carbon nanotubes. basic concepts and physical properties, 2004, p.220

S. Reich, C. Thornsen, J. Maultzsch

Carbon Nanotubes
Basic Concepts and Physical Properties


S. Reich, C. Thornsen,J. Maultzsch

Carbon Nanotubes
Basic Concepts and Physical Properties

W I LEYVCH

WILEY-VCH Verlag GmbH & CO.KGaA


Authors
Prof Christian Thornsen
Technische Universillt Berlin, Germany

Dr. Stephanie Xeich
[Jniversrty of Cambridge, UK
Dipl. Phys. J a n k i Muulczsch
Technische Univcrsitat Bcrlin, Germany

Cover Picture
The cover shows an electronic wave function of a
(19,O) nanotube; white and blue are for different
sign.The background is a contour plot ot the
conduction band in the graphenc Brillouin zone.

First Reprint 2004

Th~r
hook was carelully produced. Nevertheless,
authors and publisher do not warrant the information containcd therein to be tree of errors.
Readers are adviscd to kccp in mind that qtatemenls, dala, illustrations, procedural details or
other items may inadvertenlly be inaccurate.

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Preface

This book has evolved from a number of years of intensive research on carbon nanotubes. We
feel that the knowledge in the literature in the last five years has made a significant leap forward to warrant a comprehensive presentation, Large parts of the book are based on the Ph.D.
thesis work of Stephanie Reich and Janina Maultzsch. All of us benefited much from a close
scientific collaboration with the group of Pablo Ordej6n at the Institut de Cibncia de Materials
dc Barcelona, Spain, on density-functional theory. Many of the results presented in this book
would not have been obtained without him; the band-structure calculations we show were performed with the program Sicsta, which he co-authored. We learnt much of the theory of line
groups from an intense scientific exchange with the group of Milan Damnjanovik, Faculty of
Physic3, Belgrade, Serbia and Monte Negro. Christian Thomsen thanks Manuel Cardona from
the Max-Planck Institut fiir Festkorperforschung in Stuttgart, Germany, for introducing him to
the fascinating topic of Raman scattering in solids and for teaching him how solid-state physics
concepts can be derived from this technique. We acknowlegde the open and intense discussions with many colleagues at physics meetings and workshops, in particular the Krchberg
meetings organized by Hans Kuzmany, Wicn, Austria, for many years and thc Nanotcch series
of confcrcnces. Stephanie Reich thanks the following bodies for their financial support while
working on this book, the Berlin-Brandenburgische Akademie der Wissenschaften, Berlin,
Germany, the Oppenheimer Fund, Cambridge, UK, and Newnham College, Cambridge, UK.
Janina Maultzsch acknowledges funding from the Deutsche Forschungsgemeinschaft.
Marla Machhn, Peter Rafailov, Sabine Bahrs, Ute Habocck, Harald Schecl, Michacl Stoll,
Matthias Dworzak, Riidcgcr Kiihler (Pisa, Italy) gave us serious input by critically reading various chapters of the book. Their suggestions have made the hook clearer and better. We thank
them and all other members of the research group at the Technische Universitat Berlin, who
gave support to the research on graphite and carbon nanotubes over the years, in particular,
Heiner Perls, Bernd Scholer, Sabine Morgner, and Marianne Heinold. Michael Stoll compiled

the index. We thank Vera Palmer and Ron Schulz from Wiley-VCH for their support.

Stephanie Reich
Berlin, October 2003

Christian Thomsen

Janina Maultzsch


Contents

Preface

v

1 Introduction

1

2 Structure and Symmetry
2.1 Structure of Carbon Nanotubes . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Symmetry of Single-walled Carbon Nanotubes . . . . . . . . . . . . . . . .
2.3.1 Symmetry Operations . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Symmetry-based Quantum Numbers . . . . . . . . . . . . . . . . . .
2.3.3 Irreducible representations . . . . . . . . . . . . . . . . . . . . . . .
2.3.4 Projection Operators . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.5 Phonon Symmetries in Carbon Nanotubes . . . . . . . . . . . . . . .
2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Electronic Properties of Carbon Nanotubes

3
3
9
12
12
15
I8
21
27
30

Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3I . I Tight-binding Description of Graphcne . . . . . . . . . . . . . . . .
Zone-folding Approximation . . . . . . . . . . . . . . . . . . . . . . . . . .
Electronic Density of States . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Experimental Verifications of the DOS . . . . . . . . . . . . . . . . .
Beyond Zone Folding - Curvature Effects . . . . . . . . . . . . . . . . . . .
3.4.1 Secondary Gaps in Metallic Nitnotubes . . . . . . . . . . . . . . . .
3.4.2 Rehybridization of the cr and 7c States . . . . . . . . . . . . . . . . .
Nanotube Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.1 Low-energy Properties . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.2 Visible Energy Range . . . . . . . . . . . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31
31
33
41

44
47
50
50
53
60
60
62
64

4 Optical Properties
4.1 Absorption and Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Selection Rules and Depolarization . . . . . . . . . . . . . . . . . .
4.2 Spectra of Isolated Tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Photoluminescence Excitation - (nl,n z ) Assignment . . . . . . . . . . . . .
4.4 4-A-diameter Nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67
67
68
72
73
77

3.1
3.2
3.3

3.4

3.5

3.6


...

v~ll

Contents

4.5 Bundles of Nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Excited-state Carrier Dynamics . . . . . . . . . . . . . . . . . . . . . . . . .
4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 Electronic Transport

79
80
83

5.1 Room-temperature Conductance of Nanotubes . . . . . . . . . . . . . . . . .
5.2 Electron Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 CoulombBlockade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 LuttingerLiquid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85
85
88

93
96
99

6 Elastic Properties
6.1 Continuum Model of Isolated Nanotubes . . . . . . . . . . . . . . . . . . . .
6.I .1 Abinitio. Tight.binding. and Force-constants Calculations . . . . . .
6.2 Pressure Dependence of the Phonon Frequencies . . . . . . . . . . . . . . .
6.3 Micro-mechanical Manipulations . . . . . . . . . . . . . . . . . . . . . . . .
6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

101
101
105
107
111
114

7 Rarnan Scattering
7.1 Raman Basics and Selection Rules . . . . . . . . . . . . . . . . . . . . . . .
7.2 Tensor Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.1 Polarized Measurements . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Raman Measurements at Large Phonon q . . . . . . . . . . . . . . . . . . .
7.4 Double Resonant Raman Scattering . . . . . . . . . . . . . . . . . . . . . .
7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - .

