phonons in carbon nanotubes 2000 advances in physics

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Phonons in carbon nanotubesM. S. Dresselhaus a; P. C. Eklund b

a Department of Electrical Engineering and Computer Science and Department of Physics,Massachusetts Institute of Technology, Cambridge, MA 02139, USA. b Department of Physics,Pennsylvania State University, University Park, PA 16802, USA.

To cite this Article Dresselhaus, M. S. and Eklund, P. C.'Phonons in carbon nanotubes', Advances in Physics, 49: 6, 705 —814To link to this Article: DOI: 10.1080/000187300413184URL: http://dx.doi.org/10.1080/000187300413184

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Phonons in carbon nanotubes

M. S. DRESSELHAUS{* and P. C. EKLUND{*

{ Department of Electrical Engineering and Computer Science and Department ofPhysics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

{ Department of Physics, Pennsylvania State University, University Park, PA 16802,USA

[Received December 1999; revision received 14 March 2000; accepted 17 March 2000]

Abstract

A broad review of the unusual one-dimensional properties of phonons incarbon nanotubes is presented, including phonons in isolated nanotubes and incrystalline arrays of nanotubes in nanotube bundles. The main technique forprobing the phonon spectra has been Raman spectroscopy and the many uniqueand unusual features of the Raman spectra of carbon nanotubes are reviewed.Also included is a brief review of the thermal properties of carbon nanotubes inrelation to their unusual phonon dispersion relations and density of states.

Contents PAGE

1. Introduction 706

2. Background 7072.1. Structure and notation 7072.2. Electronic structure 712

3. Phonon modes 7183.1. Phonon dispersion relations for nanotubes 7203.2. Raman and infrared active modes of carbon nanotubes 7253.3. Inter-tube or intra-bundle interactions 729

4. Raman spectra of single-wall nanotubes 7344.1. Radial breathing mode phenomena 7384.2. Tangential stretch modes 7434.3. Temperature dependence of the Raman spectra 7484.4. High pressure e� ects on the tangential modes 7504.5. Anti-Stokes spectra 7534.6. Surface enhanced Raman spectra in carbon nanotubes 7594.7. Polarization studies in Raman scattering 7674.8. D-band and G 0-band spectra 7764.9. Overtone and combination modes 785

4.9.1. Overtones 7864.9.2. Combination modes 7894.9.3. Overtones and combination modes for the D-band and

G 0-band 7904.10. Raman studies of doped carbon nanotubes 791

5. Thermal properties 7965.1. Speci® c heat 7965.2. Thermal conductivity 800

ADVANCES IN PHYSICS, 2000, VOL. 49, NO. 6, 705 ± 814

Advances in Physics ISSN 0001± 8732 print/ISSN 1460± 6976 online # 2000 Taylor & Francis Ltdhttp://www.tandf.co.uk/journals

* e-mail : [email protected]; [email protected]

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5.3. Thermopower 805

6. Concluding remarks 807

Acknowledgements 808

References 808

1. Introduction

Carbon nanotubes provide a remarkable model one-dimensional (1D) system,one atom in thickness, a few tens of atoms in circumference, and many microns inlength. Although much has been written about the remarkable electronic propertieswhich allow a carbon nanotube to be either semiconducting or metallic depending onits diameter and chirality [1, 2], the phonon properties are also remarkable, showingunique 1D behaviour and special characteristics, such as the radial breathing mode,the twist acoustic mode and 1D phonon subbands. The strong coupling betweenelectrons and phonons in this one-dimensional system furthermore gives rise tohighly unusual resonance Raman spectra, and unique features in the Stokes and anti-Stokes spectra, as elaborated on in this article. The capability of surface-enhancedRaman spectroscopy to give rise to large enhancement factors in the Raman signal,likewise results in many unusual phenomena, not observed in other systems.

The article starts by brie¯ y reviewing the basic structure and notation used todescribe single-wall carbon nanotubes (SWNTs) in section 2.1, followed by adescription of the unique electronic structure of carbon nanotubes in section 2.2,largely linked to the unique symmetry of a graphene layer and to the 1D nature ofthe density of electronic states in a nanotube. With this background, the phononmodes in carbon nanotubes are reviewed in section 3, ® rst for isolated single-wallnanotubes section 3.1, and then for crystalline arrays of nanotubes arranged innanotube bundles in section 3.3. The group theory and the Raman and infraredselection rules for single-wall carbon nanotubes (SWNTs) are reviewed in section 3.2.The e� ect of the inter-tube interactions which are expected from the formation ofnanotube bundles is discussed in section 3.3. Throughout, emphasis is given to thedi� erences between the phonon modes in a carbon nanotube and the phonon modesof a graphene sheet which conceptually gives rise to the nanotube.

The dominant experimental technique for studying phonons in carbon nanotubeshas been Raman scattering, which is reviewed in detail in section 4. Particularemphasis is given to the phenomenon of the diameter selective resonant Ramanscattering process. The rather extensive use of the low frequency radial breathingmode for nanotube diameter characterization is reviewed in section 4.1, including thee� ect of inter-tube interactions. The higher frequency modes associated with carbonatom displacements in the cylindrical shell of the nanotube are discussed insection 4.2 with particular emphasis given to the di� erence in spectra betweensemiconducting and metallic nanotubes. Many other topics on Raman spectroscopyare then reviewed, including the temperature (section 4.3) and pressure (section 4.4)dependence of the Raman spectra, the anti-Stokes spectra (section 4.5), the surface-enhanced Raman spectra (section 4.6), polarization phenomena (section 4.7), thespecial features in the Raman spectra associated with wave vectors close to theK-point in the Brillouin zone (section 4.8), overtones and combination modes(section 4.9), and the e� ect of doping on the Raman spectra (section 4.10). Sinceour present knowledge of many of these topics is still at an early stage of

M. S. Dresselhaus and P. C. Eklund706


understanding, emphasis is given to important open issues and insights for gainingfurther understanding of phonons in this unique 1D system.

Thermodynamic and thermal transport phenomena involving phonons arediscussed in section 5, including speci® c heat (section 5.1), thermal conductivity(section 5.2), and thermopower (section 5.3). This is an area of research which is onlynow beginning to be addressed by the world-wide scienti® c community. Sincepotential practical applications may depend signi® cantly on the thermodynamicparameters, it is highly desirable for the research community to be aware of thecurrent status of developments on this topic. Finally, perspectives on the currentstatus of the ® eld are brie¯ y presented in section 6.

2. Background

This section provides a brief introduction to the structure and electronicproperties of carbon nanotubes, emphasizing those aspects that are particularlyrelevant to understanding the behaviour of the phonons in carbon nanotubes andexperiments that are carried out to study the phonon spectra, and the phonondensity of states, with particular emphasis given to the unusual 1D properties ofcarbon nanotubes.

2.1. Structure and notationA single-wall carbon nanotube (SWNT) can be described as a single layer of a

graphite crystal that is rolled up into a seamless cylinder, one atom thick, usuallywith a small number (perhaps 20± 40) of carbon atoms along the circumference and along length (microns) along the cylinder axis [1]. This nanotube is speci® ed by thechiral vector Ch

Ch ˆ na1 ‡ ma2 ² …n ;m†; …1†which is often described by the pair of indices …n ;m† that denote the number of unitvectors na1 and ma2 in the hexagonal honeycomb lattice contained in the vector Ch.As shown in ® gure 1, the chiral vector Ch makes an angle ³, called the chiral angle,with the zigzag or a1 direction. The vector Ch connects two crystallographicallyequivalent sites O and A on a two-dimensional (2D) graphene sheet where a carbonatom is located at each vertex of the honeycomb structure [3]. The axis of the zigzagnanotube corresponds to ³ ˆ 08, while the armchair nanotube axis corresponds to³ ˆ 308, and the general nanotube axis corresponds to 0 µ ³ µ 308. The seamlesscylinder joint of the nanotube is made by joining the line AB 0 to the parallel line OBin ® gure 1. In terms of the integers …n ;m†, the nanotube diameter dt is given by

dt ˆ Ch=p ˆ 31=2aC¡C…m2 ‡ mn ‡ n2†1=2=p ; …2†where aC¡C is the nearest-neighbour C± C distance (1.421 AÊ in graphite) , Ch is thelength of the chiral vector Ch and the chiral angle ³ is given by

³ ˆ tan¡1 ‰31=2m=…m ‡ 2n†Š: …3†Thus, a nanotube can be speci® ed by either its …n ;m† indices or equivalently by dt

and ³. Next we de® ne the unit cell OBB 0A of the 1D nanotube in terms of the unitcell of the 2D honeycomb lattice de® ned by the vectors a1 and a2 ( ® gure 1).

In ® gure 2 we show (a) the unit cell and (b) the Brillouin zone of two-dimensionalgraphite as a dotted rhombus and shaded hexagon, respectively, where a1 and a2 are

Phonons in carbon nanotubes 707


basis vectors in real space, and b1 and b2 are reciprocal lattice basis vectors. In thex ;y coordinates shown in ® gure 2, the real space basis vectors a1 and a2 of thehexagonal lattice are expressed as

a1 …̂31=2

2a ;

a2†; a2 …̂31=2

2a; ¡

a2†; …4†

where a ˆ ja1j ˆ ja2j ˆ 1:42 £ 31=2 ˆ 2:46 AÊ is the lattice constant of two-dimen-sional graphite. Correspondingly the basis vectors b1 and b2 of the reciprocal latticeare given by:


a1

a2

O

A

B

B

T

Ch

?

R

y

x

Figure 1. The unrolled honeycomb lattice of a nanotube. When we connect lattice sites O

and A, and sites B and B 0, a nanotube can be constructed. OA¡!

and OB¡!

de® ne thechiral vector Ch and the translational vector T of the nanotube, respectively. Therectangle OAB 0B de® nes the unit cell for the nanotube. The ® gure is constructed foran …n; m† ˆ …4; 2† nanotube [2].

y

k

x

k

y

x

a

2a

1

(a) (b)

BA K

M

2

b

b

1

G

Figure 2. (a) The unit cell and (b) Brillouin zone of two-dimensional graphite are shown asthe dotted rhombus and the shaded hexagon, respectively. ai , and bi , …i ˆ 1;2† arebasis vectors and reciprocal lattice vectors, respectively. Energy dispersion relationsare obtained along the perimeter of the dotted triangle connecting the high symmetrypoints, G, K and M [4].


b1 …̂ 2p31=2a

;2pa†; b2 ˆ… 2p

31=2a;¡

2pa† …5†

corresponding to a lattice constant of 4p=31=2a in reciprocal space. The direction ofthe basis vectors b1 and b2 of the reciprocal hexagonal lattice are rotated by 308 fromthe basis vectors a1 and a2 of the hexagonal lattice in real space, as shown in ® gure 2.By selecting the ® rst Brillouin zone as the shaded hexagon shown in ® gure 2 (b), thehighest symmetry is obtained for the Brillouin zone of 2D graphite. Here we de® nethe three high symmetry points, G, K and M as the centre, the corner, and the centreof the edge, respectively. The energy dispersion relations are calculated for thetriangle GMK shown by the dotted lines in ® gure 2 (b).

To de® ne the unit cell for the 1D nanotube, we de® ne OB¡!

in ® gure 1 as theshortest repeat distance along the nanotube axis, thereby de® ning the translationvector T

T ˆ t1a1 ‡ t2a2 ² …t1 ;t2†; …6†where the coe� cients t1 and t2 are related to …n;m† by

t1 ˆ …2m ‡ n†=dR ;

t2 ˆ ¡…2n ‡ m†=dR ;…7†

where dR is the greatest common divisor of …2n ‡ m ;2m ‡ n† and is given by

dR ˆ d ; if n ¡ m is not a multiple of 3d ;

3d ; if n ¡ m is a multiple of 3d ;…8†

in which d is the greatest common divisor of …n ;m†. The magnitude of the translationvector T ˆ jTj is

jTj ˆ 31=2L =dR …9†where L is the length of the chiral vector Ch ˆ pdt and dt is the nanotube diameter.The unit cell of the nanotube is de® ned as the area delineated by the vectors T andCh. The number of hexagons, N, contained within the 1D unit cell of a nanotube isdetermined by the integers …n ;m† and is given by

N ˆ 2…m2 ‡ n2 ‡ nm†dR

: …10†

The addition of a single hexagon to the honeycomb structure in ® gure 1 correspondsto the addition of two carbon atoms. Assuming a value aC¡C ˆ 0:142 nm on acarbon nanotube, we obtain dt ˆ 1:36 nm and N ˆ 20 for a (10,10) nanotube. Sincethe real space unit cell is much larger than that for a 2D graphene sheet, the 1DBrillouin zone (BZ) for the nanotube is much smaller than the BZ for a single two-atom graphene 2D unit cell. Because the local crystal structure of the nanotube is soclose to that of a graphene sheet, and because the Brillouin zone is small, Brillouinzone-folding techniques have been commonly used to obtain approximate electronand phonon dispersion relations for carbon nanotubes …n;m† with speci® c symmetry.

Whereas the lattice vector T, given by equation (6), and the chiral vector Ch,given by equation (1), both determine the unit cell of the carbon nanotube in realspace, the corresponding vectors in reciprocal space are the reciprocal lattice vectorsK2 along the nanotube axis and K1 in the circumferential direction, which givesdiscrete k values in the direction of the chiral vector Ch. The vectors K1 and K2 are



obtained from the relation Ri ¢ Kj ˆ 2p¯ij , where Ri and Kj are, respectively, thelattice vectors in real and reciprocal space, and K1 and K2 therefore satisfy therelations

Ch ¢ K1 ˆ 2p ; T ¢ K1 ˆ 0 ;

Ch ¢ K2 ˆ 0; T ¢ K2 ˆ 2p:…11†

From equations (11) it follows that K1 and K2 can be written as:

K1 ˆ 1N

…¡t2b1 ‡ t1b2†; K2 ˆ 1N

…mb1 ¡ nb2†; …12†

where b1 and b2 are the reciprocal lattice vectors of a two-dimensional graphenesheet given by equation (5). The N wave vectors ·K1 (· ˆ 0 ; . . . ;N ¡ 1) give rise toN discrete k vectors in the circumferential direction. For each of the · discrete valuesof the circumferential wave vectors, a one-dimensional electronic energy bandappears, whereas each · gives rise to 6 branches in the phonon dispersion relations.Because of the translational symmetry of T, we have continuous wave vectors in thedirection of K2 for a carbon nanotube of in® nite length. However, for a nanotube of® nite length L t, the spacing between wave vectors is 2p=L t and e� ects associated withthe ® nite nanotube length have been observed experimentally [5].

Typically, the experimental single-wall nanotube (SWNT) samples have a distri-bution of diameters and chiral angles because of the absence of experimentaltechniques at present for producing SWNTs with a unique dt and ³. The nanotubematerial produced by the presently available synthesis methods, including laservaporization, carbon arc discharge, vapour phase deposition, and solar energysynthesis, appears in a scanning electron microscopy (SEM) image as a mat ofcarbon nanotubes bundles 10± 20 nm in diameter and up to 100 mm or more in length(see ® gure 3) and containing between 30± 500 SWNTs. The nanotubes within abundle are twisted together, thereby maximizing the bonding interaction betweenSWNTs within the bundles where the hexagons on adjacent nanotubes tend to be inthe same AB registry as in crystalline graphite [7]. The nanotube bundles attract oneanother and wrap around each other to form ropes [7]. These nanotube ropes areaccompanied by varying amounts of amorphous carbon, residual catalyst, and otherunwanted material, from which the nanotube ropes must be separated. A number ofexperimental observations have also been made on a material called `bucky paper’ ,which refers to a ¯ at tangled mat of bundles of carbon nanotubes that are collectedon ® lter paper as a suspension of SWNT bundles is passed through the ® lter paper.

Under transmission electron microscopy (TEM) examination, each nanotuberope is found to consist primarily of bundles of single-wall carbon nanotubes thatare mostly aligned along a common axis (see ® gure 3 (a)). X-ray di� raction (whichviews many ropes at once) and transmission electron microscopy (which typicallyviews one or two ropes) show that the diameters of the single-wall nanotubes intypical SWNT samples have a strongly peaked narrow distribution of diameters.Typical nanotube diameters in the ropes are between 0.9± 1.8 nm, depending on thecatalyst and growth conditions though smaller diameter nanotubes as small as0.4 nm have been reported [8]. For the synthesis conditions used in the early work,the diameter distribution was strongly peaked at 1.38§ 0:02 nm, very close to thediameter of an ideal (10,10) nanotube. X-ray di� raction measurements [6, 9] furthershowed that, within these r̀opes’ , the single-wall nanotubes form a two-dimensional



triangular lattice with a lattice constant of 1.7 nm, and an inter-tube separation of0.315 nm at closest approach within a rope, in good agreement with prior theoreticalmodelling results [10, 11]. The diameter and chiral angle of individual nanotubes aremeasured by transmission electron microscopy [12] and by scanning tunnellingmicroscopy [13, 14] techniques. Because of the dependence of the radial breathingmode frequency on the inverse nanotube diameter (see sections 3.3, 4.1 and 4.4),measurement of the radial mode frequencies at a variety of laser excitation energiesprovides a convenient secondary characterization tool for the measurement of thediameter distribution contained within a typical single-wall nanotube sample.However, it has been very di� cult to measure dt and ³ on the same SWNT that isused for property measurements. For example, it has not yet been possible tomeasure the Raman spectrum on an individual SWNT, characterized for its dt and ³.


(a)

(b)

Figure 3. (a) Ropes of single-wall carbon nanotubes observed by scanning electronmicroscopy (SEM). The ropes are 10± 20 nm thick and ¹100 mm long. (b) At highermagni® cation, the TEM image shows that each rope contains a bundle of single-wallnanotubes with diameters of ¹1.4 nm, arranged in a triangular lattice (with latticeconstant ¹1.7 nm). The lower image is seen when the rope bends, so that the ropecross section is in the image plane of the transmission electron microscope [6].D

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The Raman spectra of both single-wall and multiwall nanotubes have beeninvestigated, where the multiwall nanotubes can be described as capped coaxialcylinders separated radially by ¹ 3:4 AÊ (see ® gure 4) [1]. Each of the constituentcylinders can, in principle, be speci® ed by its chiral vector Ch through its indices…n ;m†, or equivalently by its diameter dt and chiral angle ³. Because of the strongdependence of the Raman spectra on nanotube diameter and to a lesser degree onthe chiral angle, the structural characterization of the nanotube is necessary for thecomparison of experimental results and theoretical models, and for the comparisonof Raman spectra obtained in di� erent laboratories , on di� erent Raman spectro-meters and when using di� erent laser excitation energies Elaser.

2.2. Electronic structureBecause of the strong coupling between electrons and phonons in the resonance

Raman e� ect, the remarkable electronic properties of carbon nanotubes play animportant role in discussing the unusual Raman spectra of these unique one-dimensional periodic structures. In single-wall carbon nanotubes, con® nement ofthe electronic wave function in the radial direction is provided by the monolayerthickness of the nanotube in the radial direction. Circumferentially, the periodicboundary condition applies to the enlarged unit cell that is formed in real space. Theapplication of this periodic boundary condition to the graphene electronic statesleads to the prediction of a remarkable electronic structure for carbon nanotubes ofsmall diameter.

The 1D electronic energy band structure for carbon nanotubes [16± 20] is relatedto the energy band structure calculated for the 2D graphene honeycomb sheet usedto form the nanotube. These calculations for the electronic structure of SWNTs [2]show that about 1/3 of the nanotubes are metallic and 2/3 are semiconducting,depending on the nanotube diameter dt and chiral angle ³. It can be shown thatmetallic conduction in a …n;m† carbon nanotube is achieved when


Figure 4. Transmission electron microscopy pictures of carbon nanotubes with threecommon cap terminations: (a) a symmetric polyhedral cap, (b) an asymmetricpolyhedral cap and (c) a symmetrical ¯ at cap [15].


2n ‡ m ˆ 3q ; …13†

where q is an integer. All armchair carbon nanotubes (³ ˆ 308) are metallic andsatisfy the more general equation (13).

Calculated dispersion relations based on these simple zone folding considerationsfor tight binding energy bands are shown for metallic nanotubes …n ;m† ˆ …5 ;5† and(9,0) in ® gures 5 (a) and (b), respectively, and for a semiconducting nanotube…n ;m† ˆ …10 ;0† in ® gure 5 (c) [21]. These results are consistent with more detailedab initio calculations of the band structure [23]. The calculated electronic structurecan be either metallic or semiconducting depending on the choice of …n ;m† as givenby equation (13), although there is no di� erence in the local chemical bondingbetween the carbon atoms in the nanotubes, and no doping impurities are present [4].

