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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
WILEYVCH 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~rhook 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 densityfunctional theory. Many of the results presented in this book would not have been obtained without him; the bandstructure calculations we show were performed with the program Sicsta, which he coauthored. 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 MaxPlanck Institut fiir Festkorperforschung in Stuttgart, Germany, for introducing him to the fascinating topic of Raman scattering in solids and for teaching him how solidstate 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 BerlinBrandenburgische 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 WileyVCH 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 Singlewalled Carbon Nanotubes . . . . . . . . . . . . . . . . 2.3.1 Symmetry Operations . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Symmetrybased 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 Tightbinding Description of Graphcne . . . . . . . . . . . . . . . . Zonefolding 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 Lowenergy 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 4Adiameter 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 Excitedstate Carrier Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 Electronic Transport
79 80 83
5.1 Roomtemperature 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 Forceconstants Calculations . . . . . . 6.2 Pressure Dependence of the Phonon Frequencies . . . . . . . . . . . . . . . 6.3 Micromechanical 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 Doublewalled Nanotubes . . . . . . . . . . . . . . . . . . . . . . . 149 8.3 The Defectinduced 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 Highenergy Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 8.5.1 Raman and Infrared Spectroscopy . . . . . . . . . . . . . . . . . . . 162 8.5.2 Metallic Nanotuhes . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 8.5.3 Single and DoubleresonanceInterpretation . . . . . . . . . . . . . 172 8.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 8.7 What we Can Learn from the Raman Spectra of Singlcwalled 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 singlewalled 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 onedimensional systems. The book is intended for graduate students and researchers interested in a comprehensive introduction and review of theoretical and experimental concepts in carbonnanotube research. Emphasis is put on introducing the physical conccpts that frequently differ from common understanding in solidstate physics because of the onedimensional 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 symmetryrelated techniques can be applied to onedimensional 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 CVDrelated 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 Yu and Cardona['." and the series of volumes on LightScattcring 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 carbonnanotube 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 carbonatom 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 quasione 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 electronphonon 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 singlcwalled 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 and explain the quantum numbers, irreducible representations, and their notation. 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 singlewalled nanotube (SWNT); a tube comprising scveral, concentrically arranged cylinders is rcfcrred to as a multiwall tube (MWNT). Singlewalled nanotubes, as typically investigated in the work presented hcre, are produced by laser ablation, highpressure CO conversion (HiPCO), or the arcdischarge technique and have a Gaussian distribution of diameters d with mean diameters do E 1.0  1.5nm.L2 2112.51 The chiral angles [Eq, (2.I)] are evenly distributed.[2." Singlewalled tubes form hexagonalpacked bundles during the growth process. Figure 2.1 shows a transmission electron microscopy image of such a bundle. The walltowall distance hetwccn two tubes is in the same range as the interlaycr distance in graphite (3.41 A). Multiwall nanotubes have similar lengths to singlewalled tubes, but much larger diameters. Their inner m d
Figure 2.1: Highresolution transmission electron micrc)scopy (TEM) picture of a bundle of singlewalled nanotubes. The
hexagonal packing is nicely seen in the edgeon 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 singlcwalled 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 vectors a1 and a2 and contains two carbon atoms at the positions ( a l tan)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 closeby (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 righthanded lo Icfthanded. Because of the sixfold rotational symmetry of graphene, to any other chiral vector an equivalent one exihth with 0 60". We will hence restrict ourselves to the case
K / U , the Umklapp rules must be additionally applied. Form = 0 and m = n phonons in achiral tubes, the allowcd modes are further restricted by parity quantum numbers. In general, for q # 0 the selection rules arc lcss restrictive than for rpoint phonons, since somc of the parity quantum numbers are detincd only at q = 0. If a phonon mode is forbidden at the r point but allowed for q > 0, however, continuity prevents the mode from having a largc Raman signal for small q. For an indepth treatment of thc optical transitions, see Chap. 4. Thc Ramanscattering intensity I is usually evaluated by contracting the Raman tensors '3