115
115
119
121

123
126
133

8 Vibrational Properties
135
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
8.2 Radial Breathing Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
8.2.1 The REM in Isolated and Bundled Nanotubes . . . . . . . . . . . . . 142
8.2.2 Double-walled Nanotubes . . . . . . . . . . . . . . . . . . . . . . . 149
8.3 The Defect-induced D Mode . . . . . . . . . . . . . . . . . . . . . . . . . . 152
8.3.1 The D Mode in Graphite . . . . . . . . . . . . . . . . . . . . . . . . 153
8.3.2 The D Mode in Carbon Nanotubes . . . . . . . . . . . . . . . . . . . 154
8.4 Symmetry of the Raman Modes . . . . . . . . . . . . . . . . . . . . . . . . 158
8.5 High-energy Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
8.5.1 Raman and Infrared Spectroscopy . . . . . . . . . . . . . . . . . . . 162
8.5.2 Metallic Nanotuhes . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
8.5.3 Single- and Double-resonanceInterpretation . . . . . . . . . . . . . 172
8.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
8.7 What we Can Learn from the Raman Spectra of Singlc-walled Carbon Nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
Appendix A Character and Correlation Tables of Graphene
Appendix B Raman Intensities in Unoriented Systems

181


Contents

Appendix C Fundamental Constants

Bibliography
Index


1 Introduction

The physics of carbon nanotubes has rapidly evolved into a research field since their discovery
by lijima in multiwall for111in 1991 and as single-walled tubes two years later. Since then, theoretical w d experimental studies in different fields, such as mechanics, optics, and electronics
have focused on both the fundamental physical properties and on the potential applications of
nanotubes. In all fields there has been substantial progrcss over the last decade, the first actual
applications appearing on the market now.
We prescnt a consistent picture of experimental and thcoreiical studies of carbon nanotubes
and offer the reader insight into aspects that are not only applicable to carbon nanotubes but
are uscrul physical concepts, in particular, in one-dimensional systems. The book is intended
for graduate students and researchers interested in a comprehensive introduction and review
of theoretical and experimental concepts in carbon-nanotube research. Emphasis is put on
introducing the physical conccpts that frequently differ from common understanding in solidstate physics because of the one-dimensional nature of carbon nanotubes. The two focii of the
book, electronic and vihrational properties of carbon nanotubes, rely on a basic understanding
of the symmetry of nanotubcs, and we show how symmetry-related techniques can be applied
to one-dimensional systems in general.
Preparation of nanotubes is not treated in this book, Tor an overview we refer the reader to
excellent articles, e.g., Seo rt nl.ll.'l on CVD-related processes. Wc also do not trcal multiwall
carbon nanotubes, because dimensionality affects their physical properties to be much closer
to those of graphite. Nevcttheless, for applications of carbon nanotuhes they are extrcmely
~]
valuable, and we refer to the literature for reviews on this topic, e.g., Ajayan and ~ h o u [ ' . for
more information on the topic.
The textbook Fundamentals of Semiconductors by Y and Cardona['." and the series of
u
volumes on Light-Scattcring in ~ o l i d d41 was most helpful in developing several chapters

'
in this book. We highly recommend thcse books for rurther reading and for gaining a more
basic understanding or some of the advanced conccpts presented hcrc when needcd. There are
also a number of excellent books on various topics related to carbon-nanotube research and
applications that have appeared bcfore. We mention the volume by Dresselhaus et d.,['
the
'
book by Saito eta(.,' 1.61 thc book by ~ a r r i s [71 and thc collection of articles that was edited by
Drcsselhaus ~t
They offer valuable introductions and overvicws to a number of carbon
nanotube topics not treated hcre.
Beginning with the structure and symmetry properties of carbon nanotubcs (Chap. 2),
to which many results are intimately connectcd, we present thc electronic band structure of
single isolated tubes and of nanotube bundles as one of the two focii of this book in Chap. 3.
The optical and transport properties of carbon nanotuhes arc then treated on the basis oS the


2

1 Introduction

electronic hand structure in the optical range and near the Fermi level (Chaps. 4 and 5). We
introduce the reader to thc elastic properties of nanotubes in Chap. 6 and lo basic concepts in
Raman scattering, as needed in the book, in Chap. 7. The carbon-atom vibrittions are related to
the electronic band structure through single and double resonances and constitute the second
main focus. We treat the dynamical properties of carbon nanotuhes in Chap. 8, summarizing
what we rccl can be learnt from Raman spectroscopy on nanotubes.


2 Structure and Symmetry

Carbon nanotubes are hollow cylinders of graphite sheets. Thcy can be looked at as single
molecules, regarding their small size ( w nrn in diameter and -- prn length), or as quasi-one
dimensional crystals with translational periodicity along the tube axis. There are infinitely
many ways to roll a sheet into a cylinder, resulting in different diameters and microscopic
structures of the tubes. These are defined by thc chiral angle, the angle of the hexagon helix
around the tube axis. Some properties of carbon nanotubes can be explained within a rnacroscopic model of an homogeneous cylinder (see Chap. 6); whereas others depend crucially on
the microscopic structure of thc tubes. The latter include, for instance, thc electronic band
structure, in particular, their metallic or semiconducting nature (see Chap. 3). The fairly complex microscopic structure with tens to hundreds o C atoms in the unit cell can be described in
a vcry general way with the hclp of the nanotube symmetry. This greatly simplifies calculating and understanding physical properties like oplical absorption, phonon eigenvectors, and
electron-phonon coupling.
In this chapter we first dcscribe the geometric structure of carbon nanotubes and the construction of their Brillouin zone in relation to lhal of graphite (Sect. 2.1). In Sect. 2.2 we give
an overview of experimental methods to determine the atomic structure of carbon nanotubes.
The symmetry propertieq of singlc-walled tubes are presented in Sect. 2.3. We cxplain how to
obtain the entire tube of a given chirality from one single carbon atom by applying the symmetry operations. Furthermore, we givc an introduction to the theory of line groups or carbon
nanotubc~,[~.'I explain the quantum numbers, irreducible representations, and their notaand
tion. Finally, we show how to use a graphical method of group projectors to dcrive normal
modes from ryrnrnetry (Sect. 2.3.4), and present the phonon symmetries in nanotubes.