These surprising results and the unique features of the electronic structure ofSWNTs can be understood on the basis of the electronic structure of a graphenesheet which is a zero gap semiconductor [24] with bonding and antibonding p bandsthat are degenerate at the K-point (zone corner) of the hexagonal 2D Brillouin zone[2]. The periodic boundary conditions for the 1D carbon nanotubes of smalldiameter permit only a few wave vectors to exist in the circumferential direction,and these wave vectors k satisfy the relation n¶ ˆ pdt, where ¶ ˆ 2p=k is the deBroglie wavelength. Metallic conduction occurs when one of these allowed wavevectors k passes through the K-point of the 2D Brillouin zone, where the 2D valenceand conduction bands are degenerate because of the special symmetry of the 2Dgraphene lattice [2]. As the nanotube diameter increases, more wave vectors becomeallowed for the circumferential direction, so that the nanotubes become more two-dimensional and the semiconducting band gap disappears. The band gap for isolatedsemiconducting carbon nanotubes is proportional to the reciprocal nanotubediameter 1=dt. At a nanotube diameter of dt ¹ 3 nm, the bandgap becomescomparable to thermal energies at room temperature, showing that small diameternanotubes are needed to observe 1D quantum e� ects.

Of particular importance to the discussion of the resonant Raman spectra insingle-wall carbon nanotubes is the 1D density of states plots shown in ® gure 6 for:(a) a semiconducting (10,0) zigzag carbon nanotube and (b) a metallic (9,0) zigzagcarbon nanotube. The results for the 1D electronic density of states show sharp


Figure 5. One-dimensional energy dispersion relations for (a) armchair (5,5) nanotubes, (b)zigzag (9,0) nanotubes and (c) zigzag (10,0) nanotubes. The energy bands with asymmetry are non-degenerate, while the e-bands are doubly degenerate at a generalwave vector k [4, 21, 22].


singularities associated with the …E ¡ E0†¡1=2 van Hove singularities about eachsubband edge at energy E0 (see ® gure 5). The electronic density of states plots in® gure 6 show that the metallic nanotubes have a small, but non-vanishing 1D densityof states at the Fermi level (which is at E ˆ 0 in ® gure 6), and this non-vanishingdensity of states is independent of energy until the energies of the ® rst subband edgesof the valence and conduction bands are reached. In contrast, for a 2D graphenesheet (dashed curve), the 2D density of states is zero at the Fermi level (where alsoE ˆ 0 in ® gure 6), and varies linearly with energy, as we move away from the Fermilevel. Furthermore, the density of states for the semiconducting 1D nanotubes is zerothroughout the bandgap, as shown in ® gure 6 (a), and their bandgap energy Eg isequal to the energy di� erence E11…dt† between the two van Hove singularities in the1D density of states that span the Fermi level, where it is noted that Eg isproportional to the reciprocal nanotube diameter Eg / 1=dt. Because of thesesingularities in the density of states, high optical absorption is expected when thephoton energy matches the energy separation between an occupied peak in theelectron density of states and one that is empty. This situation occurs at the band gapfor the semiconducting nanotubes, but also at higher energies for transitions from anoccupied subband edge state to the corresponding unoccupied subband edge state.Such transitions between subband edge states can occur for both semiconducting


-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 Energy/ 0

0.0

0.5

1.0

DO

S [s

tate

s/un

it ce

ll of

gra

phite

]

-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 Energy/ 0

0.0

0.5

1.0

DO

S [s

tate

s/un

it ce

ll of

gra

phite

]

(a) (n,m)=(10,0)

(b) (n,m)=(9,0)

g

g

Figure 6. Electronic 1D density of states per unit cell of a 2D graphene sheet for two …n;0†zigzag nanotubes : (a) the …10 ;0† nanotube which has semiconducting behaviour,(b) the …9;0† nanotube which has metallic behaviour. Also shown in the ® gure is thedensity of states for the 2D graphene sheet (dotted line) [25].


and metallic nanotubes. Comparing the density of states curves in ® gure 6, we seethat the smallest energy separation between subband edge states for the semicon-ducting nanotube (10, 0) is much smaller than the corresponding separation betweensubband edges for the metallic (9, 0) nanotube. Because of the very large electrondensity of states at the van Hove singularities (or subband edges) the intensity of theinterband optical transitions Eii…dt† are exceptionally strong, giving rise to excep-tionally high intensities for the resonant Raman e� ect associated with the 1D densityof electronic states. Thus we can expect the resonance Raman e� ect for carbonnanotubes to be much stronger than for 3D crystalline materials, based on the lowdimensionality of carbon nanotubes.

The general characteristics that are predicted for the 1D electronic density ofstates of carbon nanotubes have recently been con® rmed by low temperature STM/STS (scanning tunnelling microscopy/spectroscopy) studies carried out on isolatedsingle-wall carbon nanotubes [13, 14, 26], as discussed below. Insight into thevariation of the electronic density of states with nanotube diameter dt is providedby ® gure 7, where the density of states is presented for (8, 8), (9, 9), (10, 10), (11, 11)and (12, 12) armchair nanotubes and for (14, 0), (15, 0), (16, 0), (17, 0) and (18, 0)zigzag nanotubes [28]. Referring to ® gure 7, we see that the lowest energy transitionfor the armchair nanotube, denoted by EM

11…dt†, varies continuously from 2.4 eV forthe (8, 8) nanotube to 1.6 eV for the (12, 12) armchair nanotube.

The interpretation of the interband transitions between van Hove singularitiesfor zigzag nanotubes and chiral nanotubes can be understood by plotting theenergies for the transitions between the van Hove singularities in the valence and


Figure 7. Electronic 1D density of states (DOS) calculated with a tight binding model for(8, 8), (9, 9), (10, 10), (11, 11) and (12, 12) armchair nanotubes and for (14, 0), (15,0), (16, 0), (17, 0) and (18, 0) zigzag nanotubes and assuming a nearest neighbourcarbon± carbon interaction energy ®0 ˆ 3:0 eV [27]. Wavevector conserving opticaltransitions can occur between mirror image singularities in the 1D density of states,i.e. v1 ! c1 and v2 ! c2, etc. and these optical transitions are given in the ® gure inunits of eV. These interband transitions are denoted in the text by E11, E22, etc. andare responsible for the resonant Raman e� ect discussed extensively in this review [27].


conduction bands Eii…dt† of all possible …n ;m† nanotubes, including not onlyarmchair and zigzag nanotubes, as shown in ® gure 7, but also all chiral nanotubes.Such a plot of Eii…dt† versus dt is shown in ® gure 8. In this plot, the nearest neighbourcarbon± carbon transfer energy ®0 was taken to be 2.9 eV which provides a good ® t toa variety of experiments, including resonance Raman scattering, optical absorptionand scanning tunnelling spectroscopy studies as discussed below. Figure 8 is usedextensively to interpret resonance Raman spectra in carbon nanotubes. The twolowest energy transitions ES

11…dt† and ES22…dt† are for semiconducting nanotubes,

while the next higher energy transition is EM11…dt† for metallic nanotubes, followed by

ES33…dt†, as indicated in the ® gure [29, 32]. The bounds on Eii…dt† at constant dt and i

are delineated by zigzag nanotubes and are due to trigonal warping e� ects [32]. Forexample, the (15,0) zigzag nanotube with diameter dt ˆ 1:17 nm has a lower andupper bound for EM

11…dt† ˆ 2:0 and 2.4 eV, as shown in ® gure 7. In relating ® gure 8 tothe resonance Raman e� ect, we note that resonance occurs when Eii…dt† is inresonance with either the incident or scattered photon.

Since measurements of dI=dV in the STS (scanning tunnelling spectroscopy)mode of a scanning tunnelling microscope yields a signal (shown in ® gure 9), which isproportional to the 1D density of states, the STS technique has become a powerfultool for studying the electronic structure of both metallic and semiconducting single-wall carbon nanotubes [13, 14]. The top 4 traces in ® gure 9 show that the bandgap orenergy separation is ES

11…dt† ’ 0:6 eV for the indicated semiconducting nanotubes,while the lower 3 traces show energy separations of EM

11…dt† ’ 1:8 eV for metallicnanotubes. The combined STM/STS studies [13, 33] are consistent with: (1) about2/3 of the nanotubes being semiconducting and 1/3 being metallic ; (2) the density ofstates exhibiting van Hove singularities, characteristic of the expectations for 1D


0.0 1.0 2.0 3.0dt [nm]

0.0

1.0

2.0

3.0?E

[eV

]

1.491.58

1.92

1.8

1.96

)ii

t

1.350.0 1.0 2.0 3.00.0

1.0

2.0

3.0(d

E

22S

S33

M11

SE11

E

EE

Figure 8. Calculation [29± 31] of the energy separations Eii…dt† for all …n; m† values versusnanotube diameter in the range 0:7 < dt < 3:0 nm using ®0 ˆ 2:9 eV. Semiconductingand metallic nanotubes are indicated by crosses and open circles, respectively, and thefour lowest energy transitions are labelled by ES

11…dt†, ES22…dt†, EM

11…dt†, and ES33…dt†,

where S and M, respectively, refer to semiconducting and metallic nanotubes. The® lled squares denote zigzag tubes. The vertical lines denote dt ˆ 1:35 § 0:20 nm for aparticular single-wall carbon nanotube sample [32].


systems; (3) energy gaps for the semiconducting nanotubes that are proportional to1=dt. Using the approximate relation Eg ’ 2®0aC¡C=dt for the band gap of single-wall nanotubes, the nearest neighbour overlap energy ®0 (or the transfer integral of atight binding model) can be found. The STS experiments con® rm that the density ofelectronic states near the Fermi level is zero for semiconducting nanotubes, and non-zero for metallic nanotubes [13]. These electronic density of states curves in ® gure 6and the plot of interband transition energies in ® gure 8 are also important forexplaining the resonance Raman experiments on carbon nanotubes discussed insection 4 of this review.

Subsequent atomic resolution STM/STS studies coupled with tight-bindingcalculations [34] were able to make a detailed identi® cation of the features in thedI=dV versus voltage spectrum of an individual (13,7) metallic carbon nanotubewith speci® c interband transitions. Comparison to the tight-binding calculations ledto the ® rst experimental demonstration of the trigonal warping e� ect in metallicnanotubes [34], consistent with a systematic theoretical study [32] of this phenomena


Figure 9. Derivative of the current± voltage …dI=dV † curves obtained by scanning tunnellingspectroscopy on various isolated single-wall carbon nanotubes with diameters near1.4 nm. Nanotubes #1± 4 are semiconducting and #5± 7 are metallic [13].


in both metallic and semiconducting nanotubes. This topic awaits further systematicexperimental studies.

A second method for determining the interband transitions Eii…dt† comes fromoptical spectra, where the measurements are made on ropes of single-wall carbonnanotubes, so that appropriate corrections should be made for inter-tube inter-actions in interpreting the experimental data. Optical transmission spectra weretaken for single-wall nanotubes synthesized using four di� erent catalysts [29, 35],namely NiY (1.24± 1.58 nm), NiCo (1.06± 1.45 nm), Ni (1.06± 1.45 nm) and RhPd(0.68± 1.00 nm), where the range in dt for each catalyst is indicated [29, 35]. For theNiY catalyst, three large absorption peaks were observed at 0.68, 1.2 and 1.7 eV,yielding a value of ®0 ˆ 3:0 § 0:2 eV, using the approximate relations ES

11 ’2®0aC¡C=dt and EM

11 ’ 6®0aC¡C=dt, for semiconducting and metallic nanotubes,respectively. These approximate relations denote the centre of the width of theEii…dt† transition energies in ® gure 8 at constant dt [32]. Optical spectra were alsoreported for nanotubes produced with the NiCo, Ni and RhPd catalysts, but thepeak values for the absorption bands were not explicitly quoted [29, 35, 36]. Ininterpreting the optical transmission data, corrections for the diameter distributionof the nanotubes, for trigonal warping e� ects and for the non-linear k dependence ofE…k† away from the K-point in the Brillouin zone need to be considered [30, 32]. Thecalculated Eii…dt† transition energies in ® gure 8 provide a ® rm basis for a detailedanalysis of the optical data [29, 35].

Optical absorption measurements on bromine and caesium-doped single-wallnanotubes (1:24 < dt < 1:58 nm) [37] showed an upshift of EF on doping with Cs anda downshift of EF on doping with Br2 due to charge transfer e� ects between thedopants and the nanotubes. From observation of the reduction in the opticalabsorption intensity upon doping, the shift of EF with Br2 doping could bemonitored, showing that at full Br2 saturation concentration, EF had decreasedsu� ciently so that optical transitions from an occupied to an unoccupied state couldonly occur for Elaser ¶ 2 eV, which lies above the metallic window for the EM

11…dt†transition for the NiY catalysed sample used in the experiments. Whereas one reportbased on resonant Raman spectra for semiconducting nanotubes [38] claimedreversibility in the spectra upon iodine doping and undoping, another study basedon resonant Raman spectra for metallic nanotubes upon bromine doping andundoping claimed some irreversible behaviour [39, 40]. Further studies as a functionof Elaser and dt are needed to account for the connection between these twoobservations [38± 40] (see section 4.10).

3. Phonon modes

The phonon dispersion relations in a carbon nanotube can be obtained fromthose of the 2D graphene sheet by using the same zone folding approach [41± 43] aswas used to ® nd the 1D electronic dispersion relations [2, 44]. Zone folding andsymmetry-based force constant models were used by some authors [41, 43, 45, 46] forcalculations of the phonon dispersion relations, tight binding calculations by otherauthors [47± 49], and some ab initio calculations [50, 51] were also reported. Becauseof the very weak interplanar interactions, the phonon dispersion relations forgraphite in the basal plane (see ® gure 10) provide a good ® rst approximation forthe 2D phonon dispersion relations of an isolated graphite plane, which is called agraphene sheet. The calculated phonon dispersion curves of ® gure 10 were ® t to the



experimental points obtained by electron energy loss spectroscopy, inelastic neutronscattering, velocity of sound and other techniques [41, 52, 53]. The inclusion of forceconstants taking into account fourth-neighbour interaction terms (see table 1) havebeen su� cient for reproducing the experimental data for graphite in ® gure 10.

The three phonon dispersion curves (or branches) , which originate from the G-point of the Brillouin zone with ! ˆ 0 (see ® gure 10), correspond to acoustic modes:an out-of-plane mode, an in-plane tangential (bond-bending) mode and an in-planeradial (bond-stretching ) mode, listed in order of increasing energy, respectively. Theremaining three branches correspond to optical modes: one non-degenerate out-of-plane mode and two in-plane modes that remain degenerate as we move away fromk ˆ 0.

It is noted that the out-of-plane (transverse) acoustic branch for a graphene sheetshows a special k2 energy dispersion relation around the G-point, while the other two


Figure 10. The phonon dispersion relations for graphite plotted along high-symmetry in-plane directions. Experimental points from neutron scattering and electron energyloss spectra were used to obtain values for the force constants (see table 1) and todetermine the phonon dispersion relations throughout the Brillouin zone [41].

Table 1. Force constant parameters for 2D graphite in units of104 dyn cm¡1 [41]. Here the subscripts r, ti and to refer toradial (bond stretching), transverse in-plane and trans-verse out-of-plane (bond bending) force constants,respectively (see ® gures 10 and 11).

Radial Tangential

¿…1†r ˆ 36:50 ¿

…1†ti ˆ 24:50 ¿

…1†to ˆ 9:82

¿…2†r ˆ 8:80 ¿

…2†ti ˆ ¡3:23 ¿

…2†to ˆ ¡0:40

¿…3†r ˆ 3:00 ¿

…3†ti ˆ ¡5:25 ¿

…3†to ˆ 0:15

¿…4†r ˆ ¡1:92 ¿

…4†ti ˆ 2:29 ¿

…4†to ˆ ¡0:58


in-plane acoustic branches show a linear k dependence, as is normally seen foracoustic modes. The optical out-of-plane transverse branch (at ! ¹ 865 cm¡1 at the Gpoint) also shows a k2 dependence. Because of this special k2 dependence of !…k† forthe out-of-plane acoustic branch near the G-point, there is neither a phase velocitynor a group velocity for the z component for these acoustic vibrations.

At low frequencies the in-plane acoustic branches for a graphene sheet give rise toa density of states g…!† linear in !, while the out-of-plane k2 acoustic branch givesrise to a small constant contribution (independent of !) to the density of states,which is signi® cant up to ¹400 cm¡1 and a� ects the thermal properties (see section 5)up to ¹50 K, as shown in ® gures 10 and 11.

3.1. Phonon dispersion relations for nanotubesAs a ® rst approximation, the phonon dispersion relations for an isolated single-

wall carbon nanotube can be determined by zone folding the phonon dispersioncurves !m

2D…k† of a two-dimensional graphene layer (see ® gure 1), where m ˆ 1 ; . . . ;6labels the 3 acoustic and 3 optic modes and k is a vector in the layer plane. Sincethere are 2N carbon atoms in this unit cell (see equation (10)), we will have N pairs ofbonding p and anti-bonding p¤ electronic energy bands. Similarly the phonondispersion relations will consist of 6N branches resulting from a vector displacementof each carbon atom in the nanotube unit cell.

The phonon dispersion relations of a carbon nanotube depend on the indices…n ;m† or equivalently on the diameter and chiral angle of the carbon nanotube, dt

and ³, since the phonon wave vector in the circumferential direction becomes discreteand is described by each K1 vector (see equation (12)), in accordance with theperiodic boundary conditions of the chiral vector Ch.

In the context of zone folding, the one-dimensional phonon energy dispersionrelations for !

m·1D…k† for SWNTs are related to the !m

2D…k† by:


Figure 11. The phonon density of states versus phonon energy for a 2D graphene sheet inunits of states/C-atom/cm¡1 £ 10¡2 [41].


!m·1D…k† ˆ !m

2D kK2

jK2j ‡ ·K1… †;m ˆ 1 ; . . . ;6 ;

· ˆ 0 ; . . . ;N ¡ 1 ;and ¡ p

T< k µ p

T… †;

…14†

where k is a one-dimensional wave vector, K2 is the reciprocal lattice vector alongthe nanotube axis (see equation (12)), K1 is the reciprocal lattice vector in thecircumferential direction (see equation (12)) and T is the magnitude of the one-dimensional translation vector T given in equation (9).

The zone folding procedure yields the appropriate one-dimensional !m·1D…k† for

almost all the phonon branches of a carbon nanotube. An example of the phononbranches for an isolated SWNT is shown in ® gure 12 for a (10, 10) nanotube. In this® gure T denotes the magnitude of the basis vector along the nanotube axis (seeequation (9)). For the 2N ˆ 40 carbon atoms per circumferential strip for the (10,10) nanotube, there are 120 vibrational degrees of freedom, but because of modedegeneracies there are only 66 distinct phonon branches, of which 12 modes are non-degenerate and 54 are doubly degenerate.


Figure 12. (a) The calculated phonon dispersion relations of an armchair carbon nanotubewith Ch ˆ …10; 10†. The number of degrees of freedom is 120 and the number ofdistinct phonon branches is 66. (b) The corresponding phonon density of states for a(10, 10) nanotube [2]. (c) A comparison between the phonon density of states g1D…!†for a (10, 10) nanotube shown as the solid curve and g2D…!† for a graphene sheetshown by the points [31].


In ® gure 12 (b) we show the corresponding phonon density of states forthe (10,10) nanotube in units of states per C atom per cm¡1. When we integratethe phonon density of states with respect to the energy, we get 3 states/C-atom as thetotal number of states. Since we use the same units for the phonon density of statesfor nanotubes and for 2D graphite, we can directly compare the phonon density ofstates for the (10,10) nanotube (® gure 12 (b)) and for 2D graphite (see ® gure 11). Thiscomparison given in ® gure 12 (c) [31] shows that the phonon density of states for the(10,10) nanotube is close to that for 2D graphite, since the phonon dispersionrelations are, in principle, given by the zone-folding of those for 2D graphite. Thedi� erences in the nanotube phonon density of states relative to that for 2D graphitepertain to the one-dimensional van Hove singularities for the optical phononsubbands and to the 4 acoustic modes of the nanotubes and their special propertiesat low ! discussed below.