where ei and e, are the polarization vectors of incident and scattered light, respectively. The Raman tensors of the modes given in Table 7.1 can be found by the group projcctor technique introduced in Chap. 2. Ofien, an inspection of the table together with symmetry arguments is sufficient to find the general form of the Raman tcnsors. The A 1, representation appears only for parallel polarization?, hence all nondiagonal elemcnts of its Raman tensor must be zero. Various symmetry operations tramform the xx and the yycomponents into each other in nanotubes. To obtain an A 1, rcprc\entation these two entries of the tensor must be the same. The 77component, on the other hand, is linearly independent, because all line group symmetry operations preserve the zaxis. Wc thus obtain diag[a,a,b], i.e., a purely diagonal matrix with & = a,, = a and a, = b, see Table 7.3, as the general form of an A , , Raman tensor of singlewalled nanotubeq. E l g can only have xz, u,yz, zy non7ero elements. Again xz and yz transform into each other under the symmetry operations of the tubes and are not independent, whereas xz and zx arc dccoupled. Similarly, the other Raman tensors can be found; they are listed in Table 7.3. Thc experimentally obqerved selection rules arise from the 7eros in the Raman ten\or and, with single crystals of carbon nanotubes at hand, the symmetries of all Raman excitations could be determined. At present, however, such crystals are not available. Samples with a partial alignment along the zaxis or dilute nanotube samples on different substrates have been used for Raman e ~ ~ e r i m e n l s . [ ~ 181
7.2 Tensor Invariants When single crystals of a material are not available the symmetries of Raman excitations can he studied by measuring the polarization of thc scattered light in unoriented samples. One can thus determine the socalled Raman tensor invariant^,[^.'^]^[^.^^] which yield information about the symmetry, although the assignment is not always unique. Let us assume a phonon with a diagonal Rarnan tensor with three elements a , # a 2 # u 3 . Furthermore, the scattering configuration in the laboratory frame is (ZZ). To find the Raman intensity we integratc and average over all possible orientations of the crystal. Using Euler's angles17 2'19 [7 221
integrating and rearranging yields
The result in Eq. (7.8) holds for all parallel polarizations of the incoming and outgoing linearly then has a more general form (see polarized light and for every Raman tensor except that below). For perpendicular linear polarization, e.g., Ixz, the integration yields Ixz = 3 y,'2/45. Now it becomes obvious that the symmetry can also he partially deduced from experiments on unoriented materials. For example, if the three elements of the Raman tensor are the same (= a, as for cubic point groups), Raman scaltering is forbidden in crossed linear polarization, and in parallel linear polarization we find simply ILz = u2 and, thus IxZ/IZz = 0. On the other hand, a fully uniaxial Raman lensor (nI = a2 = 0 , a7 = h) results in I x z / l n = 113. In Appendix B we show how to generalize the results of the preceding paragraph for arbitrary Raman tensors. The intensity on unoriented substances follows directly from the
xt2
Table 7.3: Raman tensors of the phonons in carhon nanoluhes. They are valid for the D, and Dgh point groups with q > 3, which refers to all realistic tubes.
7 Raman Scattering
120
transformation of a tensor under rotation. A secondrank tensor can be decomposed with respect to the rotation group into a scalar (tensor of rank zero), an antisymmetric matrix (rank one), and a symmetric traceless matrix (rank two). These irreducible components have wclldefined quantum numbers and transformation properties under rotation. The matrix element for a fixed orientation is obtained from the WignerEckitrt theorem; the integration over all crystal orientations is determined by the tensor invariants. Different authors use slightly different invariants in Raman scattering. Following Ncslor and spiror72" we define the isotropic invariant
the antisymmetric anisotropy
and the symmetric anisotropy
where a,, (i, j = x , y , ~ are ) the elements of the Raman matrix as given in Table 7.3 for carbon nanotuhes. The scattering intcnsity on an unoriented sample in any scattering configuration can he expressed by a linear combination of the tensor invariants, see Appendix B. For lincar parallel (11) and perpendicular (I) polarization of the incoming and scattered light the intcnsitics are given by (apart from a constant factor; Table 8.2) Ill = 45a2 +47:
(7.14)
11 = 3x2+5x,2,, (7.1 5) which is the generalized result of Eq. (7.8). The quotient 1 1 / 1 is known as the dcpoliiri~ation ratio p. Under nonresonant conditions the antisymmetric invariant vanishes, y: = 0. Measuring the intensity under parallcl and crossed polarization is then sufficient to find the tensor invariants. In resonance, the antisymmetric scattering does not necessarily vimish, and we need at least one more independent measurement to find the tensor invarianls. Using circular instead of linear polarization the intcnsilies in backscattering configuration are 4
1
I,?c,
~=
= 45a2
+ y,2 + 5 &
(7.16) (7.17)
where IrS0 is the intensity for corotating and I , y ) for contrarotating incoming and outgoing light; I ~ J ~ 3 / l t 1is0 the reversal coefficient P. Solving the system of four equations Eqs. (7.14)(7.17) with respect to the tensor invariants we obtain as one possible solution[721J
45a2
57: 67;
=
2 11131~y1, I
(7.18)
=
IL$OO,
(7.19)
=
I(>(?.