2.1 Structure of Carbon Nanotubes
A tube made of a single graphitc layer rolled up into a hollow cylinder is called a single-walled
nanotube (SWNT); a tube comprising scveral, concentrically arranged cylinders is rcfcrred to
as a multiwall tube (MWNT). Single-walled nanotubes, as typically investigated in the work
presented hcre, are produced by laser ablation, high-pressure CO conversion (HiPCO), or the
arc-discharge technique and have a Gaussian distribution of diameters d with mean diameters
do E 1.0 - 1.5nm.L2 21-12.51 The chiral angles [Eq, (2.I)] are evenly distributed.[2." Singlewalled tubes form hexagonal-packed bundles during the growth process. Figure 2.1 shows a
transmission electron microscopy image of such a bundle. The wall-to-wall distance hetwccn
two tubes is in the same range as the interlaycr distance in graphite (3.41 A). Multiwall nanotubes have similar lengths to single-walled tubes, but much larger diameters. Their inner m d

Figure 2.1: High-resolution transmission electron micrc)scopy
(TEM) picture of a bundle of single-walled nanotubes. The

hexagonal packing is nicely seen in the edge-on picture. Taken
from Ref. 12.21.

outer diameters are around 5 and 100nm, respectively, corrcsponding to F= 30 coaxial tubes.
Confinement effccts are expected to be less dominant than in singlc-walled tubes, because of
the large circumference. Many of the properties of multiwall tubes are already quite close to
graphite. While the multiwall nanotubes have a wide range of application, they are less well
defined from their structural and hence electronic properties due to the many possible number
of layers.
Because the microscopic structure of carbon nanotubes is closely related to gaphene1,the
tubes are usually labeled in terms of the graphcne lattice vectors. In the following sections we
show that by this rcference to graphene many properties of carbon nanotubes can be derived.
Figure 2.2 shows the graphene honeycomb lattice. The unit cell is spanned by the two
1
an)
vectors a and a2 and contains two carbon atoms at the positions ( a l -t and $ ( a l a 2 ) ,
whcrc the basis vectors of length lal I = la21 = a0 = 2.461 form an angle of 60'. In carbon
nanotubcs, the graphene sheet is rolled up in such a way that a graphene lattice vector c =
nlal n2az becomes the circumfcrcnce of the tube. This circumferential vector c, which is
usually denoted by the pair of integers (n1 , n z ) , is called the chiral vector and uniquely defines
ii particular tube. We will scc below that many properties of nanotubes, like thcir electronic
band struclurc or the spatial symmetry group, vary dramatically with the chiral vector, cvcn
for tubes with similar diameter and direction of the chiral vector. For example, the (10,10)
tube contains 40 atoms in the unit cell and is metallic; the close-by (10,9) tube with 1084
atoms in the unit cell is a semiconducting tube.
In Fig. 2.2, the chiral vector c = 8al +4a2 of an (8,4) tube is shown. The circles indicate
the four points on the chiral vector that are lattice vectors of graphenc; the first and the last

circle coincide if the sheet is rolled up. The numbcr of lattice points on the chiral vector is
given by the greatest common divisor n of (nl,n 2 ) , sincc c = n ( n l /n . a1 n 2 / n . a2) = n . c'
is a multiple of another graphene latlicc vector c'.
The direction of the chiral vector is measured by the chiral angle 8, which is defined as
the angle between a1 and c. The chiral angle 8 can be calculated frotn

A

4

+

+

+

For each tube with 8 between 0" and 30" an equivalent tube with 8 between 30" and 60"
is found, but the helix of graphene lattice points around the tube changes from right-handed
lo Icft-handed. Because of the six-fold rotational symmetry of graphene, to any other chiral
vector an equivalent one exihth with 0 60". We will hence restrict ourselves to the case

<

' ~ r a ~ h c n c single, two-dimensional laycr of graphite
IS a


2.1

Structure of Carbon Nanotuhec

Figure 2.2: Graphene honeycomb
lattice with the lattice vcctors a1 and
a2. The chiral vcctor c = 8al +4a2
of the (8,4) tubc is shown with the 4
graphenc-lattice points indicated by
circles; the first and the last coincide
if the sheet is rolled up. Perpeadicular to c is the tube axis z, the minimum translational period i \ given by
the vector a = -4al +Sa2. The vectors c and a form a rectangle, which
is the unit cell of the tube, if it is
rolled along c into a cylinder. Thc
zig-zag and armchair pattcrns along
the chiral vector of 7ig-zag and armchair tubes, respectively, are highlighted.