However, zone-folding of the graphene phonon branches does not always givethe correct dispersion relation for a carbon nanotube [2, 41], especially in the lowfrequency region, and some additional physical concepts must be introduced. Forexample, when the out-of-plane tangential acoustic (TA) modes of a graphene sheetshown in ® gure 13 (a) on the left are rolled into a nanotube as shown on the right, theradial breathing mode is formed and the resulting vibration does not have ! ! 0 ask ! 0. Therefore the radial breathing mode is not an acoustic mode, but rather is anoptical mode with a non-zero frequency at k ˆ 0. For the radial breathing modeshown in ® gure 13 (a) on the right [41], both the graphite radial force constant andthe in-plane tangential force constant in the circumferential direction of thenanotube are related to the radial breathing normal mode vibration. This resultsin a ® nite frequency !0

RBM for the radial breathing mode at k ˆ 0, where thesuperscript denotes the mode frequency for an isolated SWNT, leaving !RBM todenote the mode frequency in the presence of inter-tube interactions which occur inSWNTs found in nanotube bundles (see section 3.3). Since there is no vibration in


(b)

(a)

Figure 13. (a) The out-of-plane tangential acoustic modes at k ˆ 0 (left) in a single layer ofgraphite give rise to a radial breathing mode in the carbon nanotube with non-zerofrequency (right). (b) An acoustic mode of a carbon nanotube whose vibration isperpendicular to the nanotube axis (right) corresponds to a linear combination ofboth in-plane and out-of-plane graphite-derived modes (left). These modes do notcouple in the case of a single graphite layer, but do couple for the nanotube becauseof the curvature that is introduced by rolling up the graphene sheet [2].


the direction of the nanotube axis for the breathing mode, the bond angle of thehexagon network is unchanged.

While the acoustic vibrations of a carbon nanotube in the longitudinal directioncorrespond to acoustic vibrations in the 2D graphene sheet, the two acoustic modesin the directions perpendicular to the nanotube axis, do not directly correspond toany two-dimensional graphene phonon modes. In a graphene sheet, the in-plane andout-of-plane modes are decoupled from each other. However, when the graphenestrip is rolled up into a nanotube, the graphite-derived in-plane and out-of-planemodes do couple to each other, as shown on the left-hand side of ® gure 13 (b), toform the acoustic mode of the nanotube shown on the right. To avoid di� cultiesassociated with lack of compatibility between the TA modes in a graphene sheet withthe TA modes and the radial breathing mode of SWNTs, the three-dimensionalcarbon nanotube dynamical matrix for a given nanotube described by …n ;m† hasbeen solved in the one-dimensional Brillouin zone of the nanotube [2].

The resulting lowest energy modes are shown in ® gure 14 for a (10,10) nanotubewhere the 4 acoustic modes appropriate to SWNTs are displayed. Here we see thetransverse acoustic (TA) modes, which are doubly degenerate, and have x and yvibrations perpendicular to the nanotube (z) axis (see ® gure 13 (b) right). The highestenergy acoustic mode in ® gure 14 is the longitudinal acoustic (LA) mode withvibrations in the direction of the nanotube axis. Since the displacements of thesethree acoustic nanotube modes are three dimensional, all three phonon dispersionrelations are of the form ! ˆ vk, where v is the velocity of sound appropriate for thatacoustic branch. The sound velocities of the TA and LA phonons for a (10,10)


0

5

10

15

20

0 0.1 0.2 0.3 0.4

E (

meV

)

k(1/Å )

0 4 8 1 2P

ho

no

n D

OS

E (meV)

Figure 14. Low-energy phonon dispersion relations for a (10, 10) nanotube. There are fouracoustic modes: two degenerate TA modes (v ˆ 9 km s¡1), a t̀wist’ mode(v ˆ 15 km s¡1) and one TA mode (v ˆ 24 km s¡1 ). The inset shows the low-energyphonon density of states of the nanotube (solid line) and that of graphite (dashedline) and graphene (dot-dashed line). The nanotube phonon DOS is constant below2.5 meV, then increases stepwise as higher subbands enter; there is a 1D singularity ateach subband edge [2, 54].


armchair carbon nanotube are estimated as vTA ˆ 9:43 £ 103 m s¡1 and vLA ˆ20:35 £ 103 m s¡1, respectively, from the measured Young’s modulus, Y , from whichthe LA phonon velocity is estimated by v ˆ …Y =»†1=2, in which » is the density ofcarbon atoms taken as » ˆ 1:28 £ 103 kg m¡3 and Y ˆ 532 GPa. Ab initio calcula-tions [51, 55, 56] have shown that the Young’ s modulus and Poisson ratio for typicalSWNTs are essentially the same as for a graphene layer.

In addition, there is a fourth acoustic mode with ! ˆ 0 at k ˆ 0 for isolatedsingle-wall carbon nanotubes, and this mode is related to a rigid rotation around thenanotube axis at k ˆ 0 [28, 57]. Since the driving force for this wave motion is atwisting motion of the nanotube, this mode is called the twisting mode (TW). Thesound velocity of the TW mode is taken to be vTW ˆ 15:00 £ 103 m s¡1, which is thesame as the calculated value of the in-plane TA mode for 2D graphite, sincethe displacements associated with the TW mode are in the cylindrical plane,perpendicular to the nanotube axis. Also shown in ® gure 14 are the lowest opticalsubbands obtained by the zone folding procedure for the (10,10) armchair carbonnanotube, including an E2g mode at ¹17 cm¡1, an E1g mode at ¹118 cm¡1 and theradial breathing A1g mode at ¹165 cm¡1 for k ˆ 0 (see ® gure 15). The diagram(® gure 14) shows modes of like symmetry which couple to each other and show anti-crossing behaviour, while branches with di� erent symmetries do not interact and cansimply cross.


E E1g cm-1

A1g

2g

1gE

2g

cm-1 cm-1

cm-1cm

-1cm

1g 2g

E

A E

-1

17

(a) (b)

(e)(d)

(g)(f)

165 cm-1

118(c)

368

1587 1591

1585

Figure 15. The calculated Raman mode atomic displacements, frequencies and symmetriesfor selected normal modes for the (10, 10) nanotube modes. The symmetry and thefrequencies for these modes are not strongly dependent on the chirality of thenanotube. In the ® gure, we show the displacements for only one of the two modes inthe doubly degenerate E1g and E2g modes [2].


From the phonon dispersion relations in ® gure 12 (a) for the SWNTs, we obtainthe phonon density of states g1D…!† shown in ® gure 12 (b) and the comparisonbetween g1D…!† and g2D…!† shown in ® gure 12 (c). Of particular interest is thebehaviour of g1D…!† for the isolated SWNTs (single-wall nanotubes) in the limit ofsmall ! where the acoustic modes and the lowest phonon subbands are dominant.The four phonon branches which follow the dispersion relations ! ˆ vk contribute aconstant term to the density of states, which is proportional to the inverse of thevelocity of sound for that branch. Additional contributions to g1D…!† for the isolatedSWNTs arise from the low energy optical phonons from the phonon subbandsassociated with the low dimensionality, which contribute to g1D…!† as a …! ¡ !0†¡1=2

van Hove singularity at their subband edge frequency !0. This g1D…!† is in contrastto the g2D…!† for the graphene sheet for which the ! ˆ vk branches give rise to acontribution to g2D…!† that is linear in !, while the out-of-plane ! / k2 branchcontributes to the phonon density of states g2D…!† a small term that is constant, andindependent of !.

Other than the establishment of acoustic modes of the type shown in ® gure 14, thee� ect of the nanotube curvature on the in-plane and out-of-plane radial andtangential force constants has been considered by various authors [2, 30, 31, 40,51, 58], showing these corrections to be relatively minor, but not negligible for thesmallest diameter nanotubes. Since the magnitude of the curvature is not expected toa� ect the form of !…k† for the four acoustic branches in ® gure 14 very much, nomajor e� ect of curvature on the 1D density of phonon states g1D…!† is expected.

Interactions between adjacent SWNTs in a nanotube bundle (see section 3.3) areexpected to give rise to a dispersion relation with a ® nite frequency at k ˆ 0 and a k2

dependence for the twist mode [59]. The contribution of such an inter-tube inter-action term would result in a feature with a van Hove-type singularity in the phonondensity of states such as is introduced by a phonon subband.

3.2. Raman and infrared active modes of carbon nanotubesThe special symmetry properties of 1D carbon nanotubes results in only a few

Raman-active and infrared-active vibrational modes, as described in this section.Among the 6N calculated phonon dispersion relations for carbon nanotubes whoseunit cell contains 2N carbon atoms, only a few modes are Raman or infrared (IR)active, as speci® ed by the symmetry of the phonon modes. Since only k vectors veryclose to k ˆ 0 are coupled to the incident light because of the energy-momentumconservation requirements for the photons and phonons, we need only consider thesymmetry of the nanotube zone-centre vibrations at the G-point …k ˆ 0†. Point grouptheory of the unit cell, predicts the number of Raman-active modes and IR-activemodes and their symmetry types [2].

The numbers of the Raman-active (A1g, E1g, E2g symmetries) and IR-active (A2u,E1u symmetries) modes for the nanotubes can be predicted by group theory, once thelattice structure and its symmetry are speci® ed. We list in table 2 the number andsymmetries of the Raman-active lattice modes for all the possible di� erent types ofcarbon nanotubes.

More speci® cally, we list below the number of modes with their correspondingsymmetry types for all types of nanotubes. For armchair nanotubes (with evenn ˆ 2j having Dnh symmetry) we write:



Gvib2j ˆ 4A1g ‡ 2A1u ‡ 4A2g ‡ 2A2u ‡ 2B1g

‡ 4B1u ‡ 2B2g ‡ 4B2u ‡ 4E1g ‡ 8E1u ‡ 8E2g

‡ 4E2u ‡ ¢ ¢ ¢ ‡ 4E… j¡1†g ‡ 8E… j¡1†u: …15†In equation (15), we assume that j is even. If j is odd [such as for …n ;m† ˆ …6 ;6†], the4 and 8 are interchanged in the last two terms in equation (15). For zigzag nanotubes(with even n ˆ 2j having Dnh symmetry) we write:

Gvib2j ˆ 3A1g ‡ 3A1u ‡ 3A2g ‡ 3A2u

‡ 3B1g ‡ 3B1u ‡ 3B2g ‡ 3B2u

‡ 6E1g ‡ 6E1u ‡ 6E2g ‡ 6E2u

‡ ¢ ¢ ¢ ‡ 6E… j¡1†g ‡ 6E… j¡1†u: …16†For armchair and zigzag nanotubes (with odd n ˆ 2j ‡ 1 having Dnd symmetry) wewrite:

Gvib2j‡1 ˆ 3A1g ‡ 3A1u ‡ 3A2g ‡ 3A2u

‡ 6E1g ‡ 6E1u ‡ 6E2g ‡ 6E2u

‡ ¢ ¢ ¢ ‡ 6Ejg ‡ 6Eju: …17†And ® nally for chiral nanotubes we write:

GvibN ˆ 6A ‡ 6B‡ 6E1 ‡ 6E2 ‡ ¢ ¢ ¢ ‡ 6E…N=2†¡1: …18†

Using the information in equations (15)± (18), we obtain the informationsummarized in table 2 for the number and symmetry type of the Raman and IRactive modes for achiral (i.e. armchair and zigzag) and chiral nanotubes. Among theachiral nanotubes, only armchair nanotubes with even number indexes …2j ;2j† havedi� erent numbers of Raman and IR modes from the others, such as armchairnanotubes with odd number indexes …2j ‡ 1 ;2j ‡ 1† or zigzag nanotubes …n ;0†. Fromtable 2 we deduce the remarkable Raman and infrared selection rules for SWNTs,that the numbers of Raman and infrared-active modes do not depend on thenanotube diameter and chirality, though the total number of ® nite frequency phononmodes …6N ¡ 4† is very di� erent for di� erent chiralities and diameters. Group theoryselection rules indicate that there are only 15 or 16 Raman-active modes and 6 to 9IR-active modes for a single-wall carbon nanotube, despite the large number of


Table 2. Number and symmetries of Raman-active modes for different types of carbonnanotubes.

Nanotube structure Point group Raman-active modes IR-active modes

armchair (n; n) n even Dnh 4A1g ‡ 4E1g ‡ 8E2g A2u ‡ 7E1uarmchair (n; n) n odd Dnd 3A1g ‡ 6E1g ‡ 6E2g 2A2u ‡ 5E1uzigzag (n; 0) n even Dnh 3A1g ‡ 6E1g ‡ 6E2g 2A2u ‡ 5E1uzigzag (n; 0) n odd Dnd 3A1g ‡ 6E1g ‡ 6E2g 2A2u ‡ 5E1uchiral (n;m ) n 6ˆ m 6ˆ 0 CN 4A ‡ 5E1 ‡ 6E2 4A ‡ 5E1


vibrational modes. Even though group theory may indicate that a particular mode isRaman-active, this mode may nevertheless only have a small Raman cross-section.In fact, we have only six or seven intense Raman-active modes for any nanotubechirality.

Since the number of Raman-active and infrared-active modes for a givensymmetry category is independent of nanotube diameter, the dependence of aparticular vibrational mode on nanotube diameter can be investigated. Several ofthe mode frequencies and their Raman cross-sections are found to be highly sensitiveto the nanotube diameter, while others are not. Figure 16 shows the dependence ofthe frequency of the Raman-active modes on the nanotube diameter for armchairnanotubes, expressed in terms of their …n ;n† indices [60]. Here it is seen that the A1g

mode, which occurs at about 165 cm¡1 for an isolated (10,10) nanotube, is stronglydependent on nanotube diameter, while the modes near 1580 cm¡1 are not. Adiagram similar to ® gure 16 can be constructed for the infrared-active modes forthe armchair nanotubes, and also for Raman and infrared-active modes for zigzagand chiral nanotubes [1, 60].

Another factor which simpli® es the Raman spectra for SWNTs is the lowintensity of many of the Raman-active modes in ® gure 16. For example, ® gure 15shows only the particular Raman-active modes that are expected to have signi® cantintensity, based on calculations involving a bond polarizability model [2, 28].

The normal mode displacements for the seven Raman modes for a (10,10)nanotube which have a relatively large Raman intensity are shown in ® gure 15.The honeycomb lattice shown in ® gure 1 can be described in terms of two sublatticesconsisting of A and B atoms. In the higher frequency Raman-active A1g mode(1587 cm¡1 in ® gure 15), the A and B atoms move in opposite directions (out-of-phase) in the unit cell, while in the lower frequency Raman-active A1g mode


Figure 16. The armchair index n versus mode frequency for the Raman-active modes ofsingle-wall armchair …n;n† carbon nanotubes [28]. The nanotube diameter can befound from n using equation (2).


(165 cm¡1 in ® gure 15), the A and B atoms move in the same direction (in-phase) . Itis clear from ® gures 15 (e) to (g), that the higher-frequency modes are out-of-phasebetween nearest neighbour carbon atoms, while the lower-frequency modes of® gures 15 (a) to (d ) show in-phase motion. The out-of-phase motions observed in® gures 15 (e) to (g) are similar to the motion of the Raman-active E2g mode ofgraphite at 1582 cm¡1, which corresponds to C¡C bond stretching motions for oneof the three nearest neighbour bonds in the unit cell.

The basic motions of the atomic displacements of the normal modes shown in® gure 15 are independent of the chirality of the nanotube [2]. The normal modedisplacements illustrated in ® gure 15 show that the A1g, E1g and E2g modes havezero, two and four nodes around the nanotube z axis, respectively (see ® gures 15 (e),( f ) and (g) for out-of-phase motion, and (a), (b) and (c) for in-phase motion). Thesenormal mode patterns are very useful for the interpretation of the observed Ramanspectra, especially with regard to polarization phenomena (see section 4.7).

For the lower frequency Raman-active modes (below 500 cm¡1 for (10,10)nanotubes) , the mode frequencies ! shift systematically with increasing diameter,as shown in the log± log plot of ! in ® gure 17 as a function of the carbon nanotuberadius r ˆ dt=2 for …n ;m† in the range …8 µ n µ 10 ; 0 µ m µ n†. Figure 17 clearlyshows straight line dependences of log ! on log dt for all four low frequency Ramanmodes on this log± log plot, thus indicating a power law dependence of !…dt† on dt.No signi® cant chirality dependence is found for the mode frequencies for thesemodes, which is consistent with the fact that the energy gap of a semiconductingnanotube and the strain energy depend primarily on the nanotube radius and areonly weakly dependent on the chiral angle ³ [61, 62]. From the slopes of !…dt† for thisrange of dt (see ® gure 17), we conclude that, except for the lowest E2g mode, themode frequencies are approximately proportional to 1=dt. The frequency !2g…dt† ofthe lowest E2g mode has a predicted dependence of d¡1:95§0:03

t , which is approxi-


r [ � ]

10

100

1000

w [ cm

-1 ]

E2g

E1g

A1g

E2g

1 10

Figure 17. Log± log plot of the lower Raman mode frequencies (below 500 cm¡1 for (10,10)nanotubes), as a function of carbon nanotube radius r ˆ dt=2 [2, 43].


mately quadratic (i.e. 1=d2t ). The power law predicted for the A1g radial breathing

mode frequency !0RBM…dt† for an isolated SWNT valid in the range 0.6 nm µ dt µ

1.4 nm is

!0RBM…dt† ˆ !0

…10;10†d…10;10†

dt

1:0017§0:0007

; …19†

which is very close to linear (i.e. 1=dt ) and this relation is used by experimentalists asa secondary characterization tool for the diameter distribution in isolated SWNTsamples. When the SWNTs are in bundles, inter-tube interactions become important(see section 3.3 and section 4.1) and modify the dispersion relation for !RBM…dt†. Inequation (19) !0

…10 ;10† and d…10 ;10† are, respectively, the mode frequency and diameterof isolated (10,10) armchair nanotubes, which have been given by !0

…10 ;10†= 169 cm¡1

and d…10;10†= 1.375 nm [2] and are further discussed in section 4.1, in connection withradial breathing mode measurements.

3.3. Inter-tube or intra-bundle interactionsTo date, almost all the experimental data on SWNTs, and especially Raman

spectra, have been collected on bundles of nanotubes that are produced either in anelectric arc (EA) or by the pulsed laser vaporization (PLV) method. High resolutionTEM and X-ray di� raction studies on these nanotube bundles show that theycontain on the order of 100 well-aligned nanotubes in a close-packed triangular (orhoneycomb) lattice [6]. The intra-bundle or inter-tube interactions that arise in thislattice are usually assumed to be weak and are approximated by a van der Waalsinteraction, similar to the coupling between adjacent graphene layers in 3D crystal-line graphite. However, the e� ects of this coupling on the vibrational [63± 65] andelectronic states [66± 69] have nevertheless been shown theoretically to be of su� cientstrength to signi® cantly a� ect the physical properties of SWNTs, as measured in thelaboratory.

The e� ect of these interactions on the electronic structure has been calculated [66,68] in the local density approximation. These calculations [66± 68], however, did notconsider a possible structural distortion, or hexagonal faceting of the nanotubecross-section due to the inter-tube interactions, but the calculations did allow forcharge redistribution within a three-dimensional unit cell containing adjacentnanotubes. For semiconducting and metallic nanotubes with diameters near thatof a (10,10) nanotube, a small perturbation is found in the energy di� erence betweenthe mirror image singularities in the one-electron density of states (DOS) in thevalence and conduction bands (Eii…dt† in ® gure 8), and for metallic nanotubes, apseudo-gap (¹100 meV) was predicted to open in the DOS at the Fermi energy [66,68, 69]. An absorption band at ¹120 cm¡1 has been reported recently for a `buckypaper’ sample made from PLV-derived nanotubes [70]. The authors cite thisobserved absorption band as evidence for this predicted pseudo-gap, although theobserved gap is a factor of eight smaller [70] than theoretically predicted. Any shift inthe spacing between the 1D electronic singularities in the valence and conductionbands (see section 2.2) will, of course, also a� ect the quantitative interpretation ofthe resonant Raman scattering spectra (see section 4). That is, the laser excitationenergy to promote resonant Raman scattering from a particular nanotube will beshifted depending on whether that nanotube is isolated or is located in a nanotubebundle [66, 71].



The e� ect of inter-tube interactions on the vibrational modes of SWNTs was ® rstreported by Venkateswaran et al. [64] and this e� ect was used to interpret thepressure-induced shift in both the Raman-active tangential and radial breathingmodes (see section 4.4). They used a generalized tight-binding molecular dynamics(GTBMD) scheme [72], including the e� ects of an externally applied pressure, todetermine the structural relaxation, or faceting, within the bundle. Next, using thecoordinates obtained from this calculation, and an optimized set of van der Waalsinteraction parameters between C atoms on neighbouring nanotubes, the e� ect ofthe inter-tube interactions and applied pressure on the vibrational spectrum wascalculated, as discussed in section 4.4. Here, we focus on the theoretical resultsrelevant to the corrections to the radial breathing mode (RBM) at zero appliedpressure due to inter-tube interactions [63, 64, 73, 74].

