

5.1.1. The RBM frequency

The RBM frequency of a SWNT is given by ωRBM = A/dt , with the value A = 227 cm−1 nm
being expected by elasticity theory, thereby directly connecting 1D carbon nanotubes to their 2D
counterpart graphene from which nanotubes are conceptually derived [314,315]. The experimental
values for A and the expected value for A coming from elasticity theory agree perfectly for one
specific type of SWNT [314,315], that is, for SWNTs which are ultra-long, vertically aligned,
and grown by the water-assisted CVD method [316]. This sp2 carbon material is used here as a
standard reference material from a metrology standpoint. Most of the RBM experimental results in
the literature have been fitted to the relation ωRBM =A/dt +B, with values for the parameters A and
B varying widely from paper to paper [107,109,111,153,175,188,314,317–321]. In the limit of the
tube diameter going to infinity, we ideally expect ωRBM → 0, suggesting that B is associated with
an environmental effect, and the environmental conditions differ from one experimental system
to another.
A simple model that has been used to model the RBM is a simple harmonic oscillator equation
for a cylindrical shell subjected to an inwards pressure (p(x)) given by [314,315]

ρ ∂ 2 x(t) 2 (1 − ν 2 )
(1 − ν 2 ) 2
+ x(t) = − p(x), (83)
Y ∂t dt Yh

where x(t) is the displacement of the nanotube in the radial direction, p(x) = (24K/s02 )x(t),
and K (in eV/Å2 ) gives the van der Waals interaction strength, s0 the equilibrium separation
between the SWNT wall and the surrounding environmental shell, Y the Young’s modulus
(69.74 × 1011 g/cm·s2 ), ρ the mass density per unit volume (2.31 gm/cm3 ), ν = 0.5849 is the
Poisson’s ratio and h represents the thickness of the environmental shell [315]. If there are no
environmental effects, the term p(x) vanishes and Equation (83) will become the fundamental
0
frequency ωRBM for a pristine SWNT in units of cm−1 ,
#  1/2 $
1 Y 1
ωRBM =
0
. (84)
πc ρ(1 − ν ) 2 dt

The term inside the curly bracket above gives the fundamental value of A = 227.0 cm−1 nm.
However, for a non-vanishing inward pressure p(x), the result is
 1/2
 1 6(1 − ν 2 ) K
ωRBM = 227.0 2 + . (85)
dt Yh s02

Here [6(1 − ν 2 )/Yh] = 26.3 Å2 /eV. The shift in ωRBM due to the nanotube environment is given

by ωRBM = ωRBM − ωRBM . We fit the value K/s02 in Equation (3) to the RBM frequency as
0

a function of dt . The fitted value for the environmental term for the “super-growth” sample is
sufficiently small, since K/s02 = (2.2 ± 0.1) meV/Å4 , that it can be neglected. The dt dependent
506 R. Saito et al.

Table 2. Environmental effects on the RBM frequency of different

samples, as measured by the Ce factor in Equation (86) [323].

Ce Sample Reference

0 Water-assisted CVD Araujo [314]

0.05 HiPCO@SDS Bachilo [153]
0.059 Alcohol-assisted CVD Araujo [321]
0.065 SWNT@SiO2 Jorio [111]
0.067 Free-standing Paillet [320]

behavior of the environmental effect in ωRBM reproduces well the experimental result for dt up to
dt = 3 nm [314]. A similar environmental effect is obtained for SWNTs surrounded by different
surfactants [107,175,188,317,319], in bundles [109,321], sitting on a SiO2 substrate [111], and
even for tubes suspended in air by posts [320]. This environmental effect is almost absent in
“super-growth” SWNTs, but the reason why this sample is special is not presently understood,
and the pristine-like ωRBM behavior is lost if this sample is dispersed in solution [322].
A simple relation can be proposed for all the ωRBM results in the literature, which are generally
upshifted from the pristine values observed for the “super-growth” samples due to the van der
Waals interaction with the environment. This simple relation is [314]
%
227
Lit.
ωRBM = 1 + Ce ∗ dt2 , (86)
dt

where Ce in Equation (86) represents the effect of the environment on ωRBM , i.e. Ce = [6(1 −
ν 2 )/Eh][K/s02 ] nm−2 . The several Ce values that are obtained by fitting the RBM results for
different commonly found samples in the literature are given in Table 2. The curvature effects
become important for dt < 1.2 nm, and in this case the environmental effect depends more critically
on the specific sample. For example, the Ce for SWNT samples sitting on a SiO2 substrate may
differ from sample to sample. The observed environmentally induced upshifts for the RBM from
small diameter tubes, either within bundles or wrapped by different surfactants (e.g., SDS (sodium
dodecyl sulfate) or single stranded DNA), range from 1 to 10 cm−1 . This environmental effect gets
richer in a double wall carbon nanotube (DWNT), as discussed in Section 5.1.2.
Finally, all the ωRBM dependence on the carbon nanotube structure discussed here addresses
the importance of the diameter dependence. The chiral angle has a weaker dependence on the
RBM frequency, but to fully discuss this topic, the KA has to be introduced [193,324]. This topic
is discussed in Section 5.4.

5.1.2. The RBM for double wall carbon nanotubes

Spectroscopic experiments on DWNTs have been largely performed on solution-based samples or
in bundles [325–329]. For this reason, it has been difficult to identify and learn about the Raman
spectroscopic signatures that are specific to the inner (n, m) tube of a DWNT, which can be
contained inside different possible outer (n , m ) tubes (see Figure 52(a)). The DWNTs are formed
by an inner and an outer tube, and they can be either metallic (M) or semiconducting (S). There are
four different possible configurations, namely M@M, M@S, S@S and S@M. Here S@M denotes
a DWNT with an S inner tube inside an M outer tube, following a similar notation introduced
for fullerenes [330,331]. Interesting aspects are related to the DWNT electronic properties, e.g.
the S@M configuration can be regarded as a good approximation for an isolated semiconducting
SWNT, since it is electrostatically shielded by the outer metallic tube [330]. In order to determine
 Advances in Physics 507

which specific inner and outer tubes form a given DWNT, one has to perform Raman experiments
on individual DWNTs (see Figure 52(c)).
For a well-defined experiment, the combination of electron-beam lithography, atomic force
microscopy (AFM) and Raman spectral mapping have been developed to measure the Raman
spectra from the inner and the outer tubes of an individual DWNT (see Figure 52) [330,331]. The
Raman spectra of 11 isolated DWNTs grown from annealing C60 filled SWNTs were measured
using a single laser excitation energy (Elaser = 2.10 eV [330]).Specific Elaser values were used to
select all DWNTs with (6,5) semiconducting inner tubes that were in resonance, and all with the
S@M configuration so that the RBMs from both the inner and outer tubes of individual DWNTs
could be observed. The RBM frequencies ωRBM,o for the outer tube measured for such a DWNT as
a function of ωRBM,i for the inner tube are shown in Figure 53(a). For these 11 individual isolated
DWNTs, ωRBM,o for the outer tubes varies along with ωRBM,i , thus showing that the inner and outer
tubes impose considerable stress on one another.Actually, the nominal wall-to-wall distances dt,io
between the inner (i) and outer (o) tubes of the DWNTs are less than the 0.335 nm interlayer c-axis
distance in graphite. Figure 53(b) shows dt,io values as small as 0.29 nm, with a decrease of up
to 13% in the wall to wall distance [330]. Because of the differences in the Coulomb interaction

Figure 52. (a) Raman spectra for the RBM region for two types of DWNTs, obtained with Elaser = 2.13 eV.
(b) AFM image of one individual DWNT. The inset shows the silicon substrate with gold markers showing
the location of an individual DWNT. (c) RBM Raman spectra obtained with Elaser = 2.11 eV for an isolated
individual DWNT grown from a C60 filled SWNT-bundle. (d) AFM height profile of the individual, isolated
DWNT shown in (b), with the RBM spectrum shown in (c). The vertical lines connecting (a) and (c) show
that the ωRBM of the prominent tube diameters observed in the C60 -DWNT bundles coincide with the ωRBM
of the inner and outer tubes of the isolated C60 -DWNT. F. Villalpando-Paez et al., Nanoscale 2, pp. 406–411,
2010 [330]. Adapted by permission of the Royal Society of Chemistry.
508 R. Saito et al.

expected for the four different DWNT metallicity configurations, S@M, M@S, S@S and M@M,
the detailed relation between ωRBM and 1/dt will depend on the metallicity configuration.
By adding walls to form MWNTs, the RBM signal from inner tubes with small enough diam-
eters (dt  2 nm) can be observed experimentally [332]. However, most of the usually made
MWNT samples are composed of inner tubes with diameters too large to exhibit observable RBM
features.

5.1.3. The Raman excitation profile for the RBM

In the Raman scattering experiment, when the energy of the incident or scattered light matches
an optical transition in the scattering system, resonance effects occur, and the Raman signal is
strongly enhanced [103,109,111]. Using the resonance Raman effect, it is possible to study the
electronic structure of individual SWNTs [111,336,338]. In this section, we review observations
of the Raman excitation profile for the RBM feature for an individual SWNT.
Isolated SWNTs were grown on top of a Si/SiO2 substrate by a CVD method [111,339].
Figure 54(a) shows an AFM image of the substrate with lithographic markers that were used to
locate an individual SWNT. The dashed circles in Figure 54(a,b) display the position where the
laser spot is placed. Figure 54(c,d) show anti-Stokes/Stokes Raman spectra of the sample, measured
in the excitation wavelength (energy) range 1.585 eV ≤ Elaser ≤ 1.722 eV. The Stokes/anti-Stokes
ratio are calibrated by [n(ω) + 1]/n(ω), where n(ω) = 1/[exp(ω/kB T ) − 1], ω is the RBM
frequency, kB is the Boltzmann constant, and T is the temperature, which was found not to be
higher than 325 K [338].
An RBM Raman feature appears in the spectra in Figure 54(c,d) at 173.6 cm−1 , with a clear
resonance behavior. The RBM peak appears and disappears over the tunable energy range of Elaser ,
meaning that Elaser is tuned over the whole resonance window of one optical transition energy (Eii )
for the SWNT which is sitting under the laser spot. The linewidth for the RBM peak is sharp,
(RBM = 5 cm−1 ), showing a typical value measured for isolated SWNTs deposited on substrates
[111,340]. Figure 55 shows the 173.6 cm−1 RBM peak intensity vs. Elaser for (a) the anti-Stokes
and (b) Stokes processes for the individual SWNT measured in Figure 54.