<

>

nl 2 n2 0 (or 0" 5 0 30"). Tubes of the type (n,O) (0 = 0") arc called zig-zag tubes,
because they exhibit a zig-zag pattcrn along the circumference, see Fig. 2.2. ( n ,n ) tubcs are
called armchair tubes; their chiral angle is 0 = 30". Both, zig-zag and armchair tubes are
achiral tubes, in contrast to the general chiral tubes.
The geometry of the graphene lattice and the chiral vector of the tube determine its structural parameters like diameter, unit cell, and its number of carbon atoms, as well as the size
and the shape of lhe Brillouin zone. The diameter of the tube is given by the length of the
c h i d vector:

+

with N = n: -t nl nz n;. The smallest graphene lattice vector a perpendicular to c defines
the translational period a along thc tube axis. For example, for the (8,4) tube in Fig. 2.2 the

smallest lattice vector along the tubc axis is a = -4al $ 5a2. In general, the translational
period a is determined from the c h i d indices ( n l ,n 2 ) by

and

where X = 3 if (nl - n2)/3n is integer and R = 1 otherwise. Thus, the nanotube unit cell is
formed by a cylindrical surface with height a and diameter d. For achiral tubes, Eqs. (2.2) and
(2.4) can be simplified to


6

2

Structure and Symmetry

Figure 2.3: Stmcturc of
the (17,0), the (10,O) and
the (12,s) tube. The unit a4cells of thc tubes are high-

I~ghted;the translational
period u is mdicated.

-f

For chiral tubes, a and c have to be calculated from Eqs. (2.2) and (2.4). Tubes with the same
chiral angle 8 , i.e., with the same ratio nllnz, possess the same lattice vector a. T Fig. 2.3
n
the structures of (17,0), (10, lo), and ( 1 2,s)tubes are shown, where the unit cell is highlighted
and the translational period a is indicated. Note that u varies strongly with the chirality of the

tube; chiral tubes often have very long unit cells.
The number of carbon atoms in the unit cell, n,, can be calculated from the area St = a . c
of the cylinder surface and the area Sg of the hexagonal graphene unit cell. The ratio of these
two gives the number q of graphene hexagons in the nanotuhe unit cell

Since the graphenc unit cell contains two carbon atoms, there are

carbon atoms in the unit cell of the nanotube. In achiral tubes, q = 2n. The structural paramcters given above are summarized in Table 2.1.
Table 2.1: Structural parameters of armchair (A), zigzag (Z) and chid (i) nanotuhes. The symbols
t'
are explained in the text.

Tube

N

4 = nc/2

A

h n )

3n2

2n

e

( n ,n2)
~

n;+n~nz+nz

2Nl ( n 2 )

Diameter d

Translational period a

Chiral angle 0

finao/x

ao

30"

A


2.1

Structure o f Carbon Nanotuhps

Figure 2.4: Brillouin zone of graphene
with the high-symmetry points T , K,
and M. The reciprocal latlice vectors k r , k2 in Cartesian coordinates
are k l = (0,1)4n/fiun and k2 =
(0.5fi, -0.5) 4 ~ / d % o .

Aftcr having determined the unit cell of carbon nanotubes, we now construct their Brillouin zone. For comparison, we show in Fig. 2.4 the hexagonal Brillouin zone of graphenc
K,
with the high-symmetry points T, and M and the distances between these points.
In the direction of the tube axis, which we define as the z-axis, the reciprocal lattice vector
k, corresponds to the translational pcriod u ; its length is

As the tube is regarded as infinitely long, the wave vector k, is continuous; the first Brillouin
zone in the z-direction is the interval (-x/a, E/u]. The bracket types ( ) and [ ] indicate open
and closed intervals, respectively. Along the circumference c of the tube, any wave vector k1
is quantized according to the boundary condition

+

where m is an integer taking the values -9/2
1,. . .,0, I , . . . ,q/2. This boundary condition
is understood in the following way: The wave function of a quasi-particle of the nanotube,
e.g., an electron or a phonon, must have a phase shift of an integer multiplc of 2~ around
the circumfcrcnce. All other wavelengths will vanish by interference. A wave with wave
2i
vector k l , , = ; .m has 2m nodes around the circumference. The maximum Ikl,,l (minimum
wavelength) follows from thc number of atoms (29) in the unit cell: a projection of the carbon
atoms on the circumference of the tube leads to equidistant pairs of carbon atoms; then at least
4 atoms are neccessary for defining a wavelength, i . ~ . Iml 5 412. The first Brillouin Lone then
,
consists of q lines parallel to the z-axis separated by k1 = 2/d and k E (-r/n, ~ / a ] .
The quantized wave vector kl and thc reciprocal lattice vector k, are found from the
conditions


2 Structure and Syrnrnetv

k,A

Figure 2.5: Brillouin 70ne of a (7,7) armchair and a (13,O) zig-zag tube (thick lines). The background is
a contour plot of the electromc band structure of gaphene (white indicates the maximum energy). Note
of
that the graphene Brillouin rone in the right panel is rotatcd by 30". The Bnllouin zone cons~sts 2n
( I . P . , 14 and 26, respcctively) lines parallel to k,, where k, is the reciprocal lattice vector along the tube
axis. Each line is indexed by m E [-n, n], where m = 0 corresponds to thc line through the graphcne r
point (k = 0). Note that the Bnllouin-zone boundary n/a i3 given by n/ao for armchair and n/&a0 for
ng-7ag tubes. It can be seen from the symmetry of the graphte hexagonal Bnllouin zone that hnes with
index m and -m are the same, as well as k and -k for the samc index m.

This yields

In Fig. 2.5 the Brillouin zones of a (7,7) armchair and a (1 3,O) zig-zag tube are shown for
rn E [-n,n] in relation t o the graphene Brillouin zone. The line through the graphene point
has the index m = 0. The position of the lincs with m = 0 and rn = n i s the same for all zig-zag
and all armchair tubes, respectively, independent of their diameter. With increasing diameter
the number of lines increases while their distance decreases. For chiral tubes, see Fig. 2.1 1.
To a first approximation, the properties of carbon nanotubes are related to those of graphite
by taking from graphene the lines that correspond to the nanotube Brillouin zone, according to
Eqs. (2.1 1) and (2.12). For example, the electronic band structure of a particular nanotube is
found by cutting the two-dimensional band structure of graphene (see background of Fig. 2.5)
~
into q lines of length 2 x 1 and distance 2 / d parallel to the direction of the tube axis. This
approach is called zone folding and is commonly used in nanotube research. Since the zonefolding procedure, however, neglects any effects of the cylinder geometry and curvature of the
tube walls, results obtained by zone folding have to be used with great care. We discuss the

r

2.2 Experiments

9

zone-folding approximation for the electronic and vibrational properties in the corresponding
chapters in detail.
We now know the most basic properties of single-walled carbon nanotubes, their diametcr,
chirality, and the lattice vector along the tube axis. We also found expressions for the lattice
vectors in reciprocal space. To obtain geometrically the atomic positions of a given (n I ,n2)
nanotuhe, we could construct the corresponding graphene rectangle and roll it up into a cylinder. The atomic positions are, on the other hand, determined by the chiral indices (n ,,n z ) as
well. We explain in Sect. 2.3, how they can be calculatcd with the help of the high symmetry
of carbon nanotubes.