The GTBMD calculations [63] show that for bundles of nanotubes havingindividual nanotubes with diameters in the range 0:7 < dt < 1:5 nm, the simple1=dt frequency dependence of the radial breathing mode frequency that was derivedfor isolated nanotubes can be approximately amended by the addition of a nearlyconstant 7% upshift, independent of dt in this range of dt [42, 63]. The frequencyupshift is the result of an additional tube-wall restoring force due to next-neighbour,nanotube ± nanotube interactions. The relation between !RBM and dt is important forthe characterization of the diameter distribution of actual nanotube samples, sinceRaman spectroscopy provides a quick and convenient technique for such samplecharacterizations (see section 4.1).

The general features of the model calculations of this perturbation e� ect [63]starts with the calculation for the relaxed structure and the vibrational spectrum ofthe (10,10) nanotube lattice at zero applied pressure [42]. The (10,10) nanotubelattice with D10h symmetry relaxes by a distortion of the circular nanotube symmetrydue to inter-tube interactions to a monoclinic system with space group P2=m. Byputting the uncoupled (10,10) tubes on a triangular lattice associated with thesymmetry of the nanotube bundle, a formal lowering of the symmetry (orthorhombicspace-group Cmmm ) occurs relative to the D10h symmetry of the individualnanotubes. The resulting lowering of the symmetry to P2=m symmetry, changes,in principle, the Raman activity of the vibrational modes of the SWNTs. However,since the inter-tube coupling is weak, it is not expected that many new Raman modeswill actually be observed, and the spectral intensity should still be dominated by theselection rules for the isolated SWNT (see section 3.1). The calculations show thatthe relaxed coupled nanotube lattice di� ers only slightly from the lattice found foruncoupled nanotubes, with lattice constants for the coupled lattice being a ˆ 16:68 AÊ

and b ˆ 16:72 AÊ (perpendicular to the nanotube axis) and c ˆ 2:453 AÊ (parallel tothe nanotube axis) and cell angles ¬ ˆ ˆ 908 and ® ˆ 59:908, using the usualnotation for the monoclinic system. The shortest carbon± carbon inter-tube distanceis found to be relatively small (between 3.17 and 3.26 AÊ ) and this small distance isattributed to the curved nanotube surface and the variation in inter-tube distance isattributed to the missing 6-fold symmetry of the individual nanotubes. (Within thesame theoretical model, the shortest carbon± carbon inter-layer distance in graphite iscalculated to be 3.38 AÊ as compared to the experimental value of 3.35 AÊ [63].)

The e� ect of the van der Waals inter-tube coupling on the vibrational spectrum!…k† is displayed in ® gure 18 for (10, 10) SWNTs on a triangular lattice. As can beseen by the presence of weakly negative ! eigenvalues in ® gure 18 (a), the relaxationprocedure did not result in an overall stable solution to the eigenvalue problem,



insofar as some of the normal modes in the G ± M ± K± G plane (wave vectorsperpendicular to the tube axis), appear as decaying modes (complex frequencies)which are displayed in ® gure 18 (a) as modes with slightly negative frequencies. Thisweak instability is a sign that the chosen set of basis atoms is too small to allow for afull relaxation to a stable con® guration. By doubling the unit cell along an in-planelattice vector, part of the instabilities (around the M-point) were removed. However,relaxing the structure with the doubled unit cell did not noticeably change (except fora folding of the phonon branches) the dispersion relations for the modes withfrequencies >20 cm¡1 that were obtained using the smaller unit cell. The authors [63]therefore concluded that the main e� ects of inter-tube coupling are well describedwithin the small set of basis atoms.

The inter-tube coupling was found to lead to additional dispersion in the planeperpendicular to the nanotube axis, and this is shown along the G± M ± K± Gdirections in ® gure 18 (a). The dispersion curves !…k† for the uncoupled-tube lattice,which corresponds to the isolated nanotube discussed in section 3.1 and is shown for


Figure 18. Phonon dispersion curves in the low frequency region for (a) the 3-dimensional(10, 10) nanotube triangular lattice and (b) an isolated (10, 10) nanotube. G ± A is thek-space direction parallel to the nanotube axis, while the dispersion relations !…k† forwave vectors perpendicular to the nanotube axis are shown in the G ± M ± K± G portionof (a). Solid arrows point to the G-point frequency of the breathing modes in therespective systems, while open arrows point to the E4g modes, as discussed in the text.The carbon atom displacements corresponding to the modes labelled ¬, and ® inthe dispersion curves are given in (c) [63].


comparison in ® gure 18 (b), would be ¯ at (zero slope) in the region G± M ± K± G.Compared to the uncoupled or isolated nanotube, the frequencies of some of themodes for a nanotube within a bundle are shifted, and modes that were 2-folddegenerate for the isolated nanotube are split for a nanotube in a bundle. At the Kor M points, where neighbouring nanotubes vibrate with opposite phases, thefrequencies of some of the modes are close to the G-point frequencies of thecorresponding vibrations in the isolated nanotube and this occurrence gives rise tocoupled modes. As expected, the in¯ uence of the inter-tube coupling on thevibrational spectrum was found to diminish with increasing phonon frequency,and this coupling is essentially negligible above ! ˆ 500 cm¡1. The lowest, non-zerofrequency mode at the G-point with ! º 10± 12 cm¡1 is the librational mode (rigidsmall angle rotations) of the individual nanotubes in the lattice, obtained within adensity functional theory framework [66, 68].

The A1g symmetry breathing mode for the individual nanotube is marked as ¬ in® gure 18 (b) at its zone centre frequency value (which was found to be 156 cm¡1 forthis calculation for a (10, 10) nanotube [63]). The van der Waals coupling betweenthe nanotubes was found to upshift this mode by ¹14 cm¡1 (9%) as marked by in® gure 18 (a) and the corresponding normal mode displacements are shown in® gure 18 (c). However, a frequency shift is not the only e� ect that occurs when thetriangular lattice describing a nanotube bundle is formed. The lower symmetry of thetriangular crystalline lattice allows the A1g mode of the isolated nanotube to mixwith one partner of a doubly degenerate E4g pair of the normal modes for the (10,10)nanotube in equation (15). As a result, an additional pair of modes is introduced,( ;® ), which resemble the displacements of the breathing mode for the isolatednanotube. The corresponding displacement patterns are shown in ® gure 18 (c). Whilemode ® has larger displacement amplitudes in the region between two neighbouringnanotubes, mode shows larger displacements towards the (open) channels betweenthe isolated nanotubes and thus has the lower frequency. The ® mode is, therefore,more strongly a� ected by the van der Waals interaction. We further note that thesecond partner of the E4g pair (open arrows in the region between ® gures 18 (a) and(b)) is not a� ected by the inter-tube coupling. According to these calculations [63] themixing of the A1g breathing mode with an E`g mode occurs for all …n;m† nanotubelattices, with ` assuming the largest possible even numbered value for the speci® cnanotube type. For example, from equation (15) we see that ` ˆ 4 for a (10, 10)nanotube. However, calculations on larger nanotubes show that the amount ofmixing and the e� ect of the inter-tube coupling decreases with increasing nanotubediameter (see ® gure 19) [63]. Even though the basis atoms of the nanotube triangularlattice relax slightly away from their corresponding positions in the individualnanotube, the projections h j¬i and h® j¬i of the normal displacement vectors canbe de® ned in terms of the coupled modes projected onto the zero coupling breathingmode. It is found that h j¬i2 ‡ h® j¬i2 º 1 with h j¬i2 ˆ 0:89. If it is assumed thatthe Raman intensity of the breathing mode is carried over from the case of theisolated nanotube to the case of the coupled nanotubes, then it was proposed that theratio h® j¬i2=h j¬i2 corresponds roughly to the intensity ratio one expects for the twoRaman peaks for ® and (up to a frequency factor ! =!® ) and, in the case ofresonance Raman scattering, a small di� erence in the resonance condition for and® is expected to occur. As stated above, the intensity ratio I…! †=I…!®† decreases withincreasing nanotube diameter (as shown in the inset to ® gure 19 (b)).



The frequency shift and the mode mixing is found to be a general feature of thebreathing mode in these model calculations and to be dependent on the nanotubediameter [63]. Figure 19 (a) shows the calculated mode frequency as a function ofnanotube diameter dt for an isolated (10, 10) nanotube in comparison to the samenanotube in a triangular lattice of (10, 10) nanotubes in a nanotube bundle. Thedotted curve gives the results for a simple spring model calculation that uses theradial breathing mode frequency for an isolated nanotube and an inter-tube coupling


Freq

uenc

y [ c

m]

Freq

uenc

y [ c

m]

to

)to

-1-1

tube diameter (d ) [A]

tube diameter (d [A]

dt

[A]o

d

o[A]

t

Figure 19. Nanotube diameter dependence of various radial breathing mode frequencies. (a)Comparison between !0

RBM…dt† for isolated nanotubes, !RBM…dt† for the samenanotubes in the triangular lattice of a bundle, and the same nanotubes with an inter-tube van der Waals coupling equal to that between adjacent graphene sheets in thegraphite lattice. The inset gives the upshift D !…dt† of the radial breathing mode in thebundle relative to the isolated nanotube as a function of dt. (b) Comparison betweenthe dt dependence of the and ® modes in the nanotube bundle (see ® gure 18) and of!E`g …dt† for the largest ` value possible for a given …n;m† nanotube (see section 3.2).The inset gives the predicted Raman scattering intensity ratio of the (solid line) andof the ® (dashed line) radial breathing modes for the nanotubes in a nanotube bundleas a function of dt [63].

(a)

(b)


of 120 cm¡1 which is set equal to the coupling energy between adjacent graphenelayers in graphite. The inset shows the frequency upshift D ! of the radial breathingmode relative to that for an isolated (10, 10) nanotube as a function of nanotubediameter dt. Also shown in ® gure 19 (a) is !RBM…dt† for an isolated (10, 10) nanotubeto which the van der Waals interaction potential appropriate to graphite is added. Itis seen that the graphite van der Waals potential underestimates the tube± tubeinteraction for small dt but greatly overestimates the interaction for large dt.

Figure 19 (b) shows the nanotube diameter dependence of the frequencies of thebreathing modes and ® for a variety of individual nanotubes and nanotube latticesof the armchair, zigzag and chiral type. The indices of the respective nanotubes arelisted on top of the ® gure centred along the x axis at the value of the correspondingdiameter. With and without the inter-tube coupling, the !RBM for and ® displayonly a dependence on the diameter dt, not on the chirality of the nanotubes. Thediameter dependence of the frequency of the and ® modes for the nanotubebundles shown in ® gure 19 (b) indicates that the frequency di� erence between the and ® modes decreases as the nanotube diameter increases. The frequency of the ®

mode for small diameter nanotubes is close to that of the E`g mode, where ` is themaximum allowed integer for the nanotube modes discussed above and in section 3.2.The inset to ® gure 19 (b) gives the relative intensities of the mode and the ® mode asa function of nanotube diameter dt, and shows that for dt < 1:2 nm, it is only the mode that should be observed, while for large diameter nanotubes dt ¶ 2:0 nm, the and ® modes are expected to have equal intensity. Another calculation [74], based ona pair potential approach, has con® rmed that an upshift in frequency is expected,and agreement was obtained for the approximate magnitude of the upshift for a(10,10) nanotube lying within a nanotube bundle. These authors [74] predict theupshift to be small for small diameter nanotubes and larger as the nanotube diameterincreases.

The e� ect of inter-tube interactions on the mode frequencies over a wide range ofnanotube diameters and chiralities has been calculated on the basis of a tight bindingformalism [65] showing very large e� ects for the lowest frequency modes with radialmode displacements, and much smaller e� ects for high frequency modes withpredominantly tangential displacements. The largest e� ects of the inter-tube inter-action are predicted for the lowest frequency E2g mode (see ® gure 15).

In summary, the inter-tube interactions are expected to lower the symmetry ofthe individual isolated SWNTs, which gives rise to mode frequency shifts associatedwith distortion e� ects so that the radial breathing mode may show some chiralitydependence. Line broadening e� ects in the Raman spectra can arise from these inter-tube interactions, because of the di� erence in local environments between ananotube in the centre of the bundle and a nanotube at the edge of a nanotubebundle.

4. Raman spectra of single-wall nanotubes

Raman spectroscopy provides a particularly valuable tool for examining themode frequencies of carbon nanotubes with speci® c diameters and thereby evaluat-ing the merits of various theoretical models for the 1D phonon dispersion relationsfor carbon nanotubes. The status of this research direction is summarized. Becauseof the remarkable properties of SWNTs, the Raman spectra for these 1D systemsgive rise to new physical phenomena, not seen previously in Raman spectroscopy.



The status of research along these lines is also reviewed. The use of Ramanspectroscopy to characterize nanotube samples in terms of the diameter distributionof the nanotubes in the sample is reviewed in section 4.1, for studying the 1D electrondensity of states in resonant Raman (section 4.2) experiments through the electron±phonon coupling mechanism [28], and in selectively exciting resonances in metallic orsemiconducting nanotubes (section 4.2). The temperature dependence (section 4.3)and the pressure dependence (section 4.4) of the Raman spectra are then reviewed,followed by a comparison of the unique di� erences between the Stokes and anti-Stokes spectra (section 4.5) and how this comparison is used to elucidate thedi� erences between metallic and semiconducting SWNTs, exploiting phenomenalinked to low dimensionality. The use of surface enhanced Raman spectroscopy(SERS) to elucidate the Raman spectra of SWNTs and new aspects of the SERStechnique are reviewed in section 4.6. Polarization phenomena and spectral featuresassociated with K-point phonons are reviewed in sections 4.7 and 4.8, respectively,while section 4.9 treats overtones and combination modes, and section 4.10 reviewsprogress on the study of doped SWNTs.

Most of the early experiments on the vibrational spectra of carbon nanotubeswere carried out on multi-wall carbon nanotubes, which were too large in diameterto observe detailed quantum e� ects associated with the 1D electronic dispersionrelations and observed through the resonant Raman e� ect [75, 76]. The ® rst Ramanstudy to show a clear signature for single-wall carbon nanotubes [77] was carried outon samples containing only a small concentration of single-wall nanotubes, andhaving a wide distribution of diameters and chiralities, so that only features pertinentto phonon modes near 1580 cm¡1, which are only weakly dependent on nanotubediameter, could be observed [77]. This early work successfully identi® ed character-istic features in the Raman spectra for single-wall nanotubes because of the relativelylarge scattering cross-section of the nanotubes due to a resonant Raman processinvolving van Hove singularities in the 1D density of electronic states discussed insection 2.2.

The ® rst de® nitive study of the Raman spectrum of single-wall carbon nanotubes(SWNTs) was carried out on a sample containing ropes of single-wall carbonnanotubes with a narrow diameter distribution in the 1.2± 1.4 nm range, andprepared by a laser vaporization technique [6]. A Raman spectrum taken on thissample at a 514.5 nm laser excitation wavelength is shown in ® gure 20 [28, 50, 78].Prominent in this spectrum are a number of features near 1580 cm¡1 (see inset to® gure 20), and a strong feature at ¹186 cm¡1. From ® gure 15, we see that there arethree mode frequencies near 1580 cm¡1 which have mode symmetries A1g, E1g andE2g for armchair nanotubes, each mode frequency being almost independent ofnanotube diameter. A similar behaviour is found for the Raman band near1580 cm¡1 for zigzag and chiral nanotubes [41]. The atomic displacements associatedwith the normal modes near ¹186 cm¡1 and ¹1580 cm¡1 for a (10, 10) nanotube areshown in ® gure 15 [2].

In contrast to the high frequency band near 1580 cm¡1, the feature near¹186 cm¡1, which is identi® ed with an A1g radial breathing mode (!RBM), is stronglydependent on the nanotube diameter, as shown in ® gure 16. Calculations [42, 43]show high intensities for this radial breathing mode and for the tangential modes(!tang ) near 1580 cm¡1, in agreement with experimental Raman spectra. The otherRaman-active modes (see ® gure 16) are predicted to have low Raman cross-sections(see ® gure 21), also in agreement with experiment (® gure 20). Bond polarizability



calculations further predict that the relative intensities of the weaker Raman-activefeatures in the experimental spectra of ® gures 20 and 21 can be increased by makingmeasurements on carbon nanotubes of small length (e.g. 100 nm), small compared toan optical wavelength for laser excitation [79]. The weak features in the Ramanspectrum in ® gure 20 at about 1350 cm¡1 (associated with resonant Ramanscattering of phonons near the K-point in the 2D Brillouin zone) and at about1740 cm¡1 are discussed further in sections 4.8 and 4.9, respectively.

One-dimensional quantum e� ects are observed in the Raman spectra of single-wall carbon nanotubes through the resonant Raman enhancement e� ect betweenincident or scattered photons and the electronic transition between the van Hovesingularities in the 1D density of states in the valence and conduction bands of thenanotubes (see section 2.2). This resonant Raman e� ect can be seen experimentallyby measuring the Raman spectra at a number of laser excitation energies, as shownin ® gure 22 [28].

By comparing the various Raman spectra in ® gure 22, which were taken atdi� erent laser excitation energies Elaser, we see large di� erences in the vibrationalfrequencies and intensities of the strong A1g radial breathing mode, consistent with aresonant Raman e� ect involving nanotubes of di� erent diameters. These experi-mental observations [28] also provided the ® rst clear con® rmation for the theoretical


Figure 20. Experimental Raman spectrum taken with 514.5 nm (2.41 eV) laser excitationfrom a sample consisting primarily of single-wall nanotube bundles with diameters dtnear that of the (10, 10) nanotube (dt ˆ 1:36 § 0:20 nm) [28].


predictions about the singularities in the 1D electronic density of states of carbonnanotubes through study of both the radial breathing mode features (section 4.1)and the tangential mode features (section 4.2). This con® rmation was soon corrob-orated by a more direct measurement of the 1D electronic density of states by STM/STS spectroscopy, as discussed in section 2.2 [13, 14].

Because of this strong resonant enhancement e� ect, only a small concentration ofsingle-wall nanotubes in a sample containing other carbon forms can give rise tospectral features showing the characteristic sharp doublet structure in the 1570±1600 cm¡1 range [77]. Resonant enhancement in the Raman scattering intensity fromcarbon nanotubes occurs when the energy of the incident or the scattered photoncorresponds to a transition between the sharp features in the one-dimensionalelectronic density of states of the carbon nanotubes, as shown in ® gure 7. Theresonant enhancement e� ect is so strong that it has been possible to observe up to® fth order Raman scattering in SWNTs (e.g. up to Raman shifts of 6885 cm¡1 using488 nm laser excitation) [80].

Since the energies of these sharp features in the 1D electronic density of states arestrongly dependent on the nanotube diameter, a change in the laser frequency bringsinto resonance a di� erent carbon nanotube with a di� erent diameter that satis® es thenew resonance condition, as shown in ® gure 8. For example, the model calculation in® gure 7 shows that the (10, 10) armchair nanotube would be expected to be resonant


Figure 21. Raman spectra (top) of a rope of single-wall carbon nanotubes taken with514.5 nm excitation at ¹2 W cm¡2. The features in the spectrum denoted by thesymbol `*’ are assigned to second-order Raman scattering. The four bottom panelsare the calculated Raman spectra (based on a bond polarizability model) forarmchair …n;n† nanotubes, n ˆ 8 to 11 and the strongest Raman-allowed features areindicated by vertical bars. The arrows in the panels indicate the calculated positionsof the remaining weak, Raman-active modes [28].


at a laser frequency of 1.9 eV, while the (9, 9) nanotube would be resonant at 2.1 eV,considering only the incident photon.

4.1. Radial breathing mode phenomenaTheoretical calculations have shown that the frequency !0

RBM of the perfectlysymmetric radial breathing mode of an isolated single-wall carbon nanotube(section 3.1) has a particularly simple dependence 1=dt on the nanotube diameter,as given by equation (19) [2, 41, 50, 51, 81]. This fact, coupled with its large(resonantly enhanced) Raman cross-section for the radial breathing mode, thesensitivity of !RBM to charge transfer and to tube± tube interactions makes the radialbreathing mode a valuable probe for the structure and properties of SWNTs andSWNT-based materials.


Figure 22. An experimental room temperature Raman spectra for puri® ed single-wallcarbon nanotubes excited at ® ve di� erent laser excitation wavelengths. The laserwavelength and power density for each spectrum are indicated, as are the vibrationalfrequencies (in cm¡1) [28]. The equivalent incident photon energies for the laserexcitation are : 1320 nm ! 0.94 eV; 1064 nm ! 1.17 eV; 780 nm ! 1.58 eV; 647.1 nm !1.92 eV; 514.5 nm ! 2.41 eV.