Figure 53. All the inner tubes for the 11 peapod-DWNTs in this figure are (6,5) semiconducting tubes. (a)
Plot of the ωRBM,i for the inner tube vs. ωRBM,o for the outer tubes which pair to form eleven different isolated
DWNTs. (b) Plot of the nominal wall-to-wall distance dt,io for each of the 11 isolated DWNTs vs. ωRBM,i
shown in (a). An increase in the ωRBM,i of the (6,5) inner tubes shown here is accompanied by a decrease
in the measured nominal wall-to-wall dt,io distance for these peapod-derived DWNTs. F. Villalpando-Paez
et al., Nanoscale 2, pp. 406–411, 2010 [330]. Adapted by permission of the Royal Society of Chemistry.
 Advances in Physics 509

The Stokes RBM peak intensity I is a function of Elaser and can be evaluated from Equation (62)
[269]. The two factors in the denominator of Equation (62) describe the resonance effect with
the incident and scattered light, respectively. A ± sign before Eph applies to Stokes/anti-Stokes
processes, while γRBM is related to the inverse lifetime for the resonant scattering process [278]. The
matrix elements in the numerator are most usually considered to be independent of energy because
of the small energy range. The theory for these matrix elements is discussed in Section 3.6.1. The
lines in Figure 55 show the fits to the experimental data for the Stokes (dashed) and anti-Stokes
(solid) resonance windows, using Eph = 21.5 meV, obtained from ωRBM = 173.6 cm−1 [338]. The
asymmetric lineshape in the resonance windows in Figure 55 was obtained in [338] by considering
not a coherent & Raman process, but an incoherent scattering process, where the sum over the
internal states ( a in Equation (62)) was taken outside the square modulus. This procedure is
indeed controversial because this asymmetry could be also generated by different γr values for
the incident and scattered resonance windows or by other resonance levels lying close in energy.
Disregarding the asymmetry aspect, Eii = 1.655 ± 0.003 eV and γRBM = 8 meV is obtained
for the spectra shown in Figure 55. A shift in the Stokes (S) and anti-Stokes (aS) resonant windows
is expected due to the resonant condition for the scattered photon, Es = Eii ± Eph , with (+) for
the Stokes and (−) for the anti-Stokes processes, and this effect is shown in the upper inset to
Figure 55. For this reason, under sharp resonance conditions the IaS /IS intensity ratio depends
sensitively on Eii − Elaser , and the IaS /IS ratio can be used to determine Eii experimentally and to
determine whether the resonance is with the incident or the scattered photon [141,338].

Figure 54. (a) AFM image of a SWNT sample showing markers that were used to localize the spot
position (dashed circle) on the substrate during the Raman experiment and for further AFM charac-
terization of the SWNTs that are located within the light spot indicated by the dashed circle in (b).
(c) anti-Stokes and (d) Stokes Raman spectra from isolated SWNTs on a Si/SiO2 substrate for sev-
eral different laser excitation energies. For more details, see Ref. [338]. Adapted with permission from
A. Jorio et al., Physical Review B 63, p. 245416, 2001 [338]. Copyright © (2001) by the American Physical
Society.
510 R. Saito et al.

5.1.4. The RBM spectra of SWNT bundles

Figure 56(a) show a RBM Raman spectrum obtained from SWNT bundles [321]. The solid line
shows the fit obtained using a sum of Lorentzians. Each Lorentzian gives the RBM from one
specific (n, m) SWNT species. The red Lorentzians represent the RBM from metallic SWNTs

Figure 55. Raman intensity vs. laser excitation energy El for the ωRBM = 173.6 cm−1 peak in Figure 54, for
both anti-Stokes and Stokes processes. Circles and squares indicate two different runs on the same sample.
The line curves indicate the resonant Raman window&predicted from Equation (1), with Eii = 1.655 eV,
γr = 8 meV, but taking the sum over internal states ( m,m ) outside the square modulus. The upper inset
compares the theoretically predicted Stokes and anti-Stokes resonant windows on an energy scale in eV, and
the lower insert shows the joint density of states (JDOS) vs. Elaser for this SWNT. Adapted with permission
from A. Jorio et al., Physical Review B 63, p. 245416, 2001 [338]. Copyright © (2001) by the American
Physical Society.

Figure 56. (a) Raman spectrum (bullets) of SWNT bundles obtained with a 644 nm laser line
(Elaser = 1.925 eV). This spectrum was fitted by using 34 Lorentzians (curves under the spectra) and the
solid line is the fitting result. (b) The Kataura plot used as a guide for the fitting procedure. Adapted with
permission from P.T. Araujo et al., Physical Review Letters 98, p. 67401, 2007 [321]. Copyright © (2007)
by the American Physical Society.
 Advances in Physics 511

Figure 57. Resonance windows for specific (n, m) SWNTs within a bundle. (a) Resonance profile (black
dots) in the near-infrared range for ωRBM = 192.7 cm−1 . The data for tube (14, 3) were fitted (solid line)
using Equation (87) with γ = 0.065 eV and Eii = 1.360 eV. (b) Resonance profile in the visible range is
shown for ωRBM = 192.5 cm−1 (tube (12, 6)), with γ = 0.045 eV and Eii = 1.920 eV. (c) Resonance pro-
file in the near-ultraviolet range is shown for ωRBM = 257.6 cm−1 (tube (11, 1)), with γ = 0.073 eV and
Eii = 2.890 eV.Adapted with permission from P.T. Araujo et al., Physical Review Letters 98, p. 67401, 2007
[321]. Copyright © (2007) by the American Physical Society.

and the green Lorentzians represent the RBMs from semiconducting SWNTs. The number of
Lorentzians used to fit each resonance spectrum can be defined by the Kataura plot [177,342,343]
(see Figure 56(b)), which plots  the Eii for all possible (n, m) SWNTs. The ωRBM values obey
the relation ωRBM = (227/dt ) 1 + Ce /dt2 , which correctly describes environmental effects by
the proper choice of Ce , and this relation is discussed in detail in Section 5.1.1. For lack of
information, we assume that all the Lorentzian peaks in one experimental spectrum share the
same FWHM value.
The RBM peak intensity I(Elaser ) for each Lorentzian peak in the RBM spectra from an
individual SWNT can be evaluated by
 2
 1 
I(Elaser ) ∝   ,
 (87)
(E laser − Eii − iγ )(Elaser − Eii ± Eph − iγ )

which is a simplification of Equation (62). Figure 57 shows the resonance profiles for three
different (n, m) SWNTs [321,335] The resonance window widths γr for SWNTs in bundles are
usually within the 40–160 meV range, which are much broader than for isolated SWNTs (see
Figure 55) and γr also depends on (n, m) [321].

5.1.5. (n, m) dependence of RBM intensity – experimental analysis

In experiments using a nearly continuous set of laser energies EL , we compare the RBM intensity
for different (n, m) SWNTs at a resonance condition (i.e., EL = Eii ). The RBM spectra from
the “super-growth” water-assisted SWNTs obey the relation ωRBM = 227/dt cm−1 nm and these
spectra have been used for (n, m)-dependent intensity analysis [344].
The dt distribution of a given super-growth sample was established by high resolution trans-
mission electron microscopy (HRTEM) [344], to calibrate the RRS (resonance Raman scattering)
intensities. For the (n, m) analysis, it was assumed that SWNTs of different chiral angles are
512 R. Saito et al.

equally abundant in the growth process. Actually, chiral SWNTs are twice as populous as achiral
ones because there are right-handed and left-handed isomers present in a typical sample. The
intensity calibrated experimental RRS map in shown in Figure 58(a). The (n, m) nanotubes in the
(2n + m) = constant family have similar diameters and Eii values to one another, and the Raman
intensity within a given (2n + m) = p = constant family has a chiral angle dependence. From the
spectral map it is clear that the RBM intensity is stronger for smaller chiral angles (near zigzag
nanotubes) as compared to those with larger chiral angles. Each spectrum (S(ω,EL ), ) is the sum of
the individual contributions of all SWNTs present in the light beam, and we can write [344]

 /2

S(ω,EL ) = Pop(n, m)I(n, m)EL , (88)
n,m
(ω − ωRBM )2 + (/2)2

where Pop(n, m) is the relative population of the (n, m) nanotube species,  = 3 cm−1 is the
experimental average value for the FWHM intensity of the Raman spectra (Lorentzian), ωRBM is
the RBM frequency and ω is the corresponding Raman shift variable. The total integrated area
(I(n, m)EL ) for the Stokes process at a given excitation laser energy (EL ) is given by
 2
 M 
I(n, m)EL =   , (89)
(EL − Eii + iγ )(EL − Eii − Eph + iγ ) 

where the superscript EL denotes the laser excitation energy, Eph = ωRBM is the energy of the
RBM phonon, Eii is the energy corresponding to the ith excitonic transition, γ is the resonance
window width and M represents the matrix elements for the Raman scattering by one RBM
phonon of the (n, m) nanotube. The values for Eii and ωRBM have to be determined experimentally.
The effective matrix element square term M and the effective resonance window width γ for
each (n, m) tube were found by fitting the experimental RBM component of the RRS map with
Equation (88) using the functions:
 
MB MC cos(3θ ) 2 γB γC cos(3θ)
M = MA + + and γ = γA + + . (90)
dt dt2 dt dt2

Figure 58. (a) Experimental RRS map for the RBM feature. The intensity calibration was made by measuring
a standard tylenol sample. (b) Modeled map obtained by using Equation (88) in the same laser excitation
energy range as (a). Adapted with permission from P.B.C. Pesce et al., Applied Physical Letters 96, p. 51910,
2010 [344]. Copyright © (2010) by the American Institute of Physics.
 Advances in Physics 513

Here Mi and γi (i = a, b, c) are fitting parameters. The best values for Mi and γi , considering
S M
the excitonic transitions E22 and the lower branch of E11 , are listed in Table 3. Here dt is given in
nm, γ in meV and M in arbitrary units.
The modeled RRS map shown in Figure 58(b) was obtained using the values thus obtained in
Equation (88). Note that the model represents the experimentally observed results in Figure 58(a)
very well.

5.1.6. The experimental Kataura plot

The resonance window analysis can be extended to all (n, m) SWNTs, from which we can study the
Eii dependence on (dt , θ ). Figure 59(a) shows a 2D RBM map for the “super-growth” (S.G.) SWNT
sample [316]. The experimental Kataura plot in Figure 59(a) was constructed using 125 different
laser excitation lines [314,335]. By applying the fitting procedure described in Section 5.1.4, the
(n, m) indices have been assigned to 197 different SWNTs.
By fitting the resonance windows extracted from the data in Figure 59(a) as a function of
S.G.
ωRBM , all EiiS.G. excitonic transition energies have been obtained experimentally, as shown in
Figure 59(b). These EiiS.G. values displayed in Figure 59(b) have been fitted using the empirical
equation [321,345]:
 
p 0.812 βp cos 3θ
Eii (p, dt ) = αp 1 + 0.467 log + . (91)
dt p/dt dt2
S S M S
Here p is defined as 1, 2, 3, . . . , 8 for E11 , E22 , E11 , . . . , E66 , and is a measure of the distance
of each cutting line from the K point in the zone-folding procedure [135,177,343]. The fitting
gives αp = 1.074 for p = 1, 2, 3 and αp = 1.133 for p ≥ 4. The βp values found for the lower
(upper) Eii branches are: −0.07(0.09), −0.18(0.14), −0.19(0.29), −0.33(0.49), −0.43(0.59),
−0.6(0.57), −0.6(0.73) and −0.65 (unknown) for p = 1, 2, 3, . . . , 8, respectively [335,345].
Equation (91) carries a linear dependence of Eii on p/dt , which is expected from tight-binding the-
ory. It also includes a logarithmic correction term in p/dt that comes from many-body interactions,
plus a chiral angle θ dependence related to electronic trigonal warping and chirality-dependent
curvature effects due to σ − π hybridization [321]. The theoretical background for all these factors
is discussed in Section 3.

5.2. Exciton environmental effect

The Eii values are now understood in terms of the bright exciton energy within the framework
of a tight binding calculation which includes curvature optimization [254,346] and many-body
effects [39–41,148,342]. The assignments of Eii for SWNTs over a large region of both diameter
(0.7 < dt < 3.8 nm) and Eii (1.2–2.7 eV) values and for a variety of surrounding materials are
now available [21,230,335], thus making it possible to accurately determine the effect of the

Table 3. Adjusted parameters Mi and γi for metallic (M),

semiconductor type 1 (S1 ) and type 2 (S2 ) tubes.