2.2 Experiments
Thc atomic structure of carbon nanotubes can be investigated either by direct imaging techniques, such as transmission electron microscopy and scanning probe microscopy, or by electron diffraction, i.e., imaging in reciprocal space. Frequently, these methods are combined,
or used togethcr with other experiments like tunneling spectroscopy for cross-checking. In
this section we illustrate some examplcs of this work on dctcrmining the structure of real
nanotubeq.
Multiwall carbon nanotubes were first discovered in high-resolution transmission elcclron
"
microscopy (HRTEM), by ~ i j i m a . [ ~In HRTEM pictures, multiwall tubes usually appear as
two sets of equally spaced dark lines at two sides of a transparent corc. The lines correspond to
the tube walls projected onto the plane perpendicular to the electron beam; from their distances
the tube diameters and the inter-wall distances can be measured. HRTEM is used to verify the
presence o f nanotubes in the sample, to perform statistics on tube diameters and diameter
Figure 2.6: High-resolution transmisqion
clcctron nlicroscopy image of an SWNT
bundle. The scalc bar is 4nm. In the

lower part of thc figure Lhe bundle is approxilnatcly parallel to the electron beam
and I S seen from the top. The bundle
consists of single-wallcd nanotubes with
z 1.4 nm diamctcr arranged in a triangular lattice. From Ref. [2.7].


10

2 Slrltcture und S y m m e i ~

Figure 2.7: Electron diffraction pnmn of a qinglcwalled carbon nanotube. (a) Experiment; the inset
show^ a TEM image of [he tuhe. Pcrpe~ldicular
to
the tubc axis the spots are elongated (horizontal di-

reclion); thc oscillation period of the intcnity along
E - E' determine\ thc inverse tube diameter. (b)
Simulation of the diffraction pattern for a ( 14,h) tubc
and atomic structure of the (14,6) tube (inset). From
Ret. r2.101.

distributions, and to identify structural defects. It complements the more surface-sensitive
canning probe techniques and is thus in general important for investigatingthree-dimensional
structures like multiwall nanotubes, singlc-walled nanotube bundles, or the structures of caps
at the end of thc tuhes. In Fig. 2.6 the HRTEM image of a SWNT bundle is shown. In the
upper part, the bundles are in the plane perpendicular to the electron beam. In the lower
part, the bundle is bcnt and a cross section of the bundle can be scen. The bundle cross
section exhibits a triangular lattice of single-walled tubes. The tube diameter determined from
HRTEM is often underestimated, depending on the orientation of the tube with respcct to the
electron heam.12,'1 The error increases with decreasing tuhe diameter. In order to obtain a

more reliable diameter, simulations of the TEM images are necessary.
Electron diffraction patterns of carbon nanotubes show the intcnnediate character of a nanotuhe between a molecule and a crystal. The hexagonal symmetry of the atomic lattice gives
rise to n hexagonal geometry in the diffraction pattern. On the other hand, perpendicular to
the tube axis the apparent (pro-jected) lattice constant decrcases towards the side of the tuhe
because of thc projection of the curved wall. The diffraction spots are therefore elongated
in this direction.r2."1-12 Figure 2.7 (a) shows the diffraction pattern of a single-walled nanotube, and (b) a simulation of the pattern for a (14,6) tube.[' "'1 The spots are sharp in the
direction of the tube axis (vertical); in the horizontal direction the spots are elongated. From
the equatorial oscillation of the intensity along E - E' the diameter of the tuhe can be determined. In a single-walled lube, the electron beam passes the tube wall twicc, the front
and back are projected onto each other. This gives rise to the hexagonal pattern of pairs of
diffraction spots. The chiral angle of the tube can be obtained from thc angle by which the
two hexagons are rotated against each 0ther.[~.'~1,~*.~~1
For small-diameter tubes the combination of diameter and chiral angle can lead to a uniquc assignment of the chiral vector ( n l ,nz).
Sincc the interpretation of diffraction experiments on nanotubes is rather complex, they are
often accompanied by simulations to confirm the assignment of the tube chiralities. A review
on electron transmission and electron diffraction of carbon nanotubes is found in Refs. 12.1S]
and [2.161.


Figure 2.8: Scanning tunneling microscopy imagec of an isolated semiconducting (a) and metallic (b)
qinglc-walled carhon nantitube on a gold substrate. Thc solid arrows arc in the direction of the tube axis;
the dmhed line indicates the zig-zag direction. Bascd on the diameter and chiral angle determined from
the STM image, the tube in (a) was assigned to a (14, - 3 ) tube and in (b) to a (12,3) tube. The semiconducting and mctallic bcbav~or the tubes, respectively, was confirmed by tunneling spectroscopy at
of
specific sites. From Ref. [2.171.