For an isolated nanotube of any chirality …n ;m†, the mode frequency !0RBM has

been shown theoretically to exhibit a simple inverse diameter relationship, i.e.!0

RBM ’ 224=dt for !0RBM in cm¡1 and dt in nm, where the proportionality factor is

somewhat sensitive to the details of the calculation [81]. The results of two suchcalculations are shown in ® gure 17 and in ® gure 23. The plot in ® gure 23 includesarmchair, zigzag and chiral nanotubes with diameters in the range of currentexperimental interest. A strong dependence of the mode frequency on nanotubediameter is in fact found for all the Raman-active low frequency modes. However,the actual measurements are usually made on nanotubes which are within a bundleof nanotubes. As discussed in sections 3.3 and 4.4, this simple relationship for !0

RBMmust be corrected for weak inter-tube interactions within a bundle to obtain themeasured mode frequency !RBM. Theoretical calculations predict that these inter-actions are responsible for a 6± 21 cm¡1 frequency upshift, depending on the detailsof the calculation (see sections 3.3 and 4.4) [63, 64, 73, 74].

However, it should be realized that the resonant enhancement of Elaser withelectronic interband transitions (see section 2.2) for the radial breathing mode(RBM) (and other modes as well) will fall o� rapidly with increasing nanotubediameter, and therefore this Raman mode is not as sensitive a probe for the case oflarger diameter nanotubes. For example, Raman studies [28, 82, 83] using severalexcitation energies Elaser were carried out on samples containing bundles ofnanotubes with a bimodal distribution of nanotube diameters, having populationmaxima in the diameter distribution near that of (10, 10) and (20, 20) nanotubes [82].The Raman spectra did not show any evidence for the RBM feature of the (20, 20)


Figure 23. Calculated dependence of the frequency of the A1g radial breathing mode onnanotube diameter dt plotted on a log± log plot. This plot is for isolated single-wallcarbon nanotubes [81]. It has been shown theoretically that the tube± tube interactionswithin a nanotube bundle upshift these radial breathing mode frequencies by ¹7±10% for dt µ 1:5 nm [63].


nanotubes [82], consistent with the expectation that the cross-section for the RBMfalls o� rapidly with increasing tube diameter. These larger diameter nanotubes wereproduced by the coalescence of two smaller nanotubes in a bundle during a hightemperature heat treatment (HTT) ¹14008C in 1 atm. of hydrogen. HRTEM imagesshowed directly that ¹30% of the nanotubes in such a bundle were nearly doubled indiameter. However, the nanotube perfection of these `coalesced’ nanotubes could notbe veri® ed, and it is possible that defects in the nanotube walls might have broadenedthe radial breathing mode line to the point that it could not be observed [82].

State-of-the-art Raman samples have a narrow distribution of diameters andchiralities, which depend sensitively on the catalysts that are used in the synthesisand the growth conditions, especially the growth temperature T g [9, 84± 86]. Forexample, when 1.2 wt% of Ni/Co catalyst is used in the sample synthesis at atemperature of 1150¯C in 500 torr Ar, the diameter range is 1.0± 1.4 nm, while thinnernanotubes with diameters of 0.8± 1.0 nm are obtained in the case of a 2.4% Rh/Pdcatalyst with growth at 1100¯C [87]. Even when the same catalysts are used, a highergrowth temperature gives a larger diameter. At a higher temperature, however,atomic vibrations prevent the formation of relatively unstable pentagonal rings,compared with hexagonal rings, so the lower growth temperatures favour the growthof smaller diameter nanotubes [2].

The approximate !0RBM ¹ 1=dt relationship has been used to estimate the range

of SWNT diameters in nanotube bundles produced by the pulsed laser vaporization(PLV) process [81], by measuring the e� ect of the growth conditions and especially ofthe average temperature T g in the growth zone on the tube diameter distribution andon the number of nanotubes per bundle. Results from measurements of the lowfrequency Raman spectra collected on SWNT mats grown by pulsed laser vaporiza-tion (PLV) at various furnace temperatures T g are displayed in ® gure 24. The spectra(a)± (d ) were recorded using 1064 nm (Elaser ˆ 1:17 eV) excitation, and therefore theT g-dependence of the diameter distribution of semiconducting nanotubes is exploredin this ® gure. As can be seen, with increasing growth temperature, the sharp peaksidenti® ed with radial breathing modes from nanotubes with various diameters shiftto lower frequency as T g is increased, indicating that the nanotube diameterdistribution shifts toward larger values with increasing temperature T g in the growthzone. This shift in dt was con® rmed using HRTEM and X-ray di� ractioncharacterization measurements [81]. The spectra (e), ( f ) and (g) in ® gure 24, takenat several values of Elaser (488 nm (2.54 eV) ; 514.5 nm (2.41 eV) ; 647 nm (1.92 eV) ; and1064 nm (1.17 eV)), provide a characterization of the nanotube diameter distributioncontained in the sample prepared at T g ˆ 10008C [81].

More precise use of the radial breathing mode (RBM) frequency to probe theproperties, or diameter distribution, of SWNTs embedded in bundles of SWNTsrequires that the inter-tube interactions be considered [64, 73]. The expression linkingthe radial breathing mode !RBM to the nanotube diameter dt has been reported to bewell approximated by the expression

!RBM ˆ D !RBM ‡ !0RBM ˆ D !RBM ‡ !0

…10;10†d…10 ;10†=dt ; …20†

where !0RBM is the radial breathing mode frequency for an isolated SWNT, D !RBM is

a frequency upshift which is a constant (see ® gure 19 (a)) for nanotube diametersnear to that of a (10, 10) armchair nanotube d…10 ;10†, and !0

…10;10† is the radialbreathing mode frequency of an isolated (10, 10) nanotube. The calculated values of



these parameters vary from one research group to another, but some typical valuesare: D !RBM ˆ 14 cm¡1 [64], 6.5 cm¡1 [73] and 6 cm¡1 [65], and !0

…10 ;10†d…10;10† ˆ224 cm¡1nm [64], 232 cm¡1nm [73] and 214 cm¡1nm [65]. Since these parametervalues were obtained by ® ts of the measured !RBM to equation (20) for a limitedrange of nanotube diameters (0.7± 1.5 nm), the values of these parameters should beconsidered in pairs. Interestingly, both sets of parameters listed above arrive at aboutthe same RBM frequency (equation (20)) for a (10, 10) nanotube in a bundle, i.e.176± 177 cm¡1. Similar results, but with somewhat larger upshifts D !RBM, were alsoobtained by Henrard et al. [74]. It has been found [42, 63, 74] that D !RBM decreasessigni® cantly with increasing nanotube diameter dt (see inset to ® gure 19 (a)).However, as already discussed, tubes with dt > 2 nm are not expected to exhibit alarge (resonantly enhanced) Raman cross-section, and their RBM band is thereforemuch more di� cult to detect.


Figure 24. Room temperature RBM spectra for bundles of SWNTs produced by pulsedlaser vaporization using an Fe/Ni catalyst in a carbon target. Spectra (a)± (d ) arecollected at ® xed laser excitation energy (1.17 eV; Nd:YAG) from samples grown atT g ˆ 780, 860, 920 and 10008C, respectively. Note that the spectral weight shifts tosmaller RBM frequencies with increasing growth temperature, T g, indicating that dtincreases with increasing T g. The intensities and frequencies of the RBM bands inspectra (e) ± (g) collected from the same sample (T g ˆ 10008C) but with di� erent laserexcitation energies (488 nm (2.54 eV) ; 514.5 nm (2.41 eV) ; 647 nm (1.92 eV) ; and1064 nm (1.17 eV)) are quite di� erent, demonstrating how di� erent diameter tubes areexcited as the excitation energy Elaser changes [81].


Alvarez et al. [73] used the radial breathing mode to evaluate the e� ect of sulphur(S) on the synthesis of SWNTs in a solar-powered oven, and the results of that studyare shown in ® gure 25, where we display room temperature RBM spectra of SWNTbundles synthesized in a solar oven using a Ni± Co± S catalyst. The RBM spectra forthese solar energy-derived nanotubes show a very broad range of RBM frequencies,indicating that the diameter distribution of these solar oven nanotubes is alsobroader than, for example, the nanotube distribution usually obtained from thePLV or electric arc synthesis methods. The spectra in ® gure 25 were collected atElaser ˆ2.54 eV (top), 2.41 eV (middle) and 1.92 eV (bottom), and the scale for dt

shown at the top of the 2.54 eV excitation spectrum is calculated according toequation (20) using D !RBM ˆ 6:5 cm¡1 and !0

…10;10†d…10 ;10† ˆ 232 cm¡1nm [73]. AsElaser is varied, di� erent nanotubes within the sample have interband transitions (see® gure 8) in resonance with the incident or scattered photons, thereby accounting for


Figure 25. Raman spectra for a SWNT sample prepared from a catalyst mixture Ni± Co± Susing solar energy and using Elaser excitation at 2.54, 2.41 and 1.92 eV. Spectra for theradial breathing mode band are shown on the left and for the tangential stretch G-band are shown on the right. The top scale indicates the nanotube diameterscorresponding to the RBM frequencies on the lower scale by use of equation (20)[73].


the di� erent spectra shown in ® gure 25. By carrying out measurements of !RBM formany Elaser values, the diameter distribution of the nanotube sample can beestimated [84± 86, 88± 92]. This procedure turns out to be a practical method forcharacterizing SWNT samples to be used for the measurement of many physicalproperties. In such characterization studies it is important that RBM spectra betaken with several di� erent laser excitation energies Elaser.

4.2. T angential stretch modesThe features in the Raman spectra of single-wall carbon nanotubes in the range

1550¡1600 cm¡1 (see ® gure 20) are identi® ed with the tangential stretch G-bandmodes and can be understood by zone-folding of the 2D graphite phonon dispersionrelations [93] as discussed in section 3.1, and therefore show a number ofcharacteristics that relate to the low dimensionality of the carbon nanotube. Thedashed lines in ® gure 26 are theoretical LO and TO energy dispersion curves ingraphite plotted along G± M in the 2D graphite Brillouin zone as a function ofwavevector in units of 109 cm¡1. For armchair nanotubes, the G ± M directioncorresponds to the 2D wave vector for the graphene sheet directed along the equatorof the nanotubes for n ˆ 1 and n ˆ 2 (® rst harmonic vibration) , respectively.Theoretical plots in ® gure 26 are made for the LO …n ˆ 1†, TO …n ˆ 1†, and TO(n ˆ 2) phonons, which were identi® ed with the A1g, E1g and E2g Raman modes,respectively [93]. For the TO …n ˆ 2† curve, the plot in ® gure 26 is made against q=2.Because of the small size of the nanotube diameters of SWNTs, there are only a fewatoms along the circumference of the nanotubes, and consequently there are only afew allowed wavevectors. The application of Born± von Karman boundary con-ditions to zone-folded phonons requires an integral number p of wavelengths ¶ to ® t


Figure 26. Plot of the observed Raman frequencies for the tangential G-band modes takenwith 2.41 eV laser excitation energy for several di� erent single-wall carbon nanotubesamples, prepared with di� erent catalysts under di� erent growth conditions yieldingdi� erent mean nanotube radii r ˆ dt=2. The dashed lines are theoretical LO and TOenergy dispersion curves for 2D graphite along G ± M in the 2D Brillouin zone as afunction of q and q=2 for the n ˆ 1 and n ˆ 2, respectively. Solid dots and the symbol`T’ correspond to the observed frequencies of the peaks and shoulders in the Ramanspectra, respectively [93].


into the circumference of the nanotube (to set up a standing wave pattern) . Thus theboundary conditions require that p¶ ˆ pdt, where ¶ ˆ 2p=q, and p ˆ 0 ;1 ;2 ; . . .. Theallowed wavevectors for nanotubes are therefore q ˆ 2p=dt, so that the magnitude ofthe lowest allowed non-zero wavevector will depend on the nanotube diameter dt.Therefore, the measured frequencies for the tangential phonon modes for di� erentdiameter nanotubes should provide experimental points on the dispersion relations,which collectively can be used to plot out the phonon dispersion curves near the G-point.

In ® gure 26, the observed Raman spectral frequencies are shown for three single-wall nanotube samples prepared with di� erent catalysts and under di� erent growthconditions, yielding nanotubes with di� erent mean diameters (radii) and di� erentdistributions of diameters (radii). Speci® cally the catalysts and resulting averagenanotube radii …r ˆ dt=2 are: (a) Fe/Ni (r ˆ 0:55 nm), (b) Co (r ˆ 0:65 nm) and (c)La (r ˆ 1:0 nm), each with a distribution D r ˆ §0:1 nm. The distribution in theexperimental nanotube radii is inferred from use of the dependence of the radialbreathing mode frequency on nanotube diameter (!RBM / 1=dt ) and measurementsof !RBM for several laser excitation energies Elaser for isolated SWNTs (seesection 4.1). The data in ® gure 26 shows a clear relationship between…1=r† ˆ …2=dt† (or wavevector) and the zone-folded phonon energy dispersion curves,arising from the small number of allowed wave vectors in the circumferentialdirection of the nanotube as discussed above. From ® gure 15, we see A1g and E2gmodes that are in the circumferential direction in the case of armchair nanotubes,while the E1g mode vibrations are along the nanotube axis. This work [93] suggeststhat the upshifted A1g mode is the most intense and the downshifted TO (n ˆ 1) E1g

mode is less intense. It is interesting to observe that the mean frequency between theTO (n ˆ 1) and LO (n ˆ 1) modes in ® gure 26 should be close to that of graphite(1582 cm¡1), consistent with experimental results on many nanotube samples.

Most studies of the tangential band for single-wall carbon nanotubes havefocused primarily on the dependence of the spectra on laser excitation energy, alongwith the spectral dependence on nanotube diameter. Shown in ® gure 27 are theRaman spectra between 1400± 1700 cm¡1 obtained from a SWNT sample with adiameter distribution dt ˆ 1:37 § 0:20 nm using di� erent laser excitation energies(0:94 µ Elaser µ 3:05 eV) [28, 93± 95]. The main features in these spectra are associ-ated with the tangential C± C stretch modes of the SWNTs (see ® gure 15). We seethat all the spectra obtained in ® gure 27 for Elaser < 1.7 eV or Elaser > 2.2 eV are quitesimilar. An example of the ® t of these spectra to a set of Lorentzian oscillators withpeaks at 1563 cm¡1, 1591 cm¡1 and 1601 cm¡1 is shown in ® gure 28 (a). In fact, allthe spectra for these ranges of Elaser (see ® gure 27) can be ® t by the same set ofLorentzian oscillators, with essentially the same oscillator strengths and linewidths.From the plot in ® gure 8, the spectrum in ® gure 28 (a) is identi® ed with a resonantRaman process involving interband transitions between singularities in the 1Ddensity of electronic states for semiconducting nanotubes in resonance with theincident and/or scattered photons.

In contrast, the spectra in ® gure 27 obtained in the narrow range 1:7 < Elaser µ2.2 eV are qualitatively di� erent, showing Raman bands that are broader, and thecentre of the band has downshifted by ¹30 cm¡1. All of these spectra have beendescribed in terms of three characteristic Lorentzian components that are not presentin the spectra for the semiconducting nanotubes excited by Elaser outside of thisenergy range. The additional characteristic components have peaks at 1515 cm¡1,



1540 cm¡1 and 1581 cm¡1, as shown in ® gure 28 (b), in addition to the threeLorentzian components identi® ed above with semiconducting nanotubes. By refer-ring to ® gure 8 these three additional broader Lorentzian components (at 1515 cm¡1,1540 cm¡1 and 1581 cm¡1) are identi® ed with metallic nanotubes, with Elaser for theincident and/or scattered photons being in resonance with the interband electronictransition for metallic nanotubes. The ® tting of the 1540 cm¡1 feature for the metallicnanotubes to a Breit± Wigner± Fano lineshape is discussed below [29, 73]. The inset to® gure 27 shows low resolution Raman spectra between 1300 and 2800 cm¡1 usingthree laser energies in the metallic tube transition region between the two regimeswhere the semiconducting nanotubes are resonant [94]. The spectra in the inset showthat the intensity of the second-order band at 2700 cm¡1 (see section 4.8) is almostindependent of Elaser, whereas the intensity of the tangential band is enhanced whenElaser approaches the metallic window near 2 eV.

Referring to ® gure 8 and section 4, we see that for Elaser ˆ 1:92 eV, a SWNTsample with dt ˆ 1:49 § 0:20 nm can be in resonance with the ES

33…dt† transition forsemiconducting nanotubes at the high end of the diameter distribution and with the


Figure 27. Raman spectra of the tangential G-band modes of carbon nanotubes withdiameters in the range dt ˆ 1:37 § 0:20 nm, obtained with several di� erent laser lines.The inset shows low resolution Raman spectra between 1300 and 2800 cm¡1 in therange of laser energies 2.00± 2.18 eV where the metallic nanotubes are dominant [94].


EM11…dt† transition for metallic nanotubes at the lower end of the diameter distri-

bution. In ® gure 8 we also can see that by going to SWNT samples with largerdiameter, the resonance of the interband transition for metallic nanotubes EM

11…dt†with the incident or scattered photons can more easily be limited to only metallicnanotubes at lower Elaser , or limited only to semiconducting nanotubes in resonancewith the ES

22…dt† transition. Figure 8 also shows that if the SWNT sample has a broaddistribution of nanotube diameters as in ® gure 25, the spectra for the tangential bandwill show resonant contributions from both semiconducting and metallic nanotubes.For example, the broad low frequency tails, especially for the traces atElaser ˆ 2:41 eV and 2.54 eV can be explained in terms of a resonant EM

11…dt†contribution from the small diameter metallic nanotubes contained in the sample.The strategic selection of Elaser to excite only metallic or only semiconductingnanotubes is discussed further in section 4.5.

For a metallic nanotube with a given diameter dt, the enhancement of its Ramanpeaks will occur every time the incident or scattered photon is in resonance with theenergy separation between the highest valence subband Ev1…dt† and the lowestconduction subband Ec1…dt†, so that E11…dt† ˆ Ec1 …dt† ¡ Ev1…dt† [42]. Since thesample used in obtaining the spectra in ® gure 28 contains metallic nanotubes withdiameters in the range 1.1± 1.6 nm (see inset of ® gure 29), the overall enhancement ofthe intensity of a particular Raman mode, such as the most intense feature identi® edwith metallic nanotubes which is at 1540 cm¡1, is given by the sum of thecontributions of each individual nanotube with a given diameter dt, weighted bythe distribution of diameters which is here assumed to be Gaussian. The expectedRaman intensity I…Elaser ;dt† for the dominant Lorentzian oscillator (at 1540 cm¡1)associated with the metallic nanotubes can then be written as [97]:

I…Elaser ;dt† ˆX

dt

A exp¡…dt ¡ d0†2

D d2t =4

" #

£ ‰…EM11…dt† ¡ Elaser†2 ‡ G2

e=4Š¡1

£ ‰…EM11…dt† ¡ Elaser ‡ Eph†2 ‡ G2

e =4Š¡1; …21†


1400 1450 1500 1550 1600 1650 1700Raman shift (cm?1)

0

Inte

ns

ity

(a

.u.)

1400 1450 1500 1550 1600 1650 1700Raman shift (cm?1)

0

Inte

ns

ity

(a

.u.)

1400 1500 1600 1700 1500 1600 1700Raman Shift (cm Raman Shift (cm

Raman Signal (a.u.)