Type MA MB MC γA γB γC

M 1.68 0.52 5.54 23.03 28.84 1.03

S1 −19.62 29.35 4.23 −3.45 65.10 7.22
S2 −1.83 3.72 1.61 −10.12 42.56 −6.84

These parameters are to be used in Equation (90) with dt in nm, yielding M in

arbitrary units and γ in meV [344].
514 R. Saito et al.

Figure 59. (a) RBM resonance Raman map for the “super-growth” (S.G.) SWNT sample [314,335,345].
(b) Kataura plot of all transition energies (EiiS.G. ) that could be experimentally obtained from the resonance
windows extracted from (a) as a function of ωRBM . (c) Kataura plot obtained from Equation (91) with
the parameters that best fit the data in (b). The stars stand for M-SWNTs, the open bullets stand for type I
S-SWNTs and the filled bullets stand for type II S-SWNTs. Reprinted figure with permission from P.T. Araujo
and A. Jorio, Physical Status Solidi B, 2008, 245, pp. 2201–2204 [335]. Copyright © Wiley–VCH Verlag
GmbH & Co. KGaA.

general dielectric constant κ on Eii . By “general” we mean that κ comprises the screening from
both the tube core electrons and from the tube environment.24 A dt -dependent effective κ value
for the exciton calculation is needed to reproduce the experimental Eii values consistently. This
dependence is important for the physics of quasi -1D and truly 1D materials generally and can be
used in interpreting optical experiments and environment effects for such materials. Environmental
effects are therefore considered further in Sections 5.2.1 to 5.2.3.

5.2.1. The effect of the dielectric constant κ on Eii

Figure 60 shows a map of experimental Eii values (black dots) [314,335] from a SWNT sample
grown by the water-assisted (“super-growth”) CVD method [34,316]. The resulting data for the
Eii transition energies are plotted as a function of the RBM frequencies ωRBM , as obtained by RRS
[314,335,345]. In Figure 60, the experimental values of Eii vs. ωRBM for the “super-growth” sample
exp
Eii are compared with the calculated bright exciton energies Eiical (open circles and stars), obtained
with the dielectric screening constant κ = 1. Although Eiical includes SWNT curvature and many-
exp
body effects [148], clearly the Eii values are red shifted when compared with theory, and this red
shift depends on both ωRBM (i.e., on dt ) and on the optical energy levels (i in Eii ). Figure 63 shows a
exp exp
comparison between the Eii from the “super-growth” SWNT sample (bullets) [335] and Eii from
the “alcohol-assisted” SWNT sample (open circles) [321]. From Figure 63, we see that besides the
exp
changes in ωRBM , the Eii values from the “alcohol-assisted” SWNTs are generally red shifted with
respect to those from the “super-growth” SWNTs.Assuming that κtube does not change from sample
 Advances in Physics 515

to sample for a particular type of SWNT sample, since the electronic structure of a given (n, m)
tube should be the same, these results indicate that the “alcohol-assisted” SWNTs are surrounded
by an environment with a larger κenv value than the “super growth” sample, thus increasing the
effective κ and decreasing Eii [229], which is consistent with Figure 61, discussed below.

5.2.2. Screening effect: a general κ function

The Eii values can be renormalized in the calculation by explicitly considering the dielectric
constant κ in the Coulomb potential energy given by Equation (54) [231]. Here, κ represents
the screening of the e–h (e–h) pair by core (1s) and σ electrons (κtube ) and by the surrounding
materials (κenv ), while ε(q) explicitly gives the polarization function for π -electrons calculated
within the RPA [120,148,275]. To fully account for the observed energy-dependent Eii redshift,
exp
the total κ values (1/κ = Cenv /κenv + Ctube /κtube ) are fitted to minimize Eii − Eiical in Figure 60.
The κ values are found to depend on the following three parameters: (1) the subband index
p where p = 1, 2, 3, 4 and 5 stands for E11S S
, E22 , E11M S
, E33 S
and E44 , respectively [177,229], (2) the
diameter of the nanotube dt and (3) the exciton size lk in reciprocal space [232]. Using the optimized
κ values which are fitted to the calculated Eii values as a function of κ and to the experimental Eii
values, a general κ function is modeled to have the following functional form [232]:
' (
1 b 1 c
κ ≈ Cκ p a
. (92)
dt lk

The parameters (a, b, c) thus determined are common for all different samples (a, b, c) = (0.8 ±
0.1, 1.6 ± 0.1, 0.4 ± 0.05) so as to both optimize the correlation between κ and (p, dt , lk ), and
to minimize differences between theory and experiment. Here, it should be mentioned that the
variable lk is involved in the κ function because of the screening by the different environments

exp
Figure 60. Black dots show Eii vs. ωRBM results obtained from resonance Raman spectra taken from a
super-growth SWNT sample. The black open circles (semiconducting; S-SWNTs) and the dark-gray stars
(metallic; M-SWNTs) give Eiical calculated for the bright exciton with a dielectric constant κ = 1 [148]. Along
the x axis, the Eiical values are calculated using the relation ωRBM = 227/dt . Due to computer time availability,
only Eii for tubes with dt < 2.5 nm (i.e., ωRBM > 91 cm−1 ) have been calculated. Transition energies EiiS
(i = 1 to 5) denote semiconducting SWNTs and EiiM (i = 1, 2) denote metallic SWNTs. Reprinted figure
with permission from P.T. Araujo et al., Physical Review Letters 103, p. 146802, 2009 [229]. Copyright ©
(2009) by the American Physical Society.
516 R. Saito et al.

Figure 61. The κ function for: (a) SG, (b) AA, and (c) HiPCO samples. (d) Data for the three different types of
samples are plotted on the same figure with a fitted slope Cκ for each sample. (e)All the κ functions collapse on
to a single line after dividing each function by the corresponding C̃κ . The following symbols are used: E11 (◦),
E22 (×), E33 () and E44 (). Black, red, and blue colors, respectively, denote metallic (mod(2n + m, 3) = 0),
semiconductor type I (mod(2n + m, 3) = 1), and type II (mod(2n + m, 3) = 2) SWNTs. Reprinted figure
with permission from A.R.T. Nugraha et al., Applied Physical Letters 97, p. 91905, 2010 [232]. Copyright
© (2010) American Institute of Physics.

which modify the exciton size. However, we will show below that the selection of lk as a variable
for κ is essential for explaining the difference between metallic and semiconducting SWNTs. In
fact, Equation (92) indicates another scaling relation for excitons, similar to the previously reported
scaling law which relates Ebd with dt , κ, and the “effective mass” μ [275]. However, it is found
that the scaling relation involving μ works well only for S-SWNTs and another scaling function
is needed for M-SWNTs. This is because Ebd for an M-SWNT is screened by free electrons even
for a similar effective mass as that for the photo-excited carriers.
In Figure 61, we show a series of results for the κ function [232] for different samples grown
by different methods: (a) super-growth [316], (b) alcohol-assisted CVD [347], and (c) the high
pressure gas-phase decomposition of CO (HiPCO) [348]. For each sample, the κ function is
S S M
successfully unified for lower energy transitions (E11 , E22 , E11 ) and for the higher energy transitions
S S
(E33 , E44 ). Considering lk explicitly is important for describing the environmental effect for both
metallic and semiconducting SWNTs simultaneously, since lk is very different between metallic
and semiconducting SWNTs, even for similar effective mass values because of the screening of
the π electrons. When we consider a SWNT, the κ values for the higher Eii transitions which have
larger lk−1 values are smaller than those for lower Eii (smaller lk−1 ). Thus, lk−1 (the exciton size
in real space) is also smaller for the higher Eii because only a small amount of the electric field
created by an e–h pair will influence the surrounding materials.25
Qualitatively, the origin of the diameter dependence of κ consists of: (1) the diameter-dependent
exciton size and (2) the amount of electric field which goes into the surrounding material. These
two factors are connected to one another and Ando gave an analytic form for an expression
connecting these two factors [349]. The development of an electromagnetic model is needed to
S
fully rationalize Equation (92). Interestingly, the similarity between the κ values found for E22 and
 Advances in Physics 517

Figure 62. δEiienv versus dt , scaled by C̃κ . Circles and triangles, respectively, denote AA and HiPCO samples
(see text). Many square symbols on the zero line denote the SG sample which is taken as the standard. The
inset shows differences between experimental (exp) and calculated (cal) Eii values for all samples, showing
good agreement between experiment and the model calculations. Reprinted figure with permission from
A.R.T. Nugraha et al., Applied Physical Letters 97, p. 91905, 2010 [232]. Copyright © (2010) American
Institute of Physics.

M
E11 shows that the difference between metallic and semiconducting tubes is satisfactorily taken
into account by lk using the random phase approximation (RPA) in calculating ε(q) [148,342].
In Figure 61(d), three sets of data contained in Figure 61(a)–(c) are merged, from which we
know that the three plots when taken together depend only on the difference of the slopes, that is Cκ
of Equation (92). Values of Cκ for the SG, AA and HiPCO samples are 0.84, 1.19 and 1.28, respec-
tively. We expect that such differences in the values for Cκ arise from the environmental effects
on the exciton energies. Therefore, we assume that each Cκ value characterizes the environmental
dielectric constant κenv for that particular sample. The SG sample has the largest Eii and hence the
smallest dielectric constant relative to any of the other samples found in the literature [230], and
so we normalize Cκ of the SG sample to be C̃κ (SG) = 1.00 for simplicity. The values of C̃κ for
the other samples can then be determined by taking the ratio of their Cκ values to that for the SG
sample. Thus, C̃κ for the SG, AA, and HiPCO samples becomes 1.00, 1.42, and 1.52, respectively.
When we use the normalized C̃κ , all points collapse on to a single line, as shown in Figure 61(e),
hence giving justification for the use of this assumption. It is interesting to see that all the lines
shown in Figure 61 cross the horizontal axis at κ = 1 at the same point. This point corresponds to
the large diameter limit beyond which the 1D exciton does not exist. The corresponding diameter
is scaled by p and lk .
In Figure 62, we plot the calculated energy shift δEiienv relative to the SG results as a function
of p/dt . The data on the horizontal axis are the data for the SG sample which should be zero from
the definition of δEiienv . The δEiienv for the AA and HiPCO sample are fitted to a function:
' (
2
p p
δEiienv = EiiSG − Eiienv ≡ C̃κ A+B +C , (93)
dt dt

where A, B and C are parameters common to all types of environments and Eiienv is calculated using
the κ function obtained previously. The best fit is found for A = −42.8 meV, B = 46.34 meV · nm
and C = −7.47 meV · nm2 . In the inset to Figure 62, we show the energy difference between
experiment and theory that is obtained by using the κ function (Equation (92)) and we see all the
data points are within 50 meV for a large range of SWNT diameters and energies.
518 R. Saito et al.

exp
Figure 63. Eii vs. ωRBM results obtained for the “super-growth” (bullets) and the “alco-
hol-assisted” (open circles) SWNT samples. Reprinted figure with permission from P.T. Araujo et al.,
Physical Review Letters 103, p. 146802, 2009 [229]. Copyright © (2009) by the American Physical Society.