The surface structure or carbon nanotubes can be studied by scanning probe techniques
such as scanning tunneling (STM) and atomic force microscopy (AFM).[~
171-[2251 In STM
measurements, atomic resolution of the structure can be achieved. In Fig. 2.8 we show STM
The dark spots correspond to

images of single-walled carbon nanotubes by Odom et al.[2.17]
the centers of the graphene hexagons. The chiral angle is measured from thc &zag direction
with respect to the direction of thc tube axis as indicated by the dashed and solid lines, respectively. There are, howcvcr, some possible source3 of error in the intcrpretation of STM images.
The diameter determination is affected by the well-known problcm of imaging a convolution
of tip and sample; the tube appears flattened. Thc image distortion by non-vertical tunncling
lcads to an overestimation of the chiral angle by 15 - 70%.L2"1 This error can he corrected in a
carclul analysis. Furthermore, the tube also interacts with the substrate and the tunncling current might contain contributions from both, tunneling between tip and nanotube and between
nanotube and substrate, or tunneling through several layers in the casc of multiwall
Finally, the STM images of carbon nanotubes exhibit an asymmetry with respect to nega'('I
tive and positive bias, which in addition depcnds on whether (nl- n2)mod3 = f1.[2.281-[2
For example, the STM image of a (1 1,7) tubc [with (nl - nz) mod3 = + l ] at positivc bias is
very similar to the image of the close-by (12,7) tube [with (nl - nz)mod3 = -I] at negative
bias.L2 701 Besides computations of the STM images, the assignment of tubc chiralities can be
checked by scanning tunncling spetroscopy (STS), which probes thc clcctronic properties and
These
allow3 determination of the metallic or semiconducting character of the tube.12 171,12.221
measurements arc discussed in detail in Chap. 3.
Atomiclscanning force microscopy (AFMISFM) is another method by which the topography of a sample can be easily investigated, r.g., the distribution of isolatcd tubes on a sub-


12

2 Srructure and Symmetry

strate andlor on electrodes.[2.2",[2.311,r2.321 contrast to STM, a conducting substrate is not
In
rccpired. Usually, the tube diameter is determincd from height measurements; the width of
investigated the effect of
the tube itppears enlarged as for STM.[~~"-[~."] r t c pt a1.[2.341
~c

l
intcraction between tube and substrate on the gcomctrical structure by AFM and molecular
mechanics simulations. They found that, with increasing diarnetcr and decreasing number of
shells, the tubes arc flattened by adsorption to a substrate and can even collapse.r2351 Simultaneously with the topographic image, information on the mechanical properties of thc tubes can
R71 In addition, conductive
be obtained or the tubes can be mechanically manipulated.L2
probe AFM allows invcstigrttion of the conductance of nanotubes, r.g., a1 clcctrical contacts
or between the tubes in a
In summary, the structure of carbon nanotubes can be invcstigated by HRTEM, electron
diffraction, and scanning probe microscopy; STM offers measurements with even atomic resolution. From both, STM and electron diffraction, the chiral vector and tube diamctcr can be
determined, and hence the chiral vcctor ( n l, n Z ) ,in principle, can be found experimcnlally. Inlcrprctation of the images is, however, delicate and oftcn requires computations of the images
and cross-checking with other experimental results.
7 h 1 7 [ 2

2.3 Symmetry of Single-walled Carbon Nanotubes
The symmetry of carbon nanotubes is described by the so-called linc groups.[2.401-[2.441
Line
groups are the full space groups of one-dimensional systems including lranslations in addition
to the point-group symmctrics like rotations or reflections. Therefore, they provide a complete
~ ~ 1 , that ~ ~ l
set of quantum numbers. Damnjanovie et a ~ . [ ~ . ~ 1 , 1 ~showed1 ~ . every nanotube with a
~
particular chirality ( n ,n2) belongs to a different line group. Only armchair and zigzag tubes
with the same n bclong to the same symmetry group. Moreovcr, by starting with a single
carbon atom and successively applying all symmetry operations of the group, the wholc lube
is constructed. Because the relation between the carbon atoms and the symmetry operations
is one-to-one, the nanotubes in fact are thc line groups. Here we introduce the reader lo thc
basic concepts of line groups and their application in carbon nanotuhe physics.

2.3.1 Symmetry Operations

In order to find the symmetry groups of carbon nitnotubes we consider the symmetry operalions of graphene.12,'1,12 Those that are preserved whcn the graphene sheet is rolled into a
451
cylinder form thc nanotuhe symmetry group. Translations by multiples of a of the graphene
sheet parallel lo a rcmitin translations of the nanotube parallel to thc tube axis, see Fig. 2.3.
They form a subgroup T containing the pure translations of the tube. Translations parallel to
the circumferential vector c (perpendicular to a ) become pure rotations of the nanolube about
its axis. Given n graphene lattice points on the chiral vector c, the nanotube can be rotalcd
by multiples of 2 z l n . Single-walled nanotubes thus have n pure rotations in their symmetry
group, which are denoted by Ci (5 = 0,1,.. . ,n - 1). Thcsc again form a subgroup C,, the
of
full sylninctry group.
Translations of the graphene sheel along any other direction are combinations of translations in the a and the c direction. Therefore, whcn thc griiphcne sheet is rolled up, they result


2.3 Symmetly of Single-walled Carhnn Nanotubes

13

in translations combined with rotations about the nanotube axis. The order of these screw axis
operations is equal to the number q of graphene lattice points in the nanotube unit cell. They
~)
are denoted by ( C : l ~ n / with~the parameter

Fr[x] is the fractional part or the rational number x, and q ( n ) is the Euler f ~ n c t i o n . 1 ~ the ~
On ~ ~
unwrapped sheet the scrcw-axis operation corresponds to the primitive graphene translation
W
-c
:a. The nanotubc line group always contains a screw axis. This can be seen from
4