-1-1)1400

1.49 eV 1.92 eV

1594

1569

1601

1591

16011581

1540

1515

)Figure 28. Lorentzian ® ts to the Stokes tangential bands for Elaser ˆ 1:49 eV and 1.92 eV for

nanotubes with dt ˆ 1:49 § 0:20 [96].


where d0 and D dt denote the centre and the half-width of the Gaussian distributiondt ˆ d0 § D dt of nanotube diameters, obtained by transmission electron microscopymeasurements, Eph is the average energy (0.20 eV) of the tangential phonons and thedamping factor Ge accounts for the width of the singularities in the electronic densityof states (DOS) and the lifetime of the excited state [42]. The factor A is for simplicityassumed to be a constant, which is equivalent to assuming that all the metallicnanotubes have equal Raman scattering cross-sections for the dominant oscillator at1540 cm¡1 (see ® gure 28). The intensity I…Elaser ;dt† in equation (21) is very sensitiveto Elaser for a given SWNT sample, and a plot of I…Elaser ;dt† versus Elaser, as in® gure 29, serves to de® ne the metallic window for a particular SWNT sample. Themetallic window is de® ned as the range in Elaser where metallic SWNTs within agiven sample contribute resonantly to the Raman spectra. The expression

EM11…dt† ˆ 6aC¡C®0

dt…22†

provides a good approximation to the centre of EM11…dt† for metallic nanotubes of

diameter dt and arbitrary chiral angle (see ® gure 8) [98, 99], where aC¡C is the nearestneighbour carbon± carbon distance and ®0 is the electronic overlap integral. In theabsence of a calibration of the absolute intensity of I…Elaser ;dt†, the intensity of thedominant component in the Raman spectra for semiconducting nanotubes (at¹1593 cm¡1) is used to normalize the intensity of the dominant component associ-ated with the metallic nanotubes (at ¹1540 cm¡1), since most spectral traces showing


Figure 29. The solid circles represent the intensity ratio of the Raman peaks at 1540 and1593 cm¡1, and the solid curve represents the ® t to the experimental data usingequations (21) and (22) [94]. The inset shows the distribution of diameters measuredby TEM [28] and the Gaussian ® t to the diameter distribution data [94].


contributions from metallic nanotubes (see ® gure 28 (b)) also show contributionsfrom semiconducting nanotubes. The resulting experimental intensity ratioI1540=I1593 is plotted in ® gure 29 versus Elaser. The best ® t of equations (21) and(22) to the plot of the intensity ratio I1540 =I1593 versus Elaser measurements in ® gure29 for a SWNT sample with d0 ˆ 1:37 nm and D dt ˆ 0:18 nm, was achievedfor ®0 ˆ 2:95 § 0:05 eV, Ge ˆ 0:04 § 0:02 eV, a full width of the distributionD EM

11…dt† ˆ 0:24 eV, and a mean value for the energy separation hEM11…dt†i ˆ

1:84 eV. These values for hEM11…dt†i and EM

11…dt† are in good agreement with thedirect measurements of EM

11…dt† by scanning tunnelling spectroscopy (STS) [13] andwith electron energy loss spectroscopy (EELS) experiments on individual metallicnanotubes [100].

At the present time, there is no detailed theory available for the explanationof the broad downshifted tangential bands associated with the metallicnanotubes, except for the general argument that these phenomena arise fromthe interaction between the conduction electrons and the tangential mode phonons[94], and perhaps involving phonon± surface plasmon interactions [101, 102].

By analysing Raman lineshapes associated only with semiconducting nanotubesor only with metallic nanotubes, it should be possible to determine whether any ofthe lineshape constituents should be ® t to a Breit± Wigner± Fano (BWF) lineshape. Itis presently believed that the resonance Raman lineshapes for semiconductingnanotubes can best be ® t by Lorentzian oscillators, while the lineshapes for thelow frequency features in the resonance Raman spectra for metallic nanotubes (thestongest feature near 1540 cm¡1 ) are better ® t by a BWF function [29, 73, 101, 102].Such a detailed lineshape study should provide information on the coupling of theelectrons to other excitations of the nanotube [29, 103], since a Breit± Wigner± Fanolineshape is an indication of a sharp Raman line interacting with a broad Ramancontinuum background, presumably associated with electronic excitations. We willfrequently return to the unusual low dimensional properties of the tangential G-bandfor carbon nanotubes in the remaining subsections of section 4.

4.3. T emperature dependence of the Raman spectraThe dependence of the Raman spectra on temperature provides valuable

information about anharmonic terms in the lattice potential energy. In general,the change in phonon frequency with temperature can be attributed to a purelythermal e� ect …@!=@T †V and to a volume-related e� ect @!=@V… †T @V =@T… †P throughlattice expansion phenomena, so that we can write

D ! ˆ @!

@T… †V

D T ‡ @!

@V… †T

@V@T… †

PD T : …23†

Direct measurements on highly oriented pyrolytic graphite (HOPG) [104]indicate that D !=D T is mainly due to a purely thermal e� ect and that thevolume-related e� ect is very small, suggesting that a similar behaviour could beexpected in SWNTs.

There are three studies thus far on the temperature dependence of the Ramanspectra in carbon nanotubes [92, 105, 106]. Two of the measurements are onMWNTs where the incident laser power level is used to vary the temperature, andthe anti-Stokes/Stokes Raman intensity ratio is then used to determine the nanotubetemperature. One preliminary measurement of the T dependence of the Raman



spectra on SWNTs is currently available, and this measurement was done on asample in a cryostat which allowed the control of the temperature between 5 and500 K and emphasis was here given to the dramatic increase in linewidth of the radialbreathing mode as the temperature was increased [92]. In experiments carried out incryostats, care must be exercised to provide He exchange gas to the sample to givegood thermal contact between the sample and the thermal reservoir to be sure thatthe temperature of the sample is properly controlled. Because of the special resonantRaman e� ect in carbon nanotubes, di� erent nanotubes are resonant in the Stokesand anti-Stokes processes at a given laser excitation energy, and therefore the anti-Stokes/Stokes intensity ratio does not in general provide an accurate determinationof the nanotube temperature, as discussed further in section 4.5. The reportedtemperature-dependent Raman spectra for MWNTs [105, 106] therefore needfurther re-examination, and consequently caution should be applied to use of thecurrent literature on the temperature dependence of the nanotube mode frequenciesand intensities [105, 106].

For all the Raman-active modes, the e� ect of increasing temperature is to lowerthe mode frequency [105, 106], consistent with results for graphite and other sp2

carbons [104, 107]. For example, values of the temperature dependence of theRaman-active G-band has been reported to be ¡0:011 cm¡1 K¡1 for highly orientedpyrolytic graphite, ¡0:028 cm¡1 K¡1 for disordered graphite, ¡0:027 cm¡1 K¡1 foractivated carbons [104] and ¡0:026 cm¡1 K¡1 for MWNTs [106]. The reporteddownshift in the frequency of the radial breathing mode for SWNTs is¹0.012 cm¡1 K¡1 [92]. The various authors ® nd di� erent relative magnitudes forthe downshifts in mode frequency for the various modes for the MWNTs, with¡0:019 cm¡1 K¡1 given for the 1340 cm¡1 D-band mode, ¡0:029 cm¡1 K¡1 given forthe 1630 cm¡1 D 0-band feature, and ¡0:034 cm¡1 K¡1 for the 2700 cm¡1 G 0-band.Although there is disagreement in the literature about the absolute magnitude of thetemperature dependence …@!=@T † for the various modes, there is general agreementabout the relative magnitudes of …@!=@T † for the tangential G-band, the D-band andthe G 0-band [105, 106]. The value of …@!=@T † for the radial breathing mode inMWNTs is found to be smaller than for the other modes [105].

The larger value of @!tang=@T for the tangential G-band in carbon nanotubesrelative to crystalline graphite can be attributed to the curvature of the nanotubes,leading to a mixing of a small out-of-plane force constant component into thetangential normal modes. Since the out-of-plane lattice constants and forceconstants of 2D graphite are strongly temperature dependent, the modes withdisplacements in the circumferential direction of carbon nanotubes are expected toexhibit a larger value for @!tang=@T relative to that for a graphene sheet. The largervalue of @!tang=@T for disordered sp2 carbons relative to graphite is related todefects. Within the Elaser range where only semiconducting nanotubes are inresonance, it is expected that once reliable experimental data on @!tang=@T areavailable for SWNTs, the mode frequencies for the tangential modes can be used asan indicator of the lattice temperature of the nanotubes.

Better measurements of …@!=@T † are needed for the various Raman-active modeson puri® ed, well characterized SWNTs where the temperature is measured directlyand the nanotube sample is in good thermal contact with the thermal reservoir. Itshould further be determined whether or not …@!=@T † for a given mode is the samefor semiconducting and metallic nanotubes.



4.4. High pressure e� ects on the tangential modesPressure-dependent studies are particularly interesting for single-wall carbon

nanotube bundles, because pressure can be used to decrease the separation betweenadjacent SWNTs and therefore to increase the inter-tube coupling. Thus pressureprovides a sensitive technique for measurement of the e� ect of inter-tube interactionsin typical SWNT samples used for the measurement of Raman spectra and the otherphonon-related measurements described in this review.

The pressure dependence of the radial breathing band and of the tangential bandof SWNTs (dt 1:31 § 0:07 nm) was measured up to 5.2 GPa using a diamond anvilcell and a methanol± ethanol (4:1 volume ratio) pressure transmission medium [64].The laser excitation (5 mW) was at 514.5 nm (Elaser ˆ 2:41 eV) inside the highpressure cell, corresponding to the regime where the interband transitions for thesemiconducting nanotubes (ES

33…dt† as shown in ® gure 8) should be in resonance withthe incident and scattered photons [64]. An increase in the nanotube mode frequencyis expected with increasing pressure due to the increase in the nearest neighbour C± Cinteractions and consequently in the force constants and also to an increase in theinter-tube interaction leading to a distortion of the nanotube cross-section. Theexperimental results for both the radial breathing (R) band and for the tangential(T 1, T 2 and T 3) bands (see ® gure 30) all show increases in mode frequency withincreasing pressure [64], in agreement with these intuitive predictions. For both thelow frequency and high frequency Raman bands, the scattering intensity decreasesand the linewidth increases with increasing pressure, and the Raman spectra arelargely (but not completely) recovered when the pressure is released. The intensitydecrease for the radial breathing mode with increasing pressure is quite severe, and


Figure 30. The pressure dependence of the room-temperature Raman spectra of SWNTbundles, for the low-frequency radial breathing band (a) and the high-frequencytangential G-bands denoted by T1, T2, T3 (b), for a sample with a mean diameterbetween that of a (9, 9) and a (10, 10) nanotube (dt ˆ 1:31 § 0:07 nm) [64].


® gure 30 shows that the mode is completely quenched at a pressure of 1.9 GPa [64].The pressure dependence of the mode frequency !…P† at 300 K was ® t by a relation

!…P† ˆ !…0†‡ aPP‡ bPP2 …24†

and the results for the coe� cients aP and bP are given in table 3, showing apredominantly linear dependence of the mode frequencies on pressure. Since thedisplacements of the radial breathing mode are more sensitive to the presence ofatoms on adjacent nanotubes, a larger percentage upshift of the radial breathingmode relative to the tangential mode is expected, in agreement with experiment (seetable 3). Calculations [65] of the expected pressure dependence of the radialbreathing mode, the low frequency E2g squash mode (17 cm¡1 in ® gure 15), andthe tangential G-band (1550± 1600 cm¡1 in ® gure 15) have been carried out. Thecalculations show that !RBM has an approximate linear increase with pressure up to¹10 cm¡1 GPa¡1, and the calculated pressure dependence of the tangential G-bandfrequency is also small at 15 cm¡1 GPa¡1, but much larger than the 5 cm¡1 GPa¡1

value observed for graphite [108]. These calculated values for SWNTs [65] areconsistent with the experimental measurements [64, 65].

To explain these results in more detail, model calculations were performed on(9, 9) SWNTs on the basis of 3 models : Model I in which the entire bundle ofnanotubes, arranged in a triangular lattice, is subjected to an external compression,Model II in which the individual nanotubes are each compressed symmetrically andinter-tube coupling is ignored, and Model III in which the pressure medium isallowed to penetrate into the interstitial channels between the nanotubes, therebyintroducing an angular dependence [64]. Models I and III included van der Waalscoupling and Model II did not. All model calculations are based on a generalizedtight binding molecular dynamics (GTBMD) calculation for an isolated nanotube[109], to which a Leonard-Jones potential is added to account for the inter-tubecoupling. A comparison between the model calculations and the experimentalmeasurements for the frequency shifts for each of the modes is shown in ® gure 31,where a reasonably good ® t to the experimental data is obtained with Model I. From® gure 31, we see that the radial breathing mode frequency for Model II whichneglects the inter-tube interaction !0

RBM is 14 cm¡1 lower in frequency than !RBM forModel I which includes this interaction, thereby giving a value for D !RBM inequation (20). It is thus concluded that for SWNTs with diameters in the rangebetween a (9, 9) and a (10, 10) nanotube (1:31 § 0:07 nm), the frequency of the radial


Table 3. Values of the ® tting parameters aP and bP inequation (24) obtained by ® tting the measured pressure-

dependent Raman frequencies in ® gure 30 [64].

!…0† aP bP(cm¡1 ) (cm¡1 GPa¡1) (cm¡1 GPa¡2)

186 § 1 7 § 1 ±1550 § 1 8 § 1 ±1564 § 1 10 § 1 ¡0:9 § 0:3

1593:0 § 0:7 7:1 § 0:8 ¡0:4 § 0:2


breathing mode for SWNTs in a bundle is given by equation (20), and D !RBM can bedetermined from calculations such as those shown in ® gure 31.

The model calculations also addressed the nanotube distortions introduced bythe inter-tube interaction. The resulting calculated pressure dependence of thegraphene lattice constant a along the nanotube axis and of the lattice constant ¹


Figure 31. The pressure dependence of the (a) radial and (b) tangential vibrational modefrequencies of a bundle of (9, 9) SWNTs. The points are experimental and the curvesare ® t to various models (see text). The pressure dependence of the E2g2 mode forgraphite is included [64].

Figure 32. The pressure dependence, as calculated from Model I, of (a) the lattice constantsand (b) the hexagonal distortion of the cross-section of an individual nanotube in thebundle [64]. The lattice constant a along the nanotube axis is normalized to a0 (atzero external pressure) which is close to the value 2.46 AÊ for a graphene sheet.Likewise ¹ and ¹ 0 are normalized to ¹0 (at zero applied pressure) which is close to1.72 nm as reported for X-ray di� raction measurements on SWNT bundles [6] (seetext).


for the nanotube bundle is shown in ® gure 32. As shown schematically by the inset to® gure 32 (b), the e� ect of pressure is to increase the inter-tube interaction within ananotube bundle and to cause some faceting and departures from a circular cross-section denoted by the equilibrium radius r0. Predictions for the ¯ attening e� ect,expressed by r…<†, is found to be about 1% per GPa of pressure as shown in® gure 32 (b), where the pressure dependence of r…>†=r0 and r…<†=r…>† are also plotted[64]. Polygonalization e� ects have been reported for carbon nanotubes not subject toexternal pressure using high resolution TEM studies [110± 112]. The e� ect of thispolygonalization on the electronic structure of SWNTs has been shown to createbandgaps in metallic nanotubes (such as (12, 0) nanotubes) , or to cause semiconduct-ing nanotubes to become metallic [113].

4.5. Anti-Stokes spectraStudy of the anti-Stokes spectra [104, 106, 107] (scattered photon having energy

Elaser ‡ Eph) in comparison to the Stokes spectra (scattered photon having energyElaser ¡ Eph) provides much new information about the resonant Raman spectra andthe electronic structure of single-wall carbon nanotubes, including new physics aboutanti-Stokes and Stokes scattering phenomena [96]. The additional informationprovided by study of both the Stokes and anti-Stokes spectra includes: (1)observation of resonant Raman scattering pro® les to which only metallic nanotubescontribute, and other scattering pro® les to which only semiconducting nanotubescontribute, thereby allowing a more meaningful analysis to be made of the lineshapesof resonance Raman phenomena in SWNTs; (2) observation of very di� erent Ramanspectra for the Stokes process as compared to that for the anti-Stokes process, whichis a new phenomenon for carbon-based materials, and relates to the unique 1Delectronic structure of SWNTs; (3) evidence that resonant Raman scattering isstronger for metallic SWNTs than for semiconducting SWNTs, thereby establishinga stronger electron± phonon coupling process for metallic SWNTs; (4) the ratio of theanti-Stokes to the Stokes intensities of the tangential band cannot be used todetermine the sample temperature T s, because di� erent nanotubes (e.g. metallicSWNTs in one case, and semiconducting SWNTs in the other case) can contribute tothe Stokes and anti-Stokes spectra.

The ® rst report of an anti-Stokes feature in the Raman spectra for carbonnanotubes was for the radial breathing mode for multi-wall carbon nanotubes wherethe same frequency (120 cm¡1 ) was found for both the Stokes and anti-Stokesfeatures. The relative intensities between the anti-Stokes and Stokes features IAS=IS

was used by these authors [105] in the traditional way to determine the sampletemperature using the relation

IAS=IS ˆ …!‡ !0†4

…! ¡ !0†4 exp …-h!0=kBT † ¡ 1‰ Š¡1; …25†

in which ! is the laser frequency and !0 is the phonon frequency. No anomalousbehaviour for the anti-Stokes spectra was reported by these authors [105]. The anti-Stokes to Stokes intensity ratio for the D-band, the tangential G-band, the D 0 bandat ¹1620 cm¡1 ; and the G 0-band (see section 4.8) were all used to determine thenanotube temperature (see section 4.3) for MWNTs [106].

A detailed study of the anti-Stokes spectra in comparison to the Stokes spectra ofSWNTs has recently been carried out [96]. Striking di� erences are observed between



the Stokes and anti-Stokes spectra of SWNTS, as shown in ® gure 33, where theStokes and anti-Stokes spectra from 1200± 2800 cm¡1 are presented for two di� erentlaser excitation energies Elaser (1.58 eV and 1.96 eV). These spectra include featuresassociated with the ® rst-order spectra (the tangential band in the region 1500±1600 cm¡1 and the D-band in the region 1280± 1330 cm¡1 ), and second-order features(combination bands at ¹1740 cm¡1 and in the 1860± 1925 cm¡1 range, and the G 0-band at 2540± 2620 cm¡1 which is an overtone of the D-band) . We discuss thesedi� erences in behaviour between the Stokes and anti-Stokes spectra in more detailbelow.

The large di� erences in lineshape between the tangential band for the Stokes andanti-Stokes spectra are shown more clearly in ® gure 34, at four di� erent Elaser valuesbetween 1.49 eV and 2.19 eV. Here the behaviour of the Stokes and anti-Stokesspectra for the tangential band changes as a function of Elaser and the spectra alsochange relative to one another as Elaser is varied. The di� erent characteristiclineshapes for the tangential band discussed below allow us to easily distinguishbetween metallic and semiconducting nanotubes. At Elaser ˆ 2:19 eV, the Stokes andanti-Stokes spectra in ® gure 34 are almost the same, and both are typical of resonantRaman spectra characteristic of semiconducting nanotubes (strongest feature at1591 cm¡1 ), while at Elaser= 1.92 eV, the Stokes and anti-Stokes spectra are verydi� erent from each other, the Stokes spectrum showing domination by metallicnanotubes (strongest feature at 1540 cm¡1 ), and the anti-Stokes spectrum showingdomination by semiconducting nanotubes. At Elaser= 1.58 eV both the Stokes andanti-Stokes spectra are both characteristic of metallic nanotubes, while atElaser= 1.49 eV, the anti-Stokes spectrum is dominated by metallic nanotubes andthe Stokes spectrum is typical of semiconducting nanotubes. Figure 35 shows themetallic windows for the Stokes and anti-Stokes processes for a SWNT sample witha diameter distribution dt ˆ 1:49 § 0:20 nm, as discussed below, and the metallic


1200 1600 2000 2400 2800Raman Shift (cm

?1)

632.8 nm(1.96 eV)

1315 17411921

26201586

1586

1319Anti Stokes

Stokes

1200 1600 2000 2400 2800Raman Shift (cm?1)

782.0 nm(1.58 eV)

1290 17391867

2566

1580

1537

1299Anti Stokes

Stokes

17351915

2602

1200 1600 2000 2400 2800 1200 1600 2000 2400 2800

Raman shift (cm -1) Raman shift (cm -1)

Figure 33. Stokes and anti-Stokes Raman spectra over a wide frequency range for SWNTswith dt ˆ 1:49 § 0:20 nm at Elaser ˆ 1:58 eV (782 nm) and 1.96 eV (632.8 nm) [96].


windows in ® gure 35 account for the observations in ® gure 34. Whereas for 2D and3D sp2 carbon materials, the Stokes and anti-Stokes tangential bands at a given Elaser

value are essentially identical, the unusual resonant Raman process for 1D carbonnanotubes gives rise to di� erences in the Stokes and anti-Stokes tangential G-bandspectra when one spectrum is excited with a laser frequency from within the metallicwindow discussed in section 4.2, and the other spectrum is not, in accordance with® gure 8. These di� erences between the Stokes and anti-Stokes spectra are unique toSWNTs in comparison to other sp2 carbon-based materials and arise from thedi� erences in the one-dimensional density of electronic states for metallic andsemiconducting nanotubes discussed in section 2.2. This observation allows selectionof Elaser (e.g. Elaser ˆ 1:49 eV) to resonantly excite only semiconducting SWNTs inthe Stokes spectra or only metallic nanotubes in the anti-Stokes spectra (see® gure 35). These di� erences between the Stokes and anti-Stokes spectra at a givenElaser is due to the unique resonant enhancement phenomena arising from the one-dimensional electronic (1D) density of states of carbon nanotubes. Figure 35 clearlyshows that only semiconducting nanotubes are in resonance with Elaser ˆ 2:19 eV,while at 1.92 eV metallic nanotubes are in resonance for the Stokes process andsemiconducting nanotubes for the anti-Stokes process. Furthermore, ® gure 35 showsthat at Elaser ˆ 1:58 eV, metallic nanotubes are in resonance for both the Stokes andanti-Stokes processes, and at Elaser ˆ 1:49 eV the semiconducting nanotubes are inresonance for the Stokes process and the metallic nanotubes for the anti-Stokesprocess, in agreement with the experimental observations in ® gure 34.