5.2.3. Effect of the environmental dielectric constant κenv on Eii

exp
Looking at Figure 61, we can observe a difference in the κ values resulting from fitting the Eii to
the “super-growth” (Figure 61(a)) sample in comparison to the “alcohol-assisted” (Figure 61(b))
SWNT sample. For E22 S
and E11 M
, we see in Figure 61(a) a clear difference for κ up to p = 3 when
S S
comparing the two samples. However, for E33 and E44 (Figure 61(d)), no difference in κ in the
range κ = 1–3 can be seen between the two samples. This means that the electric field of the
S S S
E33 and E44 excitons does not extend much outside the SWNT volume, in contrast to the E22 and
M
E11 excitons for which the κenv effect is significant. Since the effect of κenv is relatively small for
M S S
energies above E11 , it is still possible to assign the (n, m) values from E33 and E44 even if the
S S
dielectric constant of the environment is not known, and even though the E33 and E44 values are
seen within a large density of dots in the Kataura plot of Figure 63.

5.3. Splitting of the G mode

The first-order Raman-allowed G-band mode in 2D graphene is a single peak feature (ωG ≈
1584 cm−1 ) [112]. In SWNT, the G-band appears as multiple peaks centered around ωG [333,
350]. Two, four or six G-band phonons are allowed for the first-order G-band Raman feature in
SWNTs, depending on the polarization scattering geometry. Two of these mode, named the totally
symmetric A1 modes (see Figure 64), usually dominate the spectra, although this depends both
on the polarization geometry and on the resonance condition. Finally, the G-band profile depends
strongly on the metal vs. semiconducting nature of the tube, as well as on changes in the Fermi
level. In this section, we discuss the effects responsible for these dependencies, which are related
to the effect of curvature and el–ph interaction on the G-band modes.

5.3.1. The G-band eigenvectors and curvature

Because of the curvature of the graphene sheet in carbon nanotubes, the longitudinal optical (LO)
vs. in-plane transverse optical (iTO) phonons, which are degenerate in graphene, have different
frequencies in carbon nanotubes. Actually, the definition of iTO and LO is different for graphene
and nanotubes. For 2D systems, transverse and longitudinal are defined with respect to the phonon
propagation direction. In the 1D SWNTs, the longitudinal vibration denotes atomic motion along
the tube axis, where phonon momentum q can be defined, and the transverse vibration corresponds
to atomic motion perpendicular to the tube axis [100,280].
 Advances in Physics 519

Figure 64. (a) The totally symmetric G-band eigenvectors for the (8,4) semiconducting SWNT. The atomic
displacements are almost parallel to the circumference. (b) The totally symmetric G-band eigenvector for the
(9,3) metallic SWNT. The atomic displacements are almost parallel to the carbon–carbon bonds. Reprinted
figure with permission from S. Reich et al., Physical Review B 64, p. 195416, 2001 [280]. Copyright ©
(2001) by the American Physical Society.

First-principles calculations have been performed for different SWNTs by Reich et al. [280].
They found that, while for armchair and zigzag SWNTs, which have higher symmetry than chiral
tube structures, the LO and iTO vibrations of the G-band exist [280], such a definition is not
strictly valid for chiral nanotubes, where the atomic vibrations actually depend on the chiral angle.
In Figure 64(a and b) Reich et al. show the mode displacements for one totally symmetric (A1 )
G-band mode in two different chiral SWNTs. For each case, another G-band mode is expected
with the atomic vibrations perpendicular to those indicated in the figure. Note that the atomic
displacements in Figure 64 are almost perfectly aligned along the circumference in the (8,4)
S-SWNT (a), but mostly parallel to the C–C bonds in the (9,3) M-SWNT (b), and in neither
case is the strict iTO definition applicable. These findings by Reich et al. [280] are qualitatively
consistent with other calculations [86], but the quantitative definitions are model-dependent. There
is still controversy about whether the many peaks within this G-band can be assigned to LO and
iTO type mode behavior, as discussed by Piscanec et al. [193].

5.3.2. The six G-band phonons – confinement effect

While the doublet nature of the G-band discussed above is related to strain effects from the tube
curvature, 1D confinement makes one, two or three of those doublets observable, each pair having
either A1 , E1 or E2 symmetry. This happens because, when rolling up the graphene sheet to form
a nanotube, the zone folding along the circumferential direction generates a larger number of
first-order Raman-allowed modes [32,99,100,167].
The difference between A1 , E1 and E2 symmetry Raman modes is given by the number of
nodes for the atomic motion along the tube circumference, where the number of nodes is zero, two
and four, respectively. This zone-folding procedure is shown in Figure 65(b). Therefore, combined
with the (quasi) LO and iTO vibrational nature, two, four or six G-band phonons can be Raman-
allowed in the Raman spectra from SWNTs, and their observation depends on the polarization of
the scattering geometry and on the resonance condition, as explained below.
Selecting the Z-axis as the SWNT axis direction, and the Y -axis as the photon propagation
direction, light can be polarized parallel (Z) or perpendicular (X) to the nanotube axis. The possible
scattering geometries are given by pi (is)ps , using what is called the Porto notation in memory
of S.P.S. Porto, where pi and ps give the propagation directions for the incident and scattered
520 R. Saito et al.

Table 4. Selection rules for polarization dependent G-band features and

the corresponding resonance conditions.

Symmetry of phonon Scattering event Resonance condition

A1 (ZZ) Elaser = Eii , Elaser ± EG = Eii

A1 (XX) Elaser = Eii±1 , Elaser ± EG = Eii±1
E1 (XZ) Elaser = Eii±1 , Elaser ± EG = Eii
E1 (ZX) Elaser = Eii , Elaser ± EG = Eii±1
E2 (XX) Elaser = Eii±1 , Elaser ± EG = Eii±1

The Z- and Y - axes are the SWNT axis direction and the photon propagation direction, respec-
tively. The polarization of the incident and scattered light is given as well as the resonance
condition. EG is the G-band phonon energy [1].

photons, respectively, while the incident polarization i and scattered polarization s appear inside
the parenthesis (is). Since the SWNT experiments are usually made by using microscopes in the
back-scattering configuration, pi and ps are usually Y and −Y , so that the simplified notation (is)
can be applied. In this case, four different scattering geometries are possible, and these are labeled
XX, XZ, ZX and ZZ.
The first-order Raman signal from isolated SWNTs can only be seen when the excitation laser
energy is in resonance with a van Hove singularity (VHS) in the JDOS. Chiral SWNTs exhibit CN
symmetry [351,352] and, following the (is) notation introduced above, selection rules imply that:
(1) totally symmetric A phonon modes are observed for the (ZZ) scattering geometry when either
the incident or the scattered photon is in resonance with Eii , and for the (XX) scattering geometry
when either the incident or the scattered photon is in resonance with Ei,i±1 . (2) E1 symmetry
modes can be observed for the (ZX) scattering geometry for resonance of the incident photon with
the Eii VHSs, or for resonance of the scattered photon with the Ei,i±1 VHSs, while for the (XZ)
scattering geometry, for resonance of the incident photon with the Ei,i±1 VHSs, or for resonance of
the scattered photon with the Eii VHSs; (3) E2 symmetry phonon modes can only be observed for
the (XX) scattering geometry for resonance with Ei,i±1 VHSs. Therefore, it is possible to observe
two, four or six G-band peaks, depending on the resonance conditions and on the polarization
scattering geometry, as summarized in Table 4.

Figure 65. (a) Schematic picture of the G-band atomic vibrations along the nanotube circumference and
along the nanotube axis of a zigzag nanotube. (b) The Raman-active modes with A, E1 and E2 symmetries
and the corresponding cutting lines μ = 0, μ = ±1 and μ = ±2 in the unfolded 2D BZ. The  points of
the cutting lines are shown by solid dots. Reprinted from Raman Spectroscopy of Carbon Nanotubes, M.S.
Dresselhaus et al. [167]. Copyright © (2005) with permission from Elsevier.
 Advances in Physics 521

Figure 66. (a) G-band polarization dependence from one isolated semiconducting SWNT sitting on a Si/SiO2
substrate [357]. Both incident and scattered light are polarized parallel to each other and vary from ZZ (bottom)
to XX (middle) and back to ZZ (top). (b,c) G-band polarization scattering geometry dependence from two
isolated SWNTs. The Lorentzian peak frequencies are given in cm−1 . The incident angles θS and θS between
the light polarization and the SWNT axis directions are not known a priori, but have been assigned as
θS ∼ 0◦ and θS ∼ 90◦ based on the relative intensities of the polarization behavior of the G-band modes and
the expected selection rules and frequencies (see Section 5.3.3) for the A1 , E1 and E2 modes [352]. Reprinted
figure with permission from A. Jorio et al., Physical Review B 65, p. R121402, 2002 [357]. Copyright ©
(2009) by the American Physical Society.

Before showing the experimental results related to the selection rules discussed above, we
introduce a general polarization behavior that is not accounted for in these selection rules and
is responsible for the totally symmetric A1 modes in the G-band spectra which dominate these
spectra. Carbon nanotubes are nano-antennas, and both the absorption and emission of light are
suppressed when the light is polarized perpendicular to the nanotube axis. This phenomenon is
called as the depolarization effect [353,354], where photo-excited carriers screen the electric field
of the cross-polarized light inside the carbon nanotube [234,353,354]. Considering these effects, it
is clear that the Raman intensity is generally largest for the (ZZ) scattering polarization geometry,
while the signal will be suppressed for the (XX) scattering, as shown in Figure 66(a) [355–357].
In addition, remember that the resonance energies are different for light along Z (Eii ) and along
X (Eii±1 ), as described in Table 4. This means that one rarely observes the Raman signals from
parallel (ZZ) and perpendicular ((ZX), (XZ) or (XX)) polarization directions simultaneously.
Now we can discuss experimental results related to the symmetry selection rules for the dif-
ferent scattering geometries [351,357,358]. Figure 66(b) shows the G-band Raman spectra from a
semiconducting SWNT with three different directions for the incident light polarization. Consid-
ering θS the initial angle between light polarization and the nanotube axis, unknown a priori, the
three spectra in Figure 66(b) were acquired with θS , θS + 40◦ and θS + 80◦ . Six well-defined G-
band peaks are observed with different relative intensities for the different polarization geometries
and, based on the selection rules and on their frequencies (see Section 5.3.3), they are assigned
as follows: 1565 and 1591 cm−1 are A1 symmetry modes; 1572 and 1593 cm−1 are E1 symmetry
modes; and 1554 and 1601 cm−1 are E2 symmetry modes. Figure 66(c) shows the G-band Raman
spectra obtained from another semiconducting SWNT, the two spectra with θS and θS + 90◦ ,
522 R. Saito et al.