Eq. (2.6), which yields q / n 2 2. In achiral tubes, q/n = 2 and w = 1; thus the screw axis
operations in these tubes consist of a rotation by x / n followed by a translation by a / 2 , see
Fig. 2.3.
From the six-fold rotation of the hexagon about its midpoint only the two-fold rotation
remains a symmetry operation in carbon nanotubes. Rotations by any other angle will tilt the
tube axis and are therefore not symmetry operations of the nanotube. This rotational axis,
which is prescnt in both chiral and achiral tubes, is perpendicular to the tube axis and denoted
by U. In Fig. 2.9 the U-axis is shown in the (8,6), the (6,0), and the (6,6) nanotube. The U-axis
points through the midpoint of a hexagon perpendicular to the cylinder surface. Equivalent to
the U-axis is the two-fold axis U' through the midpoint between two carbon atoms.
Mirror planes perpendicular to the graphene sheet must either contain the tube axis (veror
tical mirror plane ox) must be perpendicular to it (horizontal mirror planc ah)in order to
transform the nanotube into itself. It can be seen in Fig. 2.9 that only in achiral tubes are
the vertical and horizontal mirror planes, oxand ah,present.12.'1They contain the midpoints
of the graphene hexagons. Additionally, in achiral tubcs the vertical and horizontal planes
through the midpoints between two carbon atom5 form vertical glide planes (02) and horizontal rotoreflection planes (oh,),see Fig. 2.9.
In summary, the general clement of any carbon-nanotube line group is denoted as

+

with

t

=

s=

0 , 4 1 , ... ;
O , l , . , . , n - I;

u = 0,l;

{

v=

0 , l achiral
chiral

O

and w, n, q as given above. Note that
= UD,.
These elements form the line groups L, which are given by the product of the point groups
D, and Dnhfor chiral and achiral tubes, rcspectively, and the axial group T,W:

LAz

1

= TZnDnh L2nn/mcm
=

(armchair and zig-zag)

(2.15)


14

2

Structure and Symmetry

Figure 2.9: Horizontal rotational axes and mirror and glide planes of chiral and achiral tubes. Left:
Chiral (X,6) nanotube with the line grwp T & D ~ ; of the U and U' axes are shown. Middle and
one
right: zigzag-(6,O) and armchair (6,6)nanotube belonging to the same T
~ linc~
group. Additionally
D
~
~
h
to the horizontal rotational axes, achiral tubes also have o and oxmirror planes (in the Figure as q),
the glide plane o (q), rotoreflcctionplane crht. Taken from Ref. [2.1].
y
and the

and

L ~ : T ~ D= ~ 9 ~ 2 2
=
,

(chiral tubes).

(2.16)

Here 2 x w l y determines the screw axis of the axial group. The international notation is included, although it will not be used herc, for a better reference to the Tables of Kronecker
Products in Refs. [2.42] and [2.441.~
For many applications of symmetry it is not necessary to work with the full line group.
Instead, the point group is sufficient. For example, electronic and vibrational eigenfunctions
at the r point always transform as irreducible representations of the isogonal point group.
For optical transitions or first-order Rarnan scattering the point group is sufficient as well,
because these processes do not change the wave vector k. The point groups isogonal to the
nanotube-line groups, i.e., with the same order of the principal rotational axis, where the
rotations include the screw-axis operations, arc

Dq forchiral
and DZnh for achiral tubes.
Since carbon-nanotube line groups always contain the screw axis, they are non-symmorphic
groups, and the isogonal point group is not a subgroup of the full symmetry group. We will
consider this below in more detail when we introduce the quantum numbers.
After having determined the symmetry operations that leave the whole tube invariant, we
now investigate whether they leave a single atom invariant or not. Those that do form the
site symmetry of the atom and are called stabilizers; the others form the transversal of the
group.[24h1 In principle, a calculation for all carbon atoms in the unit cell can be restricted to
those atoms that by application of the transversal form the whole system. As an example we
'care m s be taken when working with those tables, hecau~e meanings of the symhols n and q in thc referut
the
ences arc interchanged.


15

2.3 Syrnrnetv of Sin~le-walled
Carbon Nnnotubes

now show how the atomic positions of the tube can be obtained from a single carbon atom. We
start with an arbitrary carbon atom and apply the U-axis operation. The atom is mapped onto
the second atom of the graphene unit cell (hexagon). The n-fold rotation about the tube axis
then generates all other hexagons with the first atom on the circumference. The screw-axis
operations (without pure translations) map these atoms onto the remaining atoms of the unit
cell. Finally, translating all the atoms of the unit cell by the translational period a forms the
whole tube. T we know, c.g., the electronic wave function of the tube at the starting atom, we
f
know the wave function of the entire tube.
Thus, just a single atom is needed to construct the whole tube by application of the symmetry operations of the tube; such a system is called a single-orbit system. The line groups
of chiral tubes comprise, besides the identity element, no further symmetry operations but
those that have been used for the construction of the tube. Therefore, the stabilizer of a carbon
atom in chiral tubes is the identity element; its site symmetry is C1. In achiral tubes, on the
other hand, there are additional mirror planes cq, and ox.Reflections in the a plane in armh
chair tubes and in the oxplane in zig-zag tubes leave the carbon atom invariant. Thus the site
symmetry of the carbon atoms in achiral tubes is C l h . We will see later that the higher site
symmetry of achiral nanotubes imposes strict conditions on, e.g., their phonon eigenvectors
or electronic wave functions.
Using the symmetry operations of the tube, we now find the atomic positions r in the unit
cell. Let us define the position of the first carbon atom at (al a z )and choose the U-axis to
coincide with the x-axis. Then in cylindrical coordinates the position of the first carbon atom
in the nanotube is given by[2.']

4 +

where 2N = nq% = 2 (ny 4 nln2
onto the new position

rts,

=

+ n;). An element (TI
y)'C;Uu

acting on the atom maps it

nu

(C,w'C;Uu It -)roo0
4

whereu=O,I, s = 0 , 1 , ...,n-1, andt = 0 , * l , f 2 ,.... Withthe helpofEq. (2.19), thepositions of all carbon atoms can be constructed for any nanotube. We summarize the symmetry
properties in Table 2.2.

232
..