(7.20)
With the la5t quantity I(.,(., wc can measure the experimental error. The experimentally obtained tensor invariants are only accepted as significantly different from zero if they are larger = +Id  (Lo,>+I,Y,)l. than In general, the conclusions that can be drawn from experiments on unoriented samples are as follows: The only symmetry that has a nonvanishing isotropic part a is the fully symmetric representation in any point or space group. Thus, phonons of A 1 symmetry are readily distinguished from the other species. Of particular interest is the observation of antisymmetric contributions to the Raman intensity, i.e., # 0. If only the antisymmetric component is present the Raman peak originates from a phonon transforming as the totally antisymmetric representation, in the point group of nanotubcs this is A2 (Table 7 . 3 ) . Totally antisymmetric scattering is scarccly observed experimentally; more li*cquentis an antisymmetric contribution to a degenerate mode.r7,23117.251 The only possiblc modes in nanotubes showing mixed symmetric and antisymmetric scattering are E l modes. For example, a strong incoming resonance with an optical transition that is allowed in ?polarization, but forbidden in x or ypolarization yields in the matrix representation of the El Raman tensor c # d (Table 7.3). Such anti2617[7.271 for multiwall nanotubes. symmetric contributions were reportcd by Rao p t al.17.'2]%L7 Degencrate modes have a symmetric anisotropy y,' different from zero, but a2 = 0. A large ratio $/a2can serve as an indicator for scattering by El and Ez symmetry modes in carbon nanotubes.
x;,
7.2.1 Polarized Measurements The tensor invariants of the Raman scattered light can be obtained from a linear combination of the intensities in linear and circular polarization as shown in Sect. 7.2. Such a basic setup  a combination of two linear polarizing elements (a Fresnel rhomb and a polarization filter) and a ;1/4 wave plate  is shown in Fig. 7 . 2 . It allows the Raman intensities to bc rcmrded in parallel, perpendicular, corotating, and contrarotating polarization without changing the illumination level or removing any polarizing elements in the light path between the measure
Fresnel rhomb (90")
NanotubesI
Spectrometer with CCD
A14 wave plate (45")
Analyser (vertical)
Figure 7.2: Raman sctup for the measurements of the tensor invariants. The polarization direction of the incoming light is choscn by the Fresnel rhomb. The laser then passes a ?~/4 zeroorder wave plate and i s focused onto the sample. The scattcrcd light comes back through the h / 4 plate, is analyzed with a polarization filter, and focused onto thc cntrance slits of the spectrometer. Different orientations of the Fresnel rhomb and the 114plate yield the relative polarizations without inserting additional elements.
7 Raman Scattering
122
Table 7.4: For a fixed spetrometer polarization the settings or Fresnel rhomb and l.14 wave plate set up as in Pig. 7.2 allow a recording of the four different relative polarizations of incident and scattered light without inserting or removing an optical element.
~ Polarization Parallel Perpendicular Corolating Contrarotating

Ill I1
bo Ic3tj
Fresnel rhomb 0" 90" 0" 90"
0" 0"
45" 45"
vertical vertical vertical vertical
271 We illustrate the arrangement in some more detail since it is relatively ments.[7.19],[7.201,[7 unknown; yet it yields the tensor invariants with high accuracy. For the polarizing elements in positions as indicatcd in Fig. 7.2 a backscattering Raman spectrum under corotating polarization is recorded: The laser light is vertically polarized and, after passing the Fresnel rhomb (90" poqition), light is horizontally linearly polarized. After passing the A/4 wave plate (its principle axis is at 45" to the horizontal) the incoming light is circularly polarized. The backscattered beam, in general, consists of a Icft and a righthand circularly polarized part, which on passing back through the h / 4 wave plate is converted into linear polarization. The corotating part is now vertically polarized, the contraroting part horizontally (the circular polarizations are specified in the lab frame). Only vertically polarized light can pass the analyzer and is recorded. The intensities in the four different polarizations are then obtained by rotating the Fresnel rhomb and the A/4 wave plate. Let us assume that the analyzer is vertical (e.g.,because of a higher sensitivity of the spectrometer for vertically polarized light), then the Raman polarizations are given by the settings in Table 7.4. Figure 7.3 shows the Raman spectra of CC14 in (a) the two linear and (b) the circular polarizations. In this molecule the fully symmetric mode is characterized by only the isotropic invariant a2being different from zero. For the A1, peak at 460cm' a depolarization ratio p = 0 and a reversal coefficient P = 0 are expected; the other modes should show p = 0.75 and P = 6 (a2= 0, & = 0). All quantities are in excellent agreement with the measured values given in Fig. 7.3. Deviations from the theoretical value are usually found for the reversal coefficient, because the circular polarizations are much more affected by the nonideal Figure 7.3: Raman spectra of CC14 (a) under lincar and (b) circular polarization. Next to the fully symmetric modc a1 4h0cm' and the modc at 3 1 4cm' the measured depolarization ratio p and reversal coefficient P are given.