Figure 67. ωG (open symbols) vs. ωRBM (bottom axis) and vs. 1/dt (top axis) for S-SWNTs.
Explicit experimental G-band data obtained with Elaser = 1.58, 2.41 and 2.54 eV are presented. Solid
symbols connected by solid lines give results obtained from ab initio calculations [249], down-
shifted by about 1% to fit the experimental data [352]. Reprinted figure with permission from
O. Dubay and G. Kresse, Physical Review B 67, p. 35401, 2003 [249]. Copyright © (2003) by the American
Physical Society.

where θS is also unknown a priori. The spectra requires four sharp Lorentzians for a good line-
shape fitting, plus a broad feature at about 1563 cm−1 . This broad feature (FWHM ∼ 50 cm−1 )
is sometimes observed in weakly resonant G-band spectra from SWNTs, and is generated by
defect-induced DR processes. For the sharp peaks, based on the polarization Raman studies and
on their relative frequencies [351], the 1554 and 1600 cm−1 peaks are E2 symmetry modes, while
the 1571 and 1591 cm−1 peaks are unresolved (A1 + E1 ) symmetry modes, with their relative
intensities depending on the incident light polarization direction [351].
Generally speaking, the intensity ratio between the two scattering geometries ZZ : XX can
assume values either larger or smaller than 1, depending on the resonance condition. Observation
of the spectra with (XX) polarization in Figure 66(c) indicates resonance with a Ei,i±1 optical
transition. In samples with a large diameter distribution, e.g. dt from 1.3 nm up to 2.5 nm in
Ref. [351], both the Eii and Ei,i±1 transitions can occur within the resonance window of the same
laser, and an average value of ZZ : XX = 1.00 : 0.25 was then observed. However, for most
isolated SWNTs, the Raman intensities exhibit an intensity ratio IZZ : IXX ∼ 1 : 0. This “antenna
effect” is observed for samples in resonance with only Eii electronic transitions, as shown in
[355–357].
Finally, the above discussion is related to phonon confinement within the first-order single
resonance process [333,350,351]. There has been interesting discussion on whether the features
in the multi-peak G-band could all be assigned to A1 symmetry modes [355,359] originating from
a defect-induced DR Raman scattering process [145]. It seems that both cases, i.e. a multi-peak
feature from multi-symmetry scattering and from an A1 symmetry feature induced by disorder
are possible in very disordered samples [360].

5.3.3. The diameter dependence of the G band phonon frequencies

The G-band mode assignment proposed in Section 5.3.2 comes from a comparison between
experimental results and ab initio calculations of the diameter dependence of ωG [249], as shown
 Advances in Physics 523

Figure 68. Experimental intensity plot of the G-band spectrum of a metallic SWNT as a function of elec-
trochemical gate potential. For this nanotube the charge neutrality point, corresponding to the Dirac point,
is 1.2 V. Adapted figure with permission from M. Farhat et al., Physical Review Letters 99, p. 145506, 2009
[201]. Copyright © (2009) by the American Physical Society.

in Figure 67. The relation between ωRBM (bottom axis – for experimental results) and inverse
nanotube diameter (top axis – for theoretical results) were made in this figure considering 1/dt =
ωRBM /248. This ωRBM = 248/dt relation [111] was broadly used in the early years of single
nanotube spectroscopy (2001–2005). The present understanding uses Equation (86) with Ce =
0.065, which is applicable to a larger variety of samples.
The spectra in this work were fit using 2, 4 or 6 peaks with FWHM γG ∼ 5 cm−1 , which is the
natural linewidth for the G-band modes [340]. The diameter dependent downshift in frequency
comes from strain and from curvature-induced mixing of low frequency out-of-plane components.
In the time-independent perturbation picture, the ωGLO mode frequency is expected to be indepen-
dent of diameter, since the atomic vibrations are along the tube axis. In contrast, the ωGTO mode
has atomic vibrations along the tube circumference, and increasing the curvature increases the
out-of-plane component, thus decreasing the spring constant with a 1/dt2 dependence. This picture
holds for semiconducting SWNTs, where G+ now stands for the LO mode, and G− stands for
the iTO mode [171]. However, for metallic SWNTs the picture is different: G+ now stands for
the iTO mode, and G− stands for the LO mode [193,249]. The G-band profile in this case is very
different and depends strongly on doping, as shown in Figure 68, and this behavior can only be
understood within a time-dependent perturbation picture. In this section, we focus on the spectra
from semiconducting SWNTs. Metallic SWNTs are discussed in Section 5.4.
The dt dependence of the frequencies for each of the three higher frequency G+ band modes
(A1 , E1 and E2 ) observed experimentally are in very good agreement with theory [249], showing
basically no diameter dependence. For the lower frequency G− band modes, both ab initio calcu-
lations and the experimental results show a considerable dt dependence of the mode frequency.
The experimental G-band frequencies from semiconducting SWNTs can be fit with [352]

C
ωG = 1592 − β
, (94)
dt
524 R. Saito et al.

where the values for the various parameters are: β = 1.4, CA1 = 41.4 cm−1 nm, CE1 =
32.6 cm−1 nm and CE2 = 64.6 cm−1 nm.
A simpler formula for the G-band intensities has also been proposed in Ref. [171], where
the G-band fitting is not performed with six Lorentzians, but rather by considering just the peak
value for the two most intense features. In this case, the higher frequency G+ peak is diameter
independent and the lower frequency G− feature decreases in frequency with a 1/dt2 dependence.
In practice, the G band can be used for a diameter determination for both semiconducting and
metallic SWNTs by using the formula from [171]. For more detailed studies, the more complete
analysis discussed above and including dynamic effects [193] and environmental effects [232]
should be used.

5.4. Kohn Anomaly effect on the G-band and the RBM mode
In this section, the first-order spectra for metallic SWNTs are discussed giving special attention
to the dependence of these spectral features on the Fermi level position and the KA phenomena
that account for the special properties of these spectral features for metallic SWNTs [193]. First,
we discuss the characteristics of the KA for the G-band in Section 5.4.1 through Section 5.4.4,
followed by an elaboration of the KA for the RBM feature in Section 5.4.5.

5.4.1. G-band Kohn anomaly

The Fermi-level dependencies of the LO and iTO modes on SWNT bundles have been measured
[50,51,361,362], showing mainly changes in frequency and linewidth of the Breit–Wigner–Fano
(BWF)-like feature in the G-band associated with metallic nanotubes, as EF was varied. This effect
is now understood within the framework of time-dependent perturbation theory including dynamic
effects [193], and this effect has been shown more clearly with Fermi level variation experiments
performed on individual metallic carbon nanotubes [193,201,283]. The G-band spectrum of an
individual metallic nanotube changes as a function of an electrochemical gate potential as shown
in Figure 68. The higher frequency peak does not change significantly as a function of gate
voltage, but the broad lower frequency peak (BWF-like) upshifts and narrows in linewidth with
both positive and negative changes in the Fermi level. Differently from semiconducting SWNTs,
where the higher frequency peak is assigned to the LO vibration, metallic SWNTs have a lower
frequency mode with a BWF-like lineshape that is assigned to the LO mode, which downshifts in
energy significantly due to the KA effect [193].
The KA is predicted to occur in metallic carbon nanotubes [193], and to exhibit two character-
istic signatures related to non-adiabatic effects: (1) within the energy window EF < |ωLO |, the LO
phonon peak is broadened due to the creation of real e–h pairs; (2) the LO frequency vs. EF curve
shows a characteristic “W” lineshape due to the two singularities located at EF = ±ωLO . This
“W” lineshape is not resolved in Figure 68 because of inhomogeneous charging due to trapped
charges on the substrate that result in a smearing of EF . However, using pristine suspended nan-
otubes that are gated electrostatically, the “W” feature can be explicitly seen in the frequency
dependence of the LO mode [363].
The intensities of the LO and iTO modes depend also on the chiral angle, with the iTO mode
being symmetry forbidden for a zigzag nanotube [283,364]. A chiral angle dependence of the iTO
mode softening has been predicted theoretically [365]. Future experiments on structurally identi-
fied and truly isolated individual (n, m) nanotubes are needed to further advance our understanding
of these phenomena related to the KA.
Gate-dependent experiments on semiconducting nanotubes reveal that their G-band phonons
also experience energy renormalization due to el–ph coupling [51,366]. However, ωLO/iTO < E11 S
 Advances in Physics 525

for semiconducting SWNTs, which means that the G-band phonons are unable to create real e–
h excitations across a typical non-zero bandgap. Therefore, lifetime broadening of the phonon
is not expected, and the G-band peak FWHM should not change significantly. Still, virtual e–h
excitations contribute to the softening of the phonon energies [366]. Both the iTO and LO modes
couple to intermediate e–h pairs, and the iTO mode is expected to show a greater EF -dependent
frequency shift, most significantly in larger diameter nanotubes, as the band gap energy approaches
the phonon energy [366]. This behavior in semiconducting SWNTs is opposite to metallic SWNTs.

5.4.2. Chemical doping and the G band

The KA effect discussed in Section 5.4.1 is important at low doping levels. For higher doping
levels, structural distortions dominate, and these distortions depend on the level of the p vs. n
doping, which should cause upshift vs. downshift in the phonon frequencies, respectively. Such p
vs. n doping behavior has been demonstrated for the G band of single-wall and multi-wall carbon
nanotubes doped chemically with atomic species that are donors or acceptors to carbon [36,367].
For example, the doping with K, Rb and Cs (alkali metals) leads to a G-band mode softening of
35 cm−1 accompanied by significant changes in the G-band lineshape. For SWNT bundles doped
with Br2 (halogens), an upshift in the Raman-mode frequencies is observed. For more details on
the effect of chemical doping on the Raman spectra of SWNTs, see, e.g. [36].

5.4.3. Substrate interaction and the G band

The confocal image shown in Figure 69(a) was obtained by integrating the G-band Raman signal
from a SWNT serpentine sample, while raster scanning the sample [324]. These carbon serpen-
tine nanotubes are SWNTs with parallel straight segments connected by alternating U-turns, as
shown in Figure 69(a) [324]. These serpentine nanotubes were grown by catalytic CVD on miscut
single-crystal quartz wafers [368]. The resulting vicinial SiO2 (1 1 0 1) substrate is insulating,
and is terminated by parallel atomic steps. At the temperature of nanotube growth, the surface
contains exposed unpassivated Si atoms, thus enabling a strong tube–substrate interaction to occur,
especially when the nanotube lies along a step. Alternatively, when the nanotube lies across the
surface steps, the interaction is discontinuous and weaker. The orientation between the tube and
crystalline quartz varies, which means that the tube–substrate interaction is modulated along the
tube. The straight segments usually lie along the quartz steps, while the U-turns lie across the
steps. Figure 69(b) shows the G-band Raman spectra obtained from all 41 points indicated by the
pointers in Figure 69(a). Careful analysis of the individual G-band spectral features shows that
the G-band spectra changes along the SWNT serpentine. These changes are related to strain and
doping which varies along the tube due to the morphology of the tube–substrate interaction in these
samples [324]. Such a strong interaction also has a strong effect on the RBM frequencies, leading
to an environmental constant of Ce = 0.085 in Equation (86) which relates ωRBM and dt [323].

5.4.4. Theoretical approach to the Kohn Anomaly

The KA in metallic SWNTs was introduced firstly by Piscanec et al. [193]. As discussed in
Section 3.6.5, the phonon softening phenomena is understood by second-order perturbation theory
in which a phonon virtually excites an e–h pair across the metallic energy subband of a metallic
SWNT (as shown in Figure 70(a)). The effective Hamiltonian in second-order perturbation theory
is given by Equation (71) [196,205,246]. In Figure 70(b), we show the energy denominator of
Equation (71), in which the real part of h(E) is related to the frequency shift, while the imaginary
part of h(E) is related to the spectral broadening. From this figure, for the energy below (above)
the phonon energy of the G-band, the virtual excitation contributes to the phonon hardening
526 R. Saito et al.

Figure 69. (a) Confocal image of the G-band integrated intensity using a laser wavelength λlaser = 632.8 nm
(Elaser = 1.96 eV). The inset shows a schematic view of the SWNT (yellow) lying on top of the miscut quartz.
(b) The G-band Raman spectra obtained at the 41 points indicated and numbered in (a). There is a modulation
of the higher frequency G+ feature (∼1590-1605 cm−1 ) along with the appearance and disappearance of the
lower frequency G− feature (∼1540 cm−1 ). This appearance and disappearance of the G+ and G− features
are related to the tube-substrate morphology and interaction. Adapted with permission from J.S. Soares et al.,
Nono Letters 10, pp. 5043–5048, 2010 [324]. Copyright © (2010) American Chemical Society.