Symmetry-based Quantum Numbers

A given symmetry of a system always implies the conservation law for a related physical
quantity. Well-known examples in empty space are the conservation of linear momentum
caused by the translational invariance of space or the conservation of the angular mornentum (isotropy of space). The rotational symmetry of an atomic orbital reflects the conserved
angular-momentum quantum numbcr. The most famous example in solid-state physics is the
Bloch theorem, which states that in the periodic potential of the crystal lattice the wave functions are given by plane waves with an envelope function having the same periodicity as the
crystal lattice.


16

2 Structure und Symmetry

Table 2.2: Symmetry properties of armchair (A), and zig-zag (2)and chiral (e)nanotubes. The symbols arc explained in the text. From thc position rm of the first carhon atom the wholc tube can be
constructed by application of the line-group symmetry operations, see Eq. (2.19).

Tube

Line group

roo0

Isogonal
point group

A

( 11
n.

T~nDnh

D2nh

L2nn/mcm

(n70)

T~nDnh

D2nh

L2nn/mcm

(i-O?&o)

(r0,:

,a)

Likewise, any quasi-particle of the nanotube or a particle that interacts with it "feels" the
nanotube symmetry. The state of the (quasi-) particle then corresponds to a particular representation of the nanotube line group, i. e., its wave function is transformed in the same way
by the symmetry operations as the basis functions of the corresponding representation. Using this, we can calculate selection rules for matrix elements, according to which a particular
transition is allowed or not. The probability for a transition from state la)to state IP) via the
interaction X is only non-zero, if IX 1 a) and (PI have some components of their symmetry in
common. If they do not share any irreducible component, their wave functions are orthogonal
and the matrix element (p lX 1 a ) van is he^.[^.^^^-[^ '1 X can be, e.g., the dipole operator of an
optical transition.
In the next section we describe the irreducible representations of the carbon-nanotube line
groups in more detail. Here, we first introduce two types of quantum numbers to characterize
the quasi-particle stater; in carbon nanotubes and show their implications for conservation
laws.
We start by describing the general state inside the Brillouin zone by the quasi-linear momentum k along the tube axis and the quasi-angular momentum m. The first corresponds t~
the translational period; the latter to both the pure rotations and the screw-axis o p e r a t i o n ~ . l ~ . ~ ~ ]
These states Ikm) are shown by the lines forming the Brillouin zone, as depicted in Fig. 2.5
for achiral tubes. For k = 0 , 7 ~ / and m = 0 , n the state is additionally characterized by its
a
even or odd parity with respect to the U-axis. Achiral tubes have additionally the parity with
respect to the vertical and horizontal mirror planes. k is a fully conserved quantum number,
since it corrcsponds to the translations of the tube, which by themselves form a group T. In
contrast, rn arises from the isogonal point group Dq, which is not a subgroup of the nanotube

line group. Therefore, rn is not fully conserved and care has to be taken when calculating
selection rules. As long as the process remains within the first Brillouin zone ( - n / a , x / n ] for
achiral tubes and within the interval [0,x / u ] for c h i d tubes, rn can be regarded as a conserved
quantum number. But if the Brillouin zone boundary or the point is crossed, rn is no longer
conserved. Such a process is often called an Urnklapp process.
The change of m in Urnklapp processes is illustrated in Fig. 2.10. A graphene rectangle
is shown, which is an unrolled (3,3) tube. The arrows indicate the displacement vectors of
the atoms along the tube axis, In (a), the translation of the tube along the tube axis is shown.
This mode has infinite wavelength along the tube axis and along the circumference, i.e., k = 0


Figure 2.10: Phonon eigenvectors for an
~mwrapped(3,3) Lube. Thc mowc indicate thc atomic displaccments. In (a)
the translation of the tubc along the tube
axis 1s shown. This vibralion has infinite
wavelength (k = 0 ) and zero nodcs along
the circumferencc m = 0. The vibration in
(b) can be understood as a r-point mode
with 2n = 6 nodes along the circumference. Alternatively, we can view the displacement as an m = 0 vihration with A =
a (dashed line). Thus the same dihplacemen1 pattern can be understood as a mode
with k = 0 and rn = n or as a mode with
k 2 x 1 and rn = 0.
~

-

n
!
.X

m

-

a
,
3

I
]

k

(b)

= 2d a

m=O

and m = 0. If we want to relate the displacement in (b) to the z-translation, we find two
solutions. First, if we view the displacement as a P-point mode, we find 2n = 6 nodes along
the circumference. The vibrational state in (b) is then characterized by k = 0 and m = n.
Alternatively, we can regard the vibration in (h) as an m = 0 mode, as in (a). The wavelength
of the vibration is in this case A = a (dashed line). Thus the stalc with k = 2n/u and m = O is
equivalent to thc htate with k = O and m = n. In this example, m jumps by n when going into
the second Brillouin zone. This holds in general for achiral lubes. In an Urnklupp process m
changes into m' according to [he following rules[' "I
m' = ( m fn ) mod q

(achiral tubes).

(2.20)

In chiral tubes, the Umklapp rules

+

m'

= ( m p ) mod y

I,

= -m

when crossing the zone boundary at x/a
when crossing the r point,

(2.21)

where p is given by

Fr[x] is again the fractional part of thc rational number x and cp(n) the Euler unction.^^.^^]
Note that for achiral tubes p = n and Eq. (2.2 1) becomes Eq. (2.20).
Another illustration for these Urnklupp rules was shown in Fig. 2.5. Lcl us follow the
nanotube Brillouin Lone along the allowed wave vectors (white lines). We start with a line
given by k E [O, x/u] and index m . From the symmetry of the graphene hexagonal Brillouin
zone and the contour plot it is seen that following this line in the interval [ n l a ,2x/u] yields
the same parts of the Brillouin zone as following the line with rn' = n - m from k = -n/a to
k = 0. This corresponds to application of Eq. (2.20).

Instead of the conventional "linear" quantum numbers k and m, the so-called "helical"
541 The
quantum numbers, and %, can be used.12~46],r2.521-~2 helical quantum numbers are
fully conserved. In contrast to rn, the new quantum number f i corresponds to the pure rotations