To compare the Stokes and anti-Stokes spectra for semiconducting and metallicnanotubes in more detail, we show in ® gure 36 a lineshape analysis for the tangential


1300 1400 1500 1600 1700 1800 1900Raman shift (cm?1)

0

Anti Stokes

1.49

1.58

1.92

2.19 eV

1540

1537

1582

1591

1300 1400 1500 1600 1700 1800 1900Raman shift (cm?1)

Stokes

1.49

1.58

1.92

2.19 eV

1592

1540

1540 1588

1591

1300 1500 1700 1900 1300 1500 1700 1900

Figure 34. Stokes and anti-Stokes Raman spectra for SWNTs of dt ˆ 1:49 § 0:20 nm takenat 4 di� erent values of Elaser to illustrate di� erences in spectral lineshape betweenthe Stokes and anti-Stokes spectra [96]. For Elaser ˆ 2:19 eV, both the Stokes andanti-Stokes processes are in resonance with semiconducting nanotubes; forElaser ˆ 1:92 eV, the Stokes process is in resonance with metallic nanotubes andthe anti-Stokes process is in resonance with semiconducting nanotubes ; forElaser ˆ 1:58 eV, both processes are in resonance with metallic nanotubes ; and ® nallyfor Elaser ˆ 1:49 eV, the Stokes process is in resonance with semiconductingnanotubes, and the anti-Stokes process is in resonance with metallic nanotubes. Thenanotube-speci® c resonant Raman process (see ® gure 8) associated with the specialcharacteristics of the 1D density of electronic states is responsible for the observeddi� erences between the Stokes and anti-Stokes lineshapes [96].


band at Elaser ˆ 1:49 eV and 1.92 eV, which for simplicity was carried out only interms of Lorentzian components [96]. For semiconducting nanotubes, the Stokesspectrum at Elaser ˆ 1:49 eV can be interpreted in terms of three characteristicLorentzian peaks: at 1569 cm¡1, 1594 cm¡1 and 1601 cm¡1 [94]. At the sameElaser ˆ 1:49 eV, the anti-Stokes (metallic) spectrum shows only three broad Lor-entzian components at 1515 cm¡1, 1540 cm¡1 and 1581 cm¡1, none of which arepresent in the Stokes (semiconducting) spectrum at the same Elaser value. It isinteresting that features associated with semiconducting nanotubes are absent fromthis anti-Stokes spectrum at Elaser ˆ 1:49 eV. In contrast, the Stokes tangential bandat 1.92 eV is mostly dominated by three Lorentzian components characteristic ofmetallic nanotubes, but also shows Lorentzian components identi® ed with semi-conducting nanotubes in resonance with ES

33…dt† at the large dt end of the diameterdistribution (see ® gure 8). Likewise the anti-Stokes (semiconducting) spectrum atElaser ˆ 1:92 eV has a long tail at low ! (below ¹1550 cm¡1), indicating a signi® cantcontribution from metallic EM

11…dt† nanotubes at the low end of the diameterdistribution (® gure 8). Although the low ! tails often seen in the semiconductingtangential mode spectra could be identi® ed with a Breit± Wigner± Fano lineshape[29], the explanation of these tails suggested by ® gure 8 is due to the presence of somemetallic nanotubes in the sample that are resonant with either the incident orscattered photons. Study of the anti-Stokes spectra accesses greater contributionsfrom the lower right quadrant of ® gure 8, thereby providing more accurateinformation about lineshapes for metallic and semiconducting tangential bands,because the Eii…dt† bands of points at constant dt in ® gure 8 are narrower and moreseparated from one another in this limit [30, 31]. Since the Stokes and anti-Stokesspectra in ® gure 36 taken at 1.49 eV have a common incident Elaser value, someexplanation is needed for the interpretation of the Stokes spectra arising fromcontributions of resonant semiconducting nanotubes and the anti-Stokes spectra


1.25 1.55 1.85 2.15

Laser Energy (eV)

AS

S

1.69 eV

Figure 35. The calculated laser excitation energy dependence of the intensity IM…Elaser ;dt†using equation (26) for the strongest Lorentzian oscillator (1540 cm¡1) for metallicnanotubes in a SWNT sample with dt ˆ 1:49 § 0:20 nm. The parameters used inequation (26) were determined from ® ts of experimental Stokes spectra to thisequation. These plots thus de® ne the metallic resonance window for Stokes (solidcurve) and anti-Stokes (dotted curve) scattering processes, where metallic nanotubesare in resonance with the incident or scattered photons. The arrow at 1.69 eV is at thecentre of the metallic window [96].


arising from contributions of resonant metallic nanotubes. This interpretation isbased on having the incident photons non-resonant with either the metallic, or thesemiconducting nanotubes in the SWNT sample (see ® gure 8). The special selectionof Elaser so that the incident photon is essentially non-resonant with either metallic orsemiconducting nanotubes, but the scattered photon is strongly resonant withmetallic nanotubes in the anti-Stokes process and with semiconducting nanotubesfor the Stokes process provides a useful method for determining the resonant Ramanlineshapes for semiconducting and metallic nanotubes more accurately. Work todate suggests that the observed lineshapes are well ® t by Lorentzian oscillators forthe case of semiconducting nanotubes, as was done in the analysis of ® gure 36,though for metallic nanotubes the strong feature near 1540 cm¡1 appears to be better® t by a Breit± Wigner± Fano lineshape, as was reported by various groups [29, 90,101,102].

The di� erences in the metallic window for the Stokes and anti-Stokes processesfor SWNTs can be readily explained in terms of the Raman scattering intensity ofmetallic nanotubes:


1400 1450 1500 1550 1600 1650 1700Raman shift (cm?1)

0

Inte

ns

ity

(a

.u.)

1400 1450 1500 1550 1600 1650 1700Raman shift (cm?1)

0

Inte

ns

ity

(a

.u.)

1400 1450 1500 1550 1600 1650 1700Raman shift (cm

?1)

0

Inte

ns

ity

(a

.u.)

1400 1450 1500 1550 1600 1650 1700Raman shift (cm

?1)

0

Inte

ns

ity

(a

.u.)

anti-Stokes

anti-Stokes

Stokes

Stokes

1.49 eV

1.92 eV

1.49 eV

1.92 eV

1515

1581

1582 1540

151515881557

15911567

16011581

1540

1400 1500 1600 1400 1500 16001700 1700

1400 1500 1600 1700 1400 1600 1700

1569

1594

1601

1500

Raman shift (cmRaman shift (cm

Raman shift (cm

-1 -1

-1-1 ))

Inte

nsi

ty (

a.u.

)

Raman shift (cm

Inte

nsi

ty (

a.u

.)

))

Figure 36. Lorentzian ® ts to the Stokes and anti-Stokes tangential bands for Elaser ˆ 1:49 eVand 1.92 eV and dt ˆ 1:49 § 0:20 nm [96]. It is of particular interest that only metallicnanotubes are in resonance for the anti-Stokes process at Elaser ˆ 1:49 eV, and onlysemiconducting nanotubes are in resonance for the Stokes process at Elaser ˆ 1:49 eV,thus allowing more accurate lineshape analyses to be carried out.


IM…Elaser ;dt† ˆX

A exp¡…dt ¡ d0†2

D d2=4

( )

£ ‰…EM11…dt† ¡ Elaser § Eph†2 ‡ G2

e =4Š¡1

£ ‰…EM11…dt† ¡ Elaser†2 ‡ G2

e =4Š¡1 …26†

in which IM…Elaser ;dt† is the scattering intensity for the Stokes process (‡ sign) andfor the anti-Stokes process (¡ sign) for metallic nanotubes in resonance with theEM

11…dt† electronic transition between the highest lying valence band van Hovesingularity and the lowest lying conduction band singularity in their 1D electronicdensity of states [1, 94]. The remaining parameters in equation (26) have the samede® nitions as the corresponding parameters in equation (21). The quantity A inequation (26) is a dimensionless factor proportional to the e� ective cross-section fora particular phonon mode for the Stokes process and also includes a multiplicativeBoltzmann factor exp …¡Eph=kT † for the anti-Stokes process [94, 114]. The factor Awill be di� erent for the various oscillators which were used in the lineshape analysisof ® gure 36, because of the di� erences in the normal mode displacements andsymmetry classi® cations associated with the various oscillators.

A plot of IM…Elaser ;dt† for the Stokes spectra, based on a ® t of equation (26) tomeasurements on a SWNT sample with dt ˆ 1:49 § 0:20 nm at many laser energiesElaser is given in ® gure 35 (solid curve) [94, 114], along with the correspondingpredicted IM…Elaser ;dt† curve for the anti-Stokes spectra (dotted curve) using thesame Stokes parameters. The central point denoted by an arrow accurately gives thecentre of the metallic window for nanotubes with diameter dt, which occurs at1.69 eV for a SWNT sample with dt ˆ 1:49 § 0:20 nm [96] and at 1.80 eV for aSWNT sample with dt ˆ 1:35 § 0:20 nm [115]. Using the formula EM

11…dt† ˆ6aC¡C®0=dt, valid for armchair nanotubes [30], yields a value of ®0 ˆ 2:89 eVassuming aC¡C ˆ 0:142 nm, in good agreement with ®0 ˆ 2:91 eV, obtained from® ts of Stokes spectra to equation (26) for SWNTs with dt ˆ 1:37 § 0:18 nm [94].Thus the carbon± carbon overlap energy ®0 for SWNTs is sensitively determined bythe centre of the metallic window and the dt value for the SWNTs, yielding a value of®0 ˆ 2:9 § 0:1 eV [30, 32], which is less than 10% lower than ®0 for graphite [116].

Di� erences between the Stokes and anti-Stokes spectra are also found (see® gure 33) for the D-band, G 0-band (see section 4.8), and the !tang ‡ 2!RBM

combination band features with regard to frequency and relative intensity, whilethe Stokes and anti-Stokes spectra for the !tang ‡ !RBM band at Elaser ˆ 1:96 eVexcitation are almost the same. No second-order features are observed in the anti-Stokes spectra above 1700 cm¡1 for Elaser ˆ 1:96 eV and this observation is explainedby a Boltzmann factor reduction in relative intensity, where it is noted that the anti-Stokes spectrum at 1.96 eV laser excitation is dominated by semiconductingnanotubes. The di� erent frequency shifts of the G 0-band features (see section 4.8)in ® gure 33 can be explained by the di� ering energies of the scattered photons whichare involved with the 2D resonance Raman enhancement e� ects associated with theK-point in the Brillouin zone [117]. For example, an expected frequency di� erence of34 cm¡1 between the anti-Stokes and Stokes G 0-band frequencies is predicted fromthe measured …@!=@Elaser† slope of 106 cm¡1/eV for the G 0-band dispersion [118],which is to be compared with the observed frequency di� erence of 36 cm¡1 shown in® gure 33 (also see section 4.8). The relative intensities of the G 0 features in the Stokes



and anti-Stokes spectra at Elaser ˆ 1:58 eV can only, in part, be accounted for by theBoltzmann factor exp…¡Eph=kT †, which a� ects the anti-Stokes intensity strongly,but not the Stokes intensity. The experiments [96] show that the G 0-band intensity atElaser ˆ 1:58 eV, where the metallic nanotubes dominate the anti-Stokes spectrum, ismuch greater than one would expect from the Boltzmann factor argument in relationto the Stokes intensity at Elaser ˆ 1:58 eV, where the spectrum is mostly dominatedby metallic nanotubes, but some semiconducting nanotubes might also contribute tothe spectra. This additional enhancement of the intensity of the G 0-band for themetallic nanotubes relative to semiconducting SWNTs implies a stronger electron±phonon coupling for metallic nanotubes. The second-order features identi® ed with!tang ‡ 2!RBM (see section 4.9) are observed at 1915 cm¡1 and at 1921 cm¡1 in theanti-Stokes (Stokes) spectrum at Elaser ˆ 1:58 eV (1.96 eV), respectively, and havescattered photons at 1.78 eV and 1.76 eV, thereby accounting for the similar phononfrequencies that are observed in the two spectra [96, 101].

The observation of anti-Stokes scattering intensity requires population of the ® rstexcited phonon state …n1†. The ratio n1=n0, where n0 is the zero phonon ground statepopulation for a phonon of energy -h!0 is

n1=n0 ˆ exp ‰¡-h!0=kT Š; …27†

so that the intensity ratio of the anti-Stokes IAS to the Stokes IS signals at Elaser ˆ -h!is expected to be as given in equation (25). Equation (25) is commonly used forordinary sp2 carbons to determine the sample lattice temperature from the anti-Stokes to Stokes Raman intensity. Since di� erent kinds of nanotubes are resonant inthe anti-Stokes and Stokes spectra at a given Elaser, this equation cannot be used todetermine the sample temperature in SWNTs. However, when both the Stokes andanti-Stokes spectra at a given Elaser are dominated by semiconducting nanotubes,equation (25) could be used to provide a reasonable estimate of the sampletemperature.

It is signi® cant that the 1540 cm¡1 feature has a large intensity in the anti-Stokestangential G-band spectrum at Elaser ˆ 1:58 eV, and the presence of the G 0-band andthe second-order features, respectively, at !M

tang ‡ !RBM and at !Mtang ‡ 2!RBM in the

anti-Stokes spectrum are all well resolved at Elaser ˆ 1:58 eV where the metallicnanotube contribution is dominant. It is also signi® cant that these second-orderfeatures are absent in the anti-Stokes spectrum at Elaser ˆ 1:96 eV where thesemiconducting nanotubes are dominant. These observations indicate that theresonance Raman process (electron± phonon coupling) occurs more strongly formetallic nanotubes than for semiconducting nanotubes.

4.6. Surface enhanced Raman spectra in carbon nanotubesIt is well established that the Raman signal from molecules can be enhanced by

many orders of magnitude when these molecules are adsorbed on metallic nano-structures [119, 120] through the surface-enhanced Raman scattering (SERS) e� ect[121, 122]. In fact, the SERS e� ect can provide enough sensitivity (signal enhance-ment factors of up to 1014 ) to measure the Raman spectrum of single molecules whenthese molecules are attached to silver nanostructured particles, due to the veryinhomogeneous ® eld distribution near these metal nanoparticles , and to theextremely large electromagnetic ® elds associated with the excited surface plasmons[123± 128].



This sensitivity of the SERS technique provided initial stimulus to the use of theSERS technique to reduce the SWNT sample size necessary for obtaining a Ramanspectrum, thereby reducing the inhomogeneous line broadening associated with thedistribution of nanotube diameters and chiral angles within a typical SWNT sampleused for Raman spectroscopy studies. The SERS e� ect has thus been applied tostudy the Raman e� ect in SWNTs and large enhancement factors in the Ramansignal have been reported [115, 129, 130]. Both Fourier transform SERS [129] andfrequency scanned SERS [115, 130] have been applied to study SWNTs, and bothgold and silver surfaces have been used for the metal substrates. The reason for usingdi� erent metal substrates is to provide a means for distinguishing between the twomain SERS enhancement mechanisms: the `electromagnetic’ enhancement mechan-ism which is not sensitive to the metal substrate species, and the charge transfer or`chemical’ mechanism which is sensitive to the choice of the metal substrate. Forsome Ag and Au nanostructured metal substrates that have been used in SERSnanotube experiments, there is a large overlap in the broad surface plasmonabsorption bands so that the SERS enhancement e� ects can be observed over abroad range of Elaser values. The sensitivity of the SERS probe can be especially highin the case of SWNTs, since the enhancement e� ects bene® t both from the strongresonance Raman e� ect associated with the van Hove singularities in the 1Delectronic density of states, and from SERS enhancement e� ects associated withstrong electrical ® elds at the nanotube± metal interface. These two e� ects can becombined multiplicatively for laser excitation wavelengths in the visible/nearinfrared spectral region. In this review of the SERS studies on SWNTs, a comparisonis made between the spectral features in the SERS spectra with the normal resonantRaman spectra taken at the same laser excitation energy Elaser. These comparisonsbetween the SERS and normal Raman spectra reveal qualitatively di� erentresponses between semiconducting and metallic nanotubes. For example, thesemiconducting nanotubes show basically the same Raman spectral shape in theSERS and resonant Raman spectra (RRS), except for a scaling factor in signal, whilethe metallic nanotubes show notable di� erences in spectral shape between the SERSand normal Raman spectra. These topics are further elaborated in this section.

In this review, the comparison between the SERS and normal Raman spectra is® rst made over a broad frequency shift range to show the big picture, followed by amore detailed comparison of the behaviour of the tangential G-band in the 1500±1650 cm¡1 range. Then the enhancement of the SERS signal with laser excitationpower is discussed and spectra taken under very high enhancement conditions arepresented. The comparison between the SERS and normal Raman spectra innanotubes reveals behaviours not present in conventional crystalline samples.

Figure 37 shows a comparison, over a broad frequency range from 500±3500 cm¡1, between the spectrum for normal resonance Raman scattering (RRS)at Elaser ˆ 1:96 eV (® gure 37 (a)) and the surface-enhanced resonant Raman spectra(SERRS) taken at the same Elaser value. These spectra were reported for bundles ofsingle-wall carbon nanotubes on a rough silver ® lm (® gure 37 (b)) and on a roughgold ® lm (® gure 37 (c)). The dominant feature in the spectra in ® gure 37 [115] isassociated with the ® rst-order tangential G-band occurring in the phonon frequencyrange 1500± 1650 cm¡1 and the lineshape for this band implies a strong resonantcontribution from metallic nanotubes. The features of lower intensity near1310 cm¡1 are identi® ed with ® rst-order Raman-active phonon modes derived fromzone-edge phonons near the K-point in the 2D graphene Brillouin zone [117] (see



section 4.8). Similar to assignments made for normal second-order resonant Ramanscattering, the feature in the 1730± 1740 cm¡1 range is tentatively associated with acombination mode [114] involving the tangential and radial breathing mode phonons(!tang + !RBM). The feature in the 1900± 1930 cm¡1 range is also tentativelyattributed to a combination mode [114] (!tang + 2!RBM), as discussed further insection 4.9. A weak feature near 2440 cm¡1 (only observed at higher magni® cation) isidenti® ed with a non-resonant overtone in the second-order spectrum of the K-pointvibration at about 1220 cm¡1 [114], as discussed further in section 4.8, and ® nally thestrong feature in the 2600± 2624 cm¡1 range is due to the G 0-band which occurs as anovertone of the D-band feature at twice the D-band frequency (see section 4.8).

Figure 37 shows that qualitatively similar spectral features are observed for theStokes process both for normal resonant Raman scattering (RRS) and for surface-enhanced resonant Raman scattering (SERRS). The di� erences in the detailedlineshape, peak frequencies, and relative intensities between the RRS and SERRSspectra are further discussed below, because these di� erences provide importantinformation about the electron± phonon coupling in SWNTs. It is also signi® cantthat the SERRS spectra for the Ag and Au ® lms in ® gure 37 are almost identical, asdiscussed further below. The features in ® gure 37 which show only a small shift inphonon frequency as a function of Elaser in the RRS spectra also show very smallfrequency shifts between features in the RRS spectra and in the correspondingSERRS spectra.

For laser excitation energies where only semiconducting nanotubes contribute tothe resonant Stokes spectra, the tangential phonon mode region yields Lorentziancomponents at 1563 cm¡1, 1591 cm¡1 and 1601 cm¡1 for both the normal resonantRaman spectra [94] and for the corresponding SERRS spectra, showing the samepeak frequencies and relative intensities for the three constituent Lorentzian com-


Raman shift (cm

Raman Intensity (a.u.)