Figure 70. (a) An intermediate e–h pair state that contributes to the energy shift of the optical phonon
modes is depicted. A phonon mode is denoted by a zigzag line and an e–h pair is represented by a loop.
The low energy e–h pair satisfying 0 ≤ E ≤ 2|EF | is forbidden at zero temperature by the Pauli principle.
(b) The energy correction to the phonon energy by an intermediate e–h pair state, especially the sign of
the correction, depends on the energy of the intermediate state as h(E). Adapted figure with permission
from K. Sasaki et al., Physical Review B 78, pp. 235405–235411, 2008 [365]. Copyright © (2008) by the
American Physical Society.

(softening). Thus, when the phonon hardening processes are suppressed by increasing the Fermi
energy, we get a large phonon softening and broadening at 2EF = ±0.2 eV. This is the origin of
the observed W lineshape [196]. Recent measurements of the G-band spectra at T = 12 K show
phonon anomalies at EF = ±ωG /2 that could be clearly distinguished experimentally [200].
Furthermore, the el–ph matrix element for this virtual excitation has a chiral angle dependence
that acts differently for iTO and LO phonons [196]. For example, the el–ph interaction for the
iTO phonon is absent (no KA effect) for armchair nanotube, while phonon hardening of the iTO
mode occurs for zigzag nanotubes. Thus, a detailed analysis of the LO and iTO phonon modes as a
function of the Fermi energy gives important information about the (n, m) assignment for metallic
carbon nanotubes. For a detailed description of these phenomena, see [193,196,205,246].
 Advances in Physics 527

Figure 71. Deviation of the experimentally observed RBM frequency (ωRBM ) from the linear depen-
dence given by (218.3/dt + 15.9), as a function of θ for a particular HiPCO nanotube sample.
Filled, open and crossed circles denote M-SWNTs, type I and type II S-SWNTs, respectively. The
dotted lines show an experimental accuracy of ±1 cm−1 . Reprinted figure with permission from
A. Jorio et al., Physical Review B 71, p. 75401, 2005 [369]. Copyright © (2005) by the American Physical
Society.

5.4.5. RBM band and G-band Kohn anomaly

The diameter dependence of the RBM frequency ωRBM can be described by elasticity theory, as
presented in Section 5.1.1. However, there are two effects that are not considered by elasticity
theory when discussing the ωRBM , the first related to the chirality-dependent distortion of the
lattice, and the second related to the el–ph coupling occurring in metallic SWNTs. The first effect
should be observable in SWNTs with dt  1 nm SWNTs, where the curvature-induced lattice
distortion is important, while the second should be observable in metallic SWNTs.
Figure 71 shows a plot of the deviations of the observed ωRBM values from the best linear
1/dt dependence that fits all the experimental data [ωRBM = ωRBM − (218.3/dt + 15.9)] as a
function of chiral angle θ, for SWNTs grown by the HiPCO (high pressure CO CVD) method
[369]. Deviations of the points from ωRBM = 0 are clearly seen, with interesting trends in the
deviations: (i) metallic SWNTs (solid bullets) exhibit systematically larger ωRBM when compared
with semiconducting SWNTs (open bullets); (ii) ωRBM depends on the chiral angle θ , showing
an increase in ωRBM with increasing θ from zigzag (0◦ ) to armchair (30◦ ) SWNTs.
Because of tube curvature, there is sp2 –sp3 mixing, and the RBM frequencies decrease with
respect to their ideal values as the SWNT diameter decreases [254,370]. In armchair tubes, the
circumferential strain is more evenly distributed between the bonds, leading to smaller bond
elongations. This is a purely geometric effect, related to the directions of the three C–C bonds
with respect to the circumferential direction. Since the RBM softening is directly related to the
bond elongation along the nanotube circumference, a larger softening of ωRBM for zigzag tubes
relative to armchair tubes is expected, as is shown in Figure 71 [369].
Finally, similar to the effect discussed for the G-band, a phonon frequency shift of the RBM
for metallic SWNTs was predicted [193,365] and observed experimentally [204] as a function of
the Fermi energy, although a much smaller shift (∼ 3 cm−1 ) due to the KA effect is expected for
ωRBM than for ωG . Furthermore, in the chiral and zigzag metallic SWNTs, a mini-gap exists and,
when the gap is larger than ωRBM , then the KA effect disappears. There is a lower limit of dt
(1–1.8 nm depending on chiral angle) below which the KA effect for the RBM phonon cannot be
observed, since the mini-gap is proportional to 1/dt2 and ωRBM is proportional to 1/dt [365].
528 R. Saito et al.

Figure 72. (a) 2D plot of the Elaser dependence for the Raman spectra of SWNT bundles in the intermedi-
ate frequency mode (IFM) range. The bright, light areas indicate high Raman scattering intensity. Arrows
point to five well-defined ωIFM features. (b) Raman spectra in the corresponding IFM range are taken
at discrete laser excitation energies Elaser = 2.05, 2.20, 2.34, and 2.54 eV. Reprinted figure with permission
from C. Fatini et al., Physical Review Letters 93, p. 87401, 2005 [371]. Copyright © (2003) by the American
Physical Society.

5.5. Double resonance effect and quantum confinement

Like in graphene, SWNTs exhibit dispersive bands related to the DR effect. However, for SWNTs
the Raman signal is dominated by resonance effects associated with van Hove singularities. The
vertical stripes in Figure 72(a) define the resonance window for a given resonance band, for a given
(n, m) SWNT within a SWNT-bundle sample. The modes appearing in the spectral region between
400 and 1200 cm−1 have been named intermediate frequency modes (IFMs), making reference to
their frequencies being between the commonly studied RBM and G modes. The IFM features are
related to combinations of out-of-plane transverse modes (oTO) and longitudinal acoustic modes
(LA), more specifically oTO ± LA [371,372] in the 2D-unfolded graphene system. Up to now, it
is not yet clear whether or not these modes are activated by defects. Theory relates the observation
of the IFMs to quantum confinement effects along the tube length [373], and some experimental
evidence has been found to support such theory [374].
The G peak [193,310,375] in the Raman spectra for SWNTs, shows a dispersive behavior with
laser excitation energy, but some unique characteristics are observed due to the 1D structure of
SWNTs. We will detail these results in the next two sections, where in Section 5.5.1 we show the
G behavior for SWNT bundles, and in Section 5.5.2 we discuss the G -band behaviour in isolated
SWNTs.
 Advances in Physics 529

Figure 73. (a) G -band data for ωG for a SWNT bundle sample taken from [222] after subtracting the
linear dispersion 2420 + 1106Elaser from the ωG vs. Elaser data shown in the inset. (b) Optical transition
energies Eii as a function of diameter for SWNTs. The vertical lines denote the diameter range of the SWNT
bundle used in the G -band dispersion experiment shown in (a). Reprinted figure with permission from
A.G. Souza Filho et al., Physical Review B 65, p. 35404, 2001 [376]. Copyright © (2001) by the American
Physical Society.

5.5.1. The G -band in SWNT bundles

For SWNT bundles, most of the (n, m) dependent 1D-related effects are averaged out, and a close
relation is observed experimentally between the G -band in SWNTs and in graphene. However,
anomalous effects are still observed, and these effects are related to the 1D structure of the material
in the nanotube bundle. The inset to Figure 73(a) shows the dispersion of the ωG frequency in
cm−1 in SWNT bundles. Fitting the observed linear dispersion of ωG for SWNTs [222] gives

ωG = 2040 + 106Elaser , (95)

which is consistent with observations in graphene and graphite. However, different from graphene
and graphite, the G -band dispersion in SWNTs exhibits a superimposed oscillatory behavior as a
function of Elaser , as shown in Figure 73(a), where the linear dispersion effect was subtracted from
the experimentally observed frequencies. The oscillatory behavior seen in Figure 73(a) is due to
the ωG dependence on tube diameter, as discussed below.
The G -band frequency (ωG ) depends on tube diameter (dt ) because of a force constant soft-
ening, which is associated with the curvature of the nanotube wall. Experiments on isolated tubes
530 R. Saito et al.

Figure 74. Cutting lines for two metallic SWNTs, one zigzag and one armchair, in the unfolded 2D BZ of
graphene. The wavevectors ki point with arrows to the locations where the van Hove singularities occur.
Reprinted figure with permission from G.G. Samsonidze et al., Physical Review Letters 90, p. 27403, 2003
[378]. Copyright © (2003) by the American Physical Society.

show that [377]

35.4
ωG = ωG0 − , (96)
dt
where ωG0 is the laser energy-dependent value observed in graphene, which can be considered
to be the limiting value of ωG for an infinite diameter tube. This diameter dependence of ωG
is the critical factor behind the oscillatory behavior observed in Figure 73(a). The vertical lines
in Figure 73(b) denote the diameter range of the SWNTs contained in the SWNT bundle used
in the G -band dispersion experiment illustrated in Figure 73. When moving along an arrow by
S
increasing the excitation laser energy, for example, within the E22 subband by changing the Elaser ,
value, just above 1 eV, different SWNTs with different diameters enter and leave the resonance
window for a given Eii optical transition. By increasing the laser energy, the diameters of the
tubes that are in resonance decrease, thus increasing the expected energy due to the DR process.
When the resonance condition with Elaser jumps from one interband transition to another, e.g.,
from E22S M
to E11 , which occurs at around Elaser = 1.5 eV, the nanotube diameter jumps to higher
values. This process modulates the ωG dispersion, as observed by the oscillatory behavior shown
in Figure 73(a).
The “continuous” frequency dispersion observed in Figure 73(b) is a result observed in SWNT
bundles where different tubes enter and leave resonance, thus probing the whole unfolded 2D BZ.
This result is clearer for the IFMs as shown in Figure 72. The 1D confinement effects on the G
spectra can only be clearly seen in experiments at the individual isolated SWNT level, discussed
further in Section 5.5.2.