Raman shift (cm')
7.3 Humun Meusurements at L n r ~ ePhonon q
123
backscattering.17"~L72n1 While the theoretical depolarization ratio is the same regardless of thc scattering geometry, the reversal coefficient, e.g., in forward scattering, is the inverse of the bitckscattering value, In CC14 the measured P for the nonfully symmetric modcs varies between 3.6 at 7 ~ 0 c m  Iand 5.3 at 220 cm A more robust indicator for the symmetry of a Raman mode than the raw valucs of p and P are the ratios between the tensor invariants. In particular, the ratio between the symmetric anisotropy and the isotropic invariant "/,2/a2 = 0.4 for Ihc 460 cmI mode, but y,'/a2 is well above 100 for all other modes.
'.
7.3 Raman Measurements at Large Phonon q The Raman experiments commonly performed in laboratories and excited with laser light in the visible or nearvisible energy regime are limited to rpoint phonons in first order due to the small wave vector of the incident light. In backscattering geometry the maximal momentum transfer and hence phonon momentum is given by ,q
4n7c
=
A;
0.01 A
I
, say at A; = 488 nm
in a material with a refractive index n = 4. This is very small compared to the cxtent of the Brillouin zone in reciprocal space, for which the lattice constants set the relevant scale, k,,
n
=  = 1.3Aa0
I
, say with
ao = 2.46A as for graphite.
(7.22)
There are several techniques of getting to larger wavc vectors that have been used in the past. One is to create an artificial, smaller Brillouin zone by growing superlattices of different ma1(' In a way, this is happening in carbon terials and studying the resulting q = 0 modes.17281317 nanotubes by wrapping up the graphene sheet and thus introducing Brillouinzone folding, see also Scct. 2.1. Most of these new q = 0 modes in nanotubes are, however, forbidden by selection rules in firstorder scattering. A second approach is to relax the perfect periodicity in single crystals. For example, the earliest reports of Raman scattering in graphite by Tuinstra and KoenigL7301 showed that the nowcalled D mode arose from a defectinduced process at large q vectors; it is not allowed in the perfect crystal. Another example is the ion bombardemcnt of crystalline silicon with Si atoms successively increasing the damage to the material. In this way, one could show how the phonon density of states of aSi evolved from that of cSi.17311 The partially oxygenated hightemperature superconductor YBa2Cun07g with 6 around 0.3 shows defectinduced peaks at 230 and 600 cmI. Interestingly, the intensity of thcsc defect peaks decr~as~s when illuminated;L7'15L7 "1 light absorption incrrases the order of oxygen atoms in one of the crystal planes (the chain plane). The optical Raman methods of largeq scattering, unfortunately, are connected with little specific information about the phonon dispersions. In secondorder Raman ~ c a t t e r i n g , ~341 ~.'~'~~ for example, two phonons arc emitted (or absorbed) at the same time, a process that is generally 10 to 100 times weaker than the firstorder spectra. Thc momenta of thc two phonons involved cancel for equal and opposite phonon vectors, and the condition ( q,n, I=I q ,  q2 1% 4nn/al is easily fulfilled for visible light throughout the Brillouin zone. As a consequence, the Raman peak is an integration over the Brillouin zone, and regions with large densities of
124
7 Raman Scattering
Figure 7.4: Overbending in carbon nanotubcs as function of chiral angle. The maximal overhending of a given tube in the range 2.8 d 50.OW was fitted to the empirical expression in Eq. (7.23), which is plottcd hcrc.
a<