5.5.2. The (n, m) dependence of the G -band; phonon trigonal warping

This section gives an appreciation of the effect of 1D confinement on the G feature in SWNTs
at the individual SWNT level. In the case of SWNTs, the resonance condition is restricted to
Elaser ≈ Eii , Eii ± Eph (the resonant transition energy between van Hove singular energies). This
fact gives rise to a ωG dependence on the SWNT diameter (see Section 5.5.1) and on the chiral
angle.
Figure 74 shows the cutting lines for two metallic SWNTs, one zigzag and one armchair, in the
unfolded 2D BZ of graphene. The van Hove singularities occur where a cutting line is tangent to
 Advances in Physics 531

Figure 75. The G -band Raman spectra for (a) a semiconducting (15, 7) and (b) a metallic (27, 3) SWNT,
showing two-peak structures [375,378,379], respectively. The vicinity of the K point in the unfolded BZ is
shown in the lower part of the figure, where the equi-energy contours for the incident Elaser = 2.41 eV and the
scattered Elaser − EG = 2.08 eV photons, together with the cutting lines and wave vectors for the resonant van
S = 2.19 eV, E S = 2.51 eV, E M(L) = 2.04 eV, E M(U) = 2.31 eV) are shown. Reprinted
Hove singularities (E33 44 22 22
figures from A.G. Souza Filho et al., Physical Review B 65, p. 85417, 2002 [375]. Copyright © (2002) by
the American Physical Society and A.G. Souza Filho et al., Chemical Physics Letters 354, pp. 62–68,
Copyright © (2002) with permission from Elsevier.

an equi-energy contour, thus causing a chiral angle dependence on the ki value where a particular
excited state ki occurs. The states at ki are those responsible for the dominant optical spectra
observed in SWNTs, including the DR features. The presence of cutting lines in carbon nanotubes
is expected to affect all the dispersive Raman features [376], but here we focus on the G -band,
because the G -band dispersion is very large and is an interesting effect.
The two-peak G -band Raman features in the Raman spectra observed from semiconducting
and metallic isolated nanotubes are shown in Figure 75(a) and (b), respectively. The presence of
two peaks in the G -band Raman feature indicates the resonance with both the incident Elaser and
scattered Elaser –EG photons, respectively, with two different van Hove singularities (VHSs) for
the same nanotube. Elaser and Elaser –EG are defined in Figure 75(a) and (b) below the G -band
spectra, by the outer and inner equi-energy contours near the 2D BZ boundary, in which the cutting
lines are shown and the trigonal warping of these constant energy contours can be seen [380]. The
two peaks in Figure 75(a) and (b) can be associated with the phonon modes of the wave vectors
qi = −2ki , where i = 3, 4, 2L, 2U relate to E33
S S
, E44 ML
, E22 MU
and E22 , respectively, and the electronic
wave vectors ki are shown in the lower part of Figure 75. For the semiconducting SWNT shown
in Figure 75(a), the resonant wave vectors k3 and k4 have different magnitudes, k4 − k3 K1 /3,
resulting in twice the difference for the phonon wave vectors, q4 − q3 2K1 /3 = 4dt /3, so that
the splitting of the G -band Raman feature arises from the phonon dispersion ωph (q) around the
K point. In contrast, for the metallic nanotube (M-SWNT) shown in Figure 75(b), the resonant
wave vectors k2L and k2U have roughly equal magnitudes and opposite directions away from the K
532 R. Saito et al.

Figure 76. Localized excitonic emission in a semiconducting SWNT. (a) Photoluminescence emission at
λem = 900 nm from a single SWNT. (b) Raman spectrum recorded from the same SWNT. The spectral
position of the RBM, ωRBM = 302 cm−1 , together with the λem = 900 nm information, leads to the (9,1)
assignment for this tube. (c) Near-field photoluminescence image of the SWNT revealing localized excitonic
emission. (d)–(e) Near-field Raman imaging of the same SWNT, where the image contrast is provided by
spectral integration over the G and D bands, respectively. (f) Corresponding topography image. The circles
indicate localized photoluminescence (c) and defect-induced (D band) Raman scattering (e). The scale bar
in (c) denotes 250 nm. (g) Evolution of the G -band spectra near the defective segment of the (9,1) SWNT.
The spectra were taken in steps of 25 nm along the nanotube, showing the defect-induced G peak (dotted
Lorentzian). The asterisks denote the spatial locations where localized photoluminescence and defect-induced
D-band Raman scattering were measured (see circles in (c) and (e), respectively). Reprinted with permission
from I.O. Maciel et al., Nature Materials 7, pp. 878–883, 2008 [293]. Copyright © (2008) American Institute
of Physics.

point, so that the splitting of the G -band Raman feature for metallic nanotubes arises from the
anisotropy of the phonon dispersion ωph (q) around the K point [378], which we identify with the
phonon trigonal warping effect. Overall, the presence of two peaks in the DR Raman features of
isolated carbon nanotubes is associated with quantum confinement effects expressed in terms of
cutting lines.
Of course the G -band is not the only feature to exhibit an (n, m) dependence for SWNT
systems. Actually, all the DR features are expected to exhibit such a chirality dependence. The
stronger the dispersive behavior, the larger is the (n, m) chirality dependence. For this reason,
SWNTs with smaller dt show larger frequency splittings and larger G -band shift effects. The
(n, m) dependence of other combination modes, such as the iTO+LA combination mode near the
 point, have also been studied in some detail (see e.g. Ref. [227]).
 Advances in Physics 533

5.6. Near-field Raman spectroscopy

The near-field technique can generate optical information with spatial resolution x below the
diffraction limit (x ∼ λlaser /2) [381,382]. The near-field Raman spectroscopy and imaging
of individual isolated SWNTs with a record spatial resolution of 25 nm was first measured by
Hartschuh et al. [382]. In sequence, this group performed several studies of local variations in
the Raman spectrum along a single SWNT [45]. Figure 76 shows near-field spectra and imaging
for an individual SWNT [293]. In particular, Figure 76(a) and (b) show the photoluminescence
and Raman spectra, respectively, with x ∼ 30 nm. The near-field microscopy maps from the
same SWNT are shown in Figure 76(c)–(f). Figure 76(c) represents the near-field photolumines-
cence image of this same SWNT, where the image contrast is provided by spectral integration
over the photoluminescence peak centered at λem = 900 nm (see Figure 76(a)). The most striking
feature in this image is the high degree of spatial localization of the photoluminescence emission
along the SWNT. This is evident by inspection of the extended topography image of the nan-
otube shown in Figure 76(f), and also of the near-field Raman image of the G-band, with a peak
intensity near 1590 cm−1 shown in Figure 76(d). While from Figure 76(d) we observe that the
G band Raman scattering is present along the entire length of the nanotube, from Figure 76(e)
we observe an increased defect-induced D band occurring at 1300 cm−1 ). The strong Raman
scattering intensity appears to be localized in the same spectral region where exciton emis-
sion was detected [387]. Defects are known to act as trapping states for e–h recombination
(i.e., exciton emission), thereby providing insights into the correlations observed between
Figure 76(c) and (a).
Interestingly, when measuring the Raman spectra across the defective spot, sudden changes
in many Raman features are observed. Maciel et al. [293] have shown that substitutional doping
in SWNTs causes changes in the G -band spectra due to charge-induced renormalization of the
electronic and vibrational energies. Figure 76(g) shows six G -band spectra measured on the same
SWNT, and these spectra were taken by moving along the SWNT to the position where the local D
band and photoluminescence emission is observed (circle in Figure 76(e)). The two spectra marked
by “*” in Figure 76(g) were obtained at this defect location, and a new peak is here observed in
the G -band. The frequency and intensity of this new peak depend on the dopant species and the
level of doping, respectively [383–385]. This makes the G -band a sensitive probe for studying and
quantifying doping, which is more accurate than the D band, since the D-band can also be related
to amorphous carbon and to any other symmetry-breaking defective sp2 structure [193,293,383–
385]. Tip-enhanced Raman spectra can show changes in chirality along a given tube [388], due to
a large increase in the local signal intensity [389].

6. Challenges of Raman spectroscopy in graphene and carbon nanotubes

We identify in this concluding section current research opportunities where graphene and carbon
nanotubes can have special impact. New research areas can be related to new structures, such
as nanoribbons, and also to still poorly developed but promising experimental techniques. For
example, near-field and time-dependent phenomena have thus far received little attention and
show promise for future in-depth development. All these developments have been the basis for
the nanometrology field, which is at an early stage of development, and carbon nanotubes have
been a focal point for nanometrological protocol development. Although much progress has been
made in the use of Raman spectroscopy to study graphene and carbon nanotubes, many research
opportunities remain, especially in the applications area. In identifying future directions we look
for both novelty and areas of special opportunity, as discussed below.
534 R. Saito et al.

6.1. The novelties of graphene

As already pointed out in this article, graphene appears as the simplest prototype material for solid
state physics, let alone the simplest in the family of sp2 carbons. We still expect a lot more to come
with the advance of all kinds of technologies for processing multi-layer graphene samples with
different stacking orders, and for generating new kinds of ribbons, edges and defects in general.
Here, the technology for producing new kinds of samples is the key issue.
When thinking about the future of graphene nanoribbons, we can now give attention to the
past developments of the nanotube field. The great rush into carbon nanotube photophysics arrived
when nanotubes with diameters close to 1 nm in diameter and smaller were first produced. Lots
of interesting many-body and quantum confinement effects could then be studied in depth, while
for larger diameter nanotubes such effects were not so evident. The new physics learned here was
then extended to larger diameter tubes. A 1 nm diameter nanotube corresponds to a nanoribbon of
about 3–4 nm in width. Since this size is still very small for present technology, we can expect, by
analogy, that the most interesting effects are yet to come from nanoribbons smaller than 10 nm in
width.
Of course this is not considering edges, which are very interesting by themselves. New and
important physics and chemistry related to edges may also come, and these are also dependent on
developing more control over the geometry and quality of the edges. For example, the scattering
at the edge becomes very important for optics and spin (or charge) transport. Progress has been
made, but not enough to lay a strong foundation for the full development of this new research area.
The study of defects in general is a very open field that might generate important new results,
both from the study of isolated defects using Raman spectroscopy, as well as through the use of
multiple characterization tools that complement Raman spectroscopy. No one has ever measured
the characteristic spectral response from a single 7-5-5-7 defect, or from a single vacancy, or even
a double vacancy, etc. It is only recently that Lφ , the spatial extent of a point-defect induced Raman
scattering process (Lφ ∼ 2 nm), was measured as is discussed in Section 4.3.3. Furthermore, Lφ
is expected to exhibit an important dependence on sample temperature. The study of Raman
spectroscopy in the transition from a low defect concentration to a high defect concentration may
provide important information on the transition from weak to strong localization. These topics are
mainly largely discussed for disordered materials, thus providing experimental information for
the development of theories linking these limits in 2D materials to studies in few-layer graphene.
Defect physics is certainly one of the most open solid state physics sub-fields and this topic can
profit greatly from studies made on nearly perfect and highly controlled 2D graphene systems.
However, the rush into studying graphene is not only because it is a prototype material, but
also because of the many fundamental connections that physicists see between graphene and
fundamental physics. Regarding graphene transport, the Raman effect probes electrons too far
from the Fermi level, thus losing out on the most interesting special properties of the massless
electrons very close to the Dirac point [61]. However, recent Raman measurements have probed
the low-energy excitation states near the Dirac point by studying phonon softening phenomena
(Section 5.4) or electronic Raman spectra [206]. As we develop probes that come closer to the
Dirac point, we expect that new physics will emerge [62,72,76–82]. For example, the Klein
paradox generates a 100% probability for electron tunneling when there is an electron wavefunction
matching that for the hole inside an energy barrier. Due to this property, it was shown that one
can modulate the electron velocity by generating a periodic potential in graphene [390]. Such an
anisotropy in the electron speed could generate measurable shifts in the G -band frequency, and
then Raman spectroscopy could be used to easily quantify aspects of the Klein paradox [76,77,81].
Finally, while many new interesting applications for Raman spectroscopy on graphene will
certainly come from the preparation of new related samples, there is still room for the development
 Advances in Physics 535

of our basic knowledge. For example, our understanding related to the intensity of the DR features
is marginal. Also, there is still controversy about the assignment of the 2450 cm−1 peak, basically
because different DR features fall at this frequency, and it is not clear which of them is responsible
for the peak. Of course, the relative intensity for the features related to defects depend on the
number of defects. However, the relative intensity of the DR features (even those unrelated to
defects) change from sample to sample. Graphene, as the prototype material, may shed light into
this and other basic issues related to inelastic light scattering.

6.2. The novelties of carbon nanotubes

The Raman spectroscopy study of carbon nanotubes and their photophysics, in general, is much
more mature than for graphene, and the well-established physics and experimental techniques for
nanotubes have been directly applied to study graphene. In a general sense, carbon nanotubes are
much more complex than graphene and thus more rich information about solid-state properties
can be obtained as a function of the nanotube structure. Though carbon nanotubes have been
studied intensively in the past decade, many open issues still remain. For example, very few works
have focused on the spectroscopy of carbon nanotube ends, junctions or defect sites even though
such spectra can easily be obtained. Double-wall carbon nanotubes are another class of carbon
nanotubes that are extremely rich and have until now only been briefly studied, and triple-wall
nanotubes have hardly been studied at all. The interlayer interactions in nanotubes are not so well
established as in graphene. One advantage of studying nanotubes is that they have a well-defined
structure, specified by two integers (n, m), and that nanotubes have no edge structure except for
the ends of a SWNT. Using this advantage, the Raman characterization of SWNTs as a function
of (n, m) will provide a family of standards for SWNTs. To establish a worldwide standard should
be an important issue for the future large scale use of SWNTs in practical applications.
As a more focused example, despite decades of studies, the many-peak structure of the Raman
G-band is still full of effects that are not yet well understood and effects related to doping and
strain are only now being studied intensively. Furthermore, the behavior of the G-band spectra
upon variation of the excitation light resonance condition is still an open issue.
Carbon nanotubes are also very interesting objects for nano electro-mechanical systems
(NEMS) because of their strength and stability. Using their strong optical response at the iso-
lated nanotube level and the very high frequency of the LO and iTO modes, SWNTs can be used
as very sensitive probes for electronic and structural changes in NEMS. In this sense, carbon
nanotubes have already become a prototype material for the development of nanometrology. The
importance of carbon nanotubes for applications is already a well-established goal [20], and here
Raman spectroscopy could play a very important role as a characterization tool of the changes
to the SWNT caused by external perturbations, due to the high degree of accuracy of Raman
spectroscopy as a characterization tool along with the relative simplicity of its use for the charac-
terization of carbon nanotube-based materials as a non-contact, non-destructive, room temperature
measurement.
The development of standard materials based on carbon nanotubes would be of extreme impor-
tance, not only for applications, but also to establish a sensitive probe of various types of SWNT
environments. The National Institute of Standards of Technology (NIST, USA) has already devel-
oped a round robin program for the measurements of the (n, m) content in a SWNT sample based on
the CoMoCAT material [391]. The water-assisted CVD-grown carbon nanotubes (called “super-
growth” tubes in the literature [316]) exhibit some results based on their Raman spectra which
indicate that this is a special material that is closest to what is expected from a pristine SWNT
(isolated from any environmental effect). These special properties are: (1) the special relation
between its ωRBM and tube diameter, (2) the highest measured energy for its optical transitions,
536 R. Saito et al.

and (3) the observation of previously elusive transitions from metallic nanotubes [230]. However,
it is not yet clear why this sample is special. Clarifying such a result could help in developing an
effective standard reference material in the graphene area.

6.3. Near-field Raman spectroscopy and microscopy

The fiber-probe-based near-field systems have the potential for high spatial resolution (50–100 nm)
and for the observation of photophysical phenomena in nanostructures. These systems thus far
have been largely utilized for luminescence studies, but have hardly given any results on Raman
spectroscopy. The near-field Raman spectroscopy and microscopy based on the tip-enhanced
Raman scattering (TERS) effect first appeared in 2003 and this work had a strong impact on
the photophysics field with the earliest TERS measurements on carbon nanotubes [381,382], and
finally achieved a record spatial resolution of 12 nm [45]. However, the instrumentation behind
this technique is still very complicated and the developments that have been achieved thus far are
still carried out by very few groups in the world. Some companies are starting to produce and
sell TERS systems, but the reliability of these systems is not yet in place. Once this technique
becomes routine, there is no doubt about its importance to the future development of the Raman
spectroscopy field as applied to nano-science. The study of subjects such as isolated defects and
edges using TERS is expected to achieve a new paradigm in the near future, which will advance
both science and technology.

6.4. Time-dependent Raman and coherent phonon spectroscopy

As pointed out in Section 2.3.6, the Fourier transformation of CP vibrations gives a spectrum
that is similar to ordinary Raman spectra. It is important to note that CP spectroscopy is not just
an alternative tool to Raman spectroscopy, but it is a special tool in its own right that allows
observation of the phase of the vibration by which the information about the adiabatic potential
for the excited states can be obtained. For example, the RBM vibration starts by opening or closing
the nanotube diameter depending of the type (Type I or Type II) of semiconducting SWNTs under
investigation [165]. Further, by using pulse-shaping techniques in which the frequency of the
repeated laser pulse can be matched to the RBM frequency of a particular SWNT, the CP spectra
only for the SWNT can be obtained among the several resonant nanotubes matched to Elaser .
The fast response of the transport properties in graphene is now operating in the 100 GHz range
[392]. Soon this frequency will become similar to the sp2 carbon phonon vibration frequency
(47 THz for the G-band). In the near future, we may be able to directly observe each physical
process contributing to the el–ph interaction and this will help us identify the energy-transfer
mechanism in these optical processes. The development of attosecond lasers might bring time-
dependent Raman spectroscopy to the level of this newly opened field in which specific electron–
electron interactions can also be observed as a function of time.

6.5. Conclusion and messages for the future

The developments of Raman spectroscopy as applied to carbon nanotubes and graphene in the
past decade has been truly impressive. To demonstrate this, one could develop an interesting set
of lectures for a course in solid-state physics by just addressing the Raman spectroscopy on sp2
carbon systems [1]. The level of information and details that have become available has allowed
the development of a very accurate theory to describe the behavior of electrons and phonons
in these systems. In this process, Raman spectroscopy has become a widely used tool for the
characterization of nanomaterials. This may be the beginning of a new era where information about
many-body effects and complex systems will start to be obtained with much better accuracies for
 Advances in Physics 537

the development of new theories to explain the newly found effects. Raman spectroscopy thus
helps the development of nanotechnology in its most basic sense, and allows study of physical
processes that are expected to occur at the nanometer level. In following the words of Professor
Richard Feynman in his lecture on 29 December 1959, that there is plenty of room for discovery
at the bottom, below 10 nm spatial resolution, below 1 meV energy resolution and below 10 fs
time resolution, and for the operation of materials at more than 1 THz frequency, 1000 T magnetic
field, and 1 TPa pressure.

Acknowledgements
The MIT authors acknowledge the support under NSF Grant DMR 10-04147. A.J. acknowledges
the financial support from the Brazilian agencies CNPq, CAPES and FAPEMIG. R.S. acknowl-
edges a Grant-in-Aid (No. 20241023) from the MEXT, Japan. We would like to thank the referee
for his/her suggestions which improved the quality of this publication.
For all figures reprinted from American Physical Society material readers may view, browse,
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ther reproduced, distributed, transmitted, modified, adapted, performed, displayed, published, or
sold in whole or part, without prior written permission from the American Physical Society.

Notes
1. The B atom in Figure 1 gives a brighter STM images than the A atom, since there are electronic energy
bands for the B atom near the Fermi energy.
2. The G peak in the Raman spectra of sp2 carbons is often called the 2D peak. It should be noted that
the mechanisms involved in the 2D double resonance (DR) processes are different from those for the G
peak which involves only two phonons. The G peak involves only two K point phonons, whereas the 2D
feature arises from the second-order D that includes a DR D-band process involving a K point phonon
and an elastic scattering process. In this review, we distinguish between the G and the 2D scattering
process and the actual values of their frequencies.
3. A Stokes process is a terminology used to denote the loss of photon energy in a scattering process. Here
the Stokes photoluminescence process is independent of the Stokes Raman process.
4. Time-dependent perturbation theory tells us that the amplitude of the wavefunctions for excited states
oscillates as a function of time if the photon energy does not match the excited-state energy, which is
the physical meaning of a virtual transition.
5. A notch filter is an optical filter that suppresses a specified range of energies of the incident light.
6. By connecting monochromators in a serial way, the resolution of a monochromator is significantly
improved although the actual signal becomes increasingly weak. Furthermore, extra monochromators
can be used as an energy-tunable filter to reject the elastically scattered light in contrast to the notch
filters, which are wavelength-specific.
7. The solution of a forced damped harmonic oscillator is not generally a Lorentzian lineshape but
approaches the Lorentzian function for ωq q . However, if ωq approaches q , the lineshape departs
from a Lorentzian function.
8. The designation π of the π band comes from its value of angular momentum which is 1.
9. Here mod denotes an integer function for evaluating the modulus for an (n, m) SWNT where we use the
notation mod (6, 3) = 0, mod (7, 3) = 1 and mod (8, 3) = 2 as an illustrative example. Some authors
use Mod 1 and Mod 2 to denote a semiconducting nanotube depending on mod(n − m, 3) = 1 or 2.
There is thus a one-to-one correspondence between type I (II) and Mod 2 (1) semiconducting nanotubes
appearing in the literature, and this is clarified in Figure 11.
10. It should be noted that there is a logarithmic 2D van Hove singularity in the density of states of graphene
at the saddle point of the energy band near the M points (center of the hexagonal edges) of the BZ.
538 R. Saito et al.

11. In time-dependent perturbation theory, the mixing (or transition) of the excited states occurs as a function
of time with some finite and often measurable probability. The virtual states are defined by such a linear
combination of excited states with some probability. The probability for the occupation of an excited
state can be large when the energy difference between the excited states and the energy of the external
field is relatively small. In such a case, we can say that the transition is resonant with the excited state.
12. Here “real absorption” means that the photo-excited electron can be in the excited states for a sufficient
time, for example, 1 ns, so that the electron can be probed in the excited state. A material can scatter
photons in a virtual process.
13. It is noted that not all even (odd) vibration modes under inversion symmetry are Raman (IR)-active
modes.
14. The two-phonon process involving one-phonon emission and one-phonon absorption does not contribute
to the Raman spectra but rather gives a correction to the effective Rayleigh spectral process.
15. Here q is the real phonon wavevector, measured from the  point, while in defining qDR , the k and k 
vectors are measured from the K point or alternatively, with respect to the K  point.
16. It is only when crystalline disorder is present that the first-order q = 0 phonons can be observed, as
discussed in Section 4.3.
17. From the matrix element α p , we can deduce another matrix element,

βp (τ ) = φμ (r)∇v(r − τ)φν (r − τ) dr
 (45a)
= φν (r)∇v(r)φμ (r + τ) dr = βp (τ )Î(βp ).

However, the integral in Equation (45a) can also √ be expressed by α terms [268].
18. The phonon amplitude is proportional to Aν (q) n̄ν (q) in which the temperature dependence of the
amplitude is expressed by n̄ν (q) given by Equation (47).
19. It is noted that the minus sign corresponds to a symmetric wavefunction and that the plus sign corresponds
to an anti-symmetric wavefunction.
20. A virtual state is a linear combination of real states. When a virtual state is close to an exciton state, the
virtual state contains a large component of the exciton states. This is the reason for the approximation
used in obtaining a representation for the virtual state.
21. A cutting line is defined by the 1D BZ of an SWNT in the 2D BZ of graphene [32,135,136].
22. Other formula for ωRBM can be used here, too. The difference is within 1–3 cm−1 .
23. The subscript htt in Thtt denotes heat treatment temperature.
24. Here core electrons refer to 1s and σ electrons. The screening by π electrons is independently considered
by the polarization function within the RPA (random phase approximation) [120,148]. See Section 5.2.1
for further details.
25. The Bethe–Salpeter equation is independently solved for each value of κ. When we obtain Eii values as
a function of κ by solving the Bethe–Salpeter equation many times, then Eii with different i values are
adapted from the different κ value. Since the Eii eigenvalues come from different cutting lines, there is
no problem with the orthogonality of the wavefunctions.

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