)-1

500 1000 1500 2000 2500 3000 3500

1.96 eV

(a)

(b)

(c)

1900

1931

1730

1931

2600

2624

2624

15411588

1588

1307

1311

13151739

1741

1591

Figure 37. Normal resonant Raman scattering (RR) spectrum (a), and surface enhancedresonant Raman scattering (SERS) spectra of single-wall nanotubes (SWNTs)adsorbed on (b) gold and (c) silver island ® lms in the 500± 3500 cm¡1 range using laserexcitation at Elaser ˆ 632:8 nm (1.96 eV) [115].


ponents. The same conclusions are reached for other Elaser excitation energies forwhich only semiconducting nanotubes contribute resonantly to the anti-Stokes RRSand SERRS spectra.

For Elaser within the 1:8 < Elaser < 2:0 eV range, the RRS and the SERRS spectrafor a SWNT sample with dt ˆ 1:35 § 0:20 nm both can be simply described by theappearance of Lorentzian components at 1515 cm¡1, 1540 cm¡1 and 1581 cm¡1,which are identi® ed (see ® gure 8) with resonantly enhanced features from metallicnanotubes in resonance with the EM

11…dt† transition [94]. In this case, signi® cantchanges can be seen when comparing the RRS spectra to the corresponding SERRSspectra, as shown in ® gure 38 at Elaser ˆ 1:96 eV. First we note the similarity betweenthe SERS spectra for nanotubes on two di� erent metal substrates, Au and Ag,indicating that the dominant SERS enhancement factor is associated with theelectromagnetic mechanism which is not sensitive to the metal substrate species.Similar results are obtained for other Elaser values within the metallic window (see


1 4 0 0 1 4 5 0 1 5 0 0 1 5 5 0 1 6 0 0 1 6 5 0 1 7 0 00

( c )

0

( b )

1 4 0 0 1 4 5 0 1 5 0 0 1 5 5 0 1 6 0 0 1 6 5 0 1 7 0 00

(a )

1515

1591

1540

1515

15911540

1515

1540

1591

1400

1500 1600

1500

1500

1600

1600

(a)

(b)

(c)

1.96 eV

Raman shift (cm -1)

1400

1700

1700

1700

(Au)

(Ag)

RRS

SERRS

SERRS

1400

Figure 38. Deconvolved spectra of the tangential vibrational band obtained with laserexcitation at Elaser ˆ 1:96 eV for (a) normal resonant Raman spectroscopy, (b) SERSon an Au substrate and (c) SERS on an Ag substrate. For all three traces, aLorentzian lineshape analysis was made in terms of the same Lorentzian oscillators at1515 cm¡1, 1540 cm¡1 and 1580 cm¡1 for the metallic nanotubes and at 1563 cm¡1,1591 cm¡1 and 1601 cm¡1 for the semiconducting nanotubes. The two dominantcomponents in the metallic nanotube regime (at 1515 and 1540 cm¡1) and thestrongest component in the semiconducting nanotube regime (at 1591 cm¡1) areexplicitly labelled and the sums of the six Lorentzian components are also explicitlyshown by the dashed curves [115].


section 4.2). Figure 38 further shows that the contributions of certain Lorentziancomponents for the metallic nanotubes (1540 and 1515 cm¡1 components) are muchmore pronounced for the SERRS spectra than for the RRS spectrum atElaser ˆ 1:96 eV (and also at other values of Elaser within the metallic window),though the peak frequencies for the three metallic Lorentzian components appear tobe the same. The same conclusions are reached if we compare the RRS and SERRSanti-Stokes spectra when the incident and/or scattered photons are in resonance withmetallic nanotubes in the anti-Stokes process [115]. It is interesting that the relativeintensity of the Lorentzian component at 1581 cm¡1, which is also associated withmetallic nanotubes, is more similar between the RRS and SERRS spectra than arethe 1540 cm¡1 and 1515 cm¡1 features. As discussed below, the di� erent relativeintensities between the RRS and SERRS spectra for the various Lorentziancomponents (1515 cm¡1, 1540 cm¡1 and 1581 cm¡1) associated with metallic nano-tubes are attributed to two factors: (1) the much larger magnitude of the SERRScharge transfer enhancement factor in the metallic nanotubes, and (2) the strongerelectron± phonon coupling that occurs in metallic nanotubes, and both of theseprocesses are sensitive to the phonon mode symmetries and normal mode displace-ments [115].

We now discuss the very large SERS enhancement factors that occur in SWNTsand the unique aspects of the SERS phenomena in SWNTs relative to other systemsin which the SERS phenomenon is observed. Figure 39 shows a comparison betweenthe surface-enhanced (SERRS) and normal resonance Raman scattering (RRS) forboth the Stokes (S) and anti-Stokes (AS) processes in SWNTs (with dt ˆ 1:35§0:20 nm) [130]. The spectra are taken at Elaser ˆ 1:49 eV excitation and in the 1500±1650 cm¡1 phonon frequency range (tangential C± C stretching mode) [130]. In® gure 39 we see that the Stokes spectra at Elaser ˆ 1:49 eV are dominated bycontributions from semiconducting nanotubes, whereas the anti-Stokes spectra aredominated by contributions from metallic nanotubes [130], as discussed insection 4.5. Also shown in ® gure 39 are the peak heights PS and PAS for the Stokesand anti-Stokes signals which are plotted as a function of incident laser intensity IL

for Elaser ˆ 1:49 eV.Figure 39 shows that the RRS and SERRS Stokes spectra at Elaser ˆ 1:49 eV are

quite similar to one another, consistent with the above discussion of the RRS andSERRS spectra for semiconducting nanotubes. On the other hand, the RRS andSERRS anti-Stokes spectra which are, in accordance with ® gure 8, dominated bycontributions from metallic nanotubes, have some similarity to one another, butcertainly di� er in detail, and are very di� erent from their Stokes counterparts at thesame Elaser value for this SWNT sample (dt ˆ 1:35 § 0:20 nm), consistent with the® ndings in ® gure 38 and with the discussion in section 4.5.

In normal resonance Raman scattering experiments (RRS), the observation ofrelatively strong anti-Stokes signals are related to sample heating through thethermal excitation of a signi® cant population of the ® rst vibrational phonon level,whereas in the SERRS experiment for nanotubes, the heating of the nanotubes ispresumably very small due to the very high thermal conductivity of SWNTs and thee� cient heat transfer to the silver substrate. This can be concluded from ® gure 39(upper right), which displays a plot of the surface-enhanced Stokes signal versus IL

the excitation laser intensity at Elaser ˆ 1:49 eV. For IL between 2 and ¹5 MW cm¡2,the Stokes SERRS signal increases linearly with IL and no frequency shift(1592 cm¡1) is observed for the Stokes line of the semiconducting nanotubes up to



about 5 MW cm¡2, indicating very little increase in temperature [106], too little toresult in a measurable frequency shift, as is observed for example in the dependenceof the normal resonance Raman spectra (RRS) of SWNTs on laser intensity due toheating [106]. The Stokes shift in ® gure 39 (top) changes abruptly from 1592 cm¡1 to1588 cm¡1 at IL ¹5 MW cm¡2 and then to 1585 cm¡1 at IL ¹8 MW cm¡2. AtIL ¹10 MW cm¡2, destruction of the nanotubes starts. The abrupt shifts in theRaman frequency have been interpreted in terms of a mode switching between thevarious phonon modes (E2g, A1g and E1g ) that constitute the tangential band, and isfurther discussed in section 4.7 [130]. This mode switching can appear through aphonon± phonon coupling mechanism that becomes important at large values of IL[2]. The plot for PS versus IL in ® gure 39 shows a linear dependence of the StokesRaman signal PS on laser intensity IL, for each of the three regions of IL delineatedin ® gure 39, but the slopes of the PS versus IL curves in ® gure 39 are found to be


I (MW/cmL 2)I (MW/cmL 2)

1400 1500 1600 1700 1800

14001500160017001800

P AS SP

15411586

1592

15951580

1540

(arb

. uni

ts)

(arb

. uni

ts)

1592 cm

1585 cm

-1

-1

16001750 1450 1600

Ram

an S

igna

l (ar

b. u

nits

)

Raman Shift (cm )

Anti-Stokes

Stokes

Anti-Stokes

Stokes

-1

1450 1750

SERRSSERRS

RRSRRS

-11588 cm

4 8 12 4 8 12

Figure 39. Lower plots show SERRS and RRS anti-Stokes and Stokes spectra of SWNTsmeasured with Elaser ˆ 1:49 eV excitation. On the top of the ® gure are surface-enhanced anti-Stokes (left) and Stokes (right) signal heights PAS and PS plotted as afunction of excitation laser intensity IL for Elaser ˆ 1:49 eV, and using nanostructuredAg particles for the metal substrate. The lines display linear (right) and quadratic(left) ® ts to the experimental data for PS and PAS , respectively (see equations (28) and(29)). The centre of the Stokes SERRS band shifts abruptly from 1592 cm¡1 to1588 cm¡1 to 1585 cm¡1with increasing IL (indicated by the vertical dashed lines), asthe PS data changes from one slope to another. At high laser excitation intensity(IL ¹ 10 MW cm¡2), irreversible destruction of the carbon nanotubes starts [130].


di� erent for each of the three regions characterized by the constant mode frequenciesof 1592 cm¡1, 1588 cm¡1 and 1585 cm¡1 [130].

Complementary information is obtained by comparison of the anti-Stokes RRSand SERRS spectra. The anti-Stokes SERRS signal for Elaser ˆ 1:49 eV in ® gure 39has been identi® ed with metallic nanotubes (see ® gure 8) and appears as a broadband from 1540± 1580 cm¡1, and its signal PAS is nonlinearly dependent on laserintensity IL between 2 and ¹10 MW cm¡2 (see ® gure 39 top left). This nonlinearitycan be due to two e� ects: increasing temperature with increasing laser intensity, and/or vibrational pumping. Through the second mechanism, a very strong Raman e� ectpopulates excited vibrational levels in excess of the normal Boltzmann population[123]. Vibrational pumping can be observed only at extremely large e� ective Ramancross-sections and is an indication of a very high SERRS enhancement level[123,126]. The carbon nanotube system is an ideal system to observe such phenom-ena because of the very high thermal conductivity of carbon nanotubes, whichsuppresses heating e� ects, and their robustness at high temperatures which allowsvery high laser excitation intensities IL to be used before the nanotubes aredestroyed.

A simple theoretical estimate for the Stokes and anti-Stokes power levels PS andPAS can be derived from a rate equation model describing the population/depopulation of the ® rst excited vibrational level by SERRS Stokes and anti-Stokestransitions [123]. In the steady state, PAS and PS can be expressed as

PAS ’ exp…¡-h!tang

kT …nL††‡ ½1nL¼SERSM NMnL¼SERRS

M ; …28†

PS ’ NSCnL¼SERRSSC ; …29†

where nL is the photon ¯ ux density of the laser, IL ˆ nLElaser=c is the intensity of thelaser beam, c is the velocity of light, NM and NSC are the numbers of metallic (M)and semiconducting (SC) nanotubes in the vibrational ground state that interactwith the laser beam, ¼SERS

M is the e� ective non-resonant SERS cross-section for themetallic nanotubes, the superscript SERRS refers to the resonant surface-enhancedRaman process, !tang is the phonon frequency, ½1 is the phonon lifetime, and T is thesample temperature. The ® rst term in the prefactor in equation (28) in bracketscorresponds to the thermal population of the ® rst vibrational level for the tangentialphonon band. Whereas the Stokes power PS remains linearly dependent on IL up to5 MW cm¡2, the experimental anti-Stokes signal PAS was found to have a quadraticdependence on laser excitation intensity because of the nL dependence of the secondterm in the prefactor and the nL dependence of the common factor on the right ofequation (28).

Figure 39 (top left) shows a ® t of the experimental data to PAS in equation (28).The results of this ® t indicate that the large anti-Stokes signal in the SERRSspectrum cannot be explained by laser heating of the sample (the ® rst term in theprefactor), but rather vibrational pumping (the second term in the prefactor) wasfound to make the dominant contribution to the observed nonlinear dependenceof the anti-Stokes signal PAS on IL. This nonlinear vibrational pumping mechanismfor converting photon energy into phonons provides a mechanism for achievinglarge intensities for the anti-Stokes process for metallic SWNTs [130]. From the® tting parameters, the product of the cross-section and the lifetime for the non-resonant Stokes Raman process for metallic nanotubes is estimated to be



½1¼SERSM ¹ 5 £ 10¡28 cm2 s. Assuming a vibrational lifetime on the order of 10 ps for

the phonons of the metallic nanotubes (in good agreement with the measuredlinewidth in ® gure 40), an e� ective Raman cross-section on the order of about10¡16 cm2 is inferred. Starting from an estimated non-resonant Raman cross-sectionin the range 10¡30 ± 10¡28 cm2 for the metallic nanotubes [131], an e� ective cross-section of 10¡16 cm2 could be reached by SERRS enhancement factors on the orderof 1012 ± 1014, as are also observed in single molecule studies [123± 127]. Enhancementfactors of this magnitude provide hope for observing Raman spectra from very smallnumbers of nanotubes, perhaps down to a single nanotube. The large SERS cross-section ¼SERS

M is associated with extremely large ® eld enhancement e� ects that can,for example, occur on fractal Ag nanostructures , especially at longer wavelengths[128, 132].

The inset to ® gure 40 shows a typical collection of colloidal silver clusters ofnanoscale particles used for obtaining very large enhancement SERRS spectra,where an extremely low concentration of SWNTs (probably only a few bundles ofSWNTS) were deposited on Ag colloidal particles. SERRS spectra could bemeasured in only a few places on the entire sample, only when a SWNT happenedto be attached to a small colloidal Ag particle [130]. In these cases, weak SERRSStokes spectra were measured using 1 s collection times. Shown in ® gure 40 are a fewselected SERRS spectra taken at di� erent spots over the entire sample, whichconsisted of many colloidal silver clusters which had no nanotubes attached to themand a very few that had attached nanotubes. Of particular interest are the smalllinewidths observed in some of the spectra in ® gure 40 for the tangential band of theStokes spectra. The smallest linewidths are much smaller than those typicallymeasured in RRS or in SERRS from nanotube bundles ( D ! ¹25 cm¡1). Thenarrowest measured linewidth for the tangential band in ® gure 40 is 9.5 cm¡1, whichis very close to the expected natural linewidth of the tangential C± C stretching modein semiconducting SWNTs deduced from a natural full width at half maximum


1400 1450 1500 1550 1600 1650 1700 1750 18001400 1450 1500 1550 1600 1650 1700 1750 1800

1596 1591

15891596

= 14 = 27

= 17= 9.5 D wD w

1450 1600 1750 1450 1600 1750Raman Shift (cm )-1

Ram

an S

igna

l (ar

b. u

nits

)

100 nm

D w D w

Figure 40. Typical fractal colloidal silver clusters (see inset) and selected SERRS Stokesspectra of the tangential band, with a linewidth D ! (in cm¡1 ) as small as 9.5 cm¡1,collected from a very few nanotubes adsorbed at such a cluster area. The spectra weretaken at Elaser ˆ 1:49 eV laser excitation with a 1 mm spot size [130].


(FWHM) linewidth of 4 cm¡1 and from the theoretical prediction that the threeunresolved E2g, A1g and E1g mode frequencies [2], that contribute to the tangentialband at ¹1590 cm¡1, di� er from one another by only 6 cm¡1 (see ® gure 15). Theselinewidth comparisons indicate that nanotubes of a unique diameter (or possibly asingle nanotube bundle) contribute to the Raman signal in ® gure 40. The stronglycon® ned electromagnetic ® eld on a silver cluster may selectively probe small numbersof nanotubes adjacent to the interface. Even stronger SERRS enhancement isexpected at low Raman frequencies [128, 132]. Thus it may be possible for SERRSto reveal the radial breathing mode spectra for individual SWNTs, free from theinhomogeneous broadening e� ects observed in the radial breathing mode spectra innormal resonant Raman scattering [133].

As mentioned above, there are two major contributions to the surface enhance-ment of the resonant Raman scattering of carbon nanotubes adsorbed on metallicsurfaces: the `electromagnetic’ mechanism and the `chemical’ mechanism, based on acharge transfer between the metallic surface and the nanotubes. The `electromag-netic’ mechanism is due to the enhanced electromagnetic ® elds associated withsurface plasmons at or near nanometre size metallic silver and gold particles.Particularly strong ® eld enhancement (up to 10 to 12 orders of magnitude) can existfor metallic substrates showing fractal cluster structures [128, 132, 134]. Theoreticalcalculations predict that these SERRS-active substrates should exhibit a particularlystrong enhancement at longer wavelength (near infrared) excitation [132, 135]. Ingeneral, the electromagnetic enhancement mechanism is independent of the chemicalnature of the species and is expected to be of similar magnitude for semiconductingand metallic nanotubes, and also to be the same for all vibrational modes. Theexistence of such large enhancement levels has been shown for both semiconductingand metallic nanotubes [130]. The second enhancement mechanism, the charge-transfer or chemical e� ect, strongly depends on the electronic structure of thenanotubes and the electronic interaction between the nanotubes and the metalsubstrate. Therefore, we expect the charge transfer mechanism to sensitivelydi� erentiate between the very di� erent 1D density of states of metallic andsemiconducting nanotubes. Fitting equations (28) and (29) to the observed depen-dence of the signal intensities PS and PAS on laser power IL indicates that at thehighest IL values used, the charge transfer or chemical e� ect contributed about twoorders of magnitude enhancement to the signal for the metallic nanotubes and verymuch less for the semiconducting nanotubes [130]. The di� erent SERRS enhance-ment factors observed for the 1540 cm¡1, the 1515 cm¡1 and the 1581 cm¡1

Lorentzian components of the SERRS spectra for metallic nanotubes relative tothe RRS spectra are attributed to the charge transfer enhancement mechanism,which is operative for the metallic nanotubes, in addition to the strong electro-magnetic ® eld e� ect discussed above. Through the charge transfer or chemical e� ectwe can account for the di� erent behaviour between the SERRS spectra of metallicand semiconducting nanotubes, which is attributed to major di� erences in theirelectronic structures and in their 1D density of states near the Fermi level, and to thestronger electron± phonon coupling in metallic nanotubes compared to semiconduct-ing nanotubes.

4.7. Polarization studies in Raman scatteringThe fundamental structural anisotropy of carbon nanotubes, due to their high

aspect ratio (length/diameter) , suggests the importance of polarized Raman spectro-



scopy studies. Only limited experimental data [136, 137] and theoretical predictions[2, 42] are presently available regarding polarization phenomena in the resonantRaman scattering from carbon nanotubes. The success of the initial reports suggestthat this subject will be an active research area for future activity.

Lattice dynamics calculations based on a bond polarization model have beencarried out under non-resonant scattering conditions for both the relative intensitiesof various Raman-active modes in isolated (10, 10) armchair nanotubes for di� erentpolarizations and polarization geometries and for their angular dependence [2, 43,138]. Calculations for nanotubes with other …n ;m† values, such as for zigzag andchiral nanotubes, are needed as well as calculations for general …n;m† nanotubesunder resonant conditions, because most experimental polarization studies are likelyto be carried out under resonance conditions, and for samples containing nanotubeswith di� erent chiralities.

The calculated sample orientation dependence of the Raman intensity showsthat, not only the symmetry, but also the direction of the normal mode displace-ments, are important in determining the angular dependence of the polarizationintensities. As shown below, this angular dependence is special for carbonnanotubes, and can in fact be used to identify the symmetry assignments for someof the Raman-active modes.

The Raman intensities for the various Raman-active modes in carbon nanotubeshave been calculated at a phonon temperature T using a non-resonant bondpolarization theory where the temperature dependence of each normal mode isgiven by a Bose± Einstein distribution function for the phonons [2]. The eigenfunc-tions for the various vibrational modes were calculated numerically at the G-point(k ˆ 0). The mode intensities were speci® cally calculated for two possible geometriesfor the polarization of the light: the VV and VH con® gurations. In the VVcon® guration, the incident and the scattered polarizations are parallel to each other,while for the VH con® guration, the incident and scattered polarizations areperpendicular to each other. Generally, the cross-section for Raman scattering is afunction of the photon scattering angle. Since the formulae for the bond polarizationtheory consider only s-scattered waves [2], the calculated results cannot distinguishbetween forward and backward photon scattering. The orientation of the nanotubeaxis is speci® ed relative to the polarization direction V.

In ® gure 41, the calculated Raman intensities for the most important Raman-allowed modes are shown for (10, 10) armchair, (17, 0) zigzag and (11, 8) chiralnanotubes, all having similar radii, which are, respectively, 6.78 AÊ , 6.66 AÊ and 6.47 AÊ[2]. Results are shown in each case for the VV and VH con® gurations discussedabove. The calculated Raman intensities for nanotube bundles containing alignedSWNTs but oriented along random directions are found by averaging over thesample orientation of the nanotube axis relative to the Poynting vector of the light,in which the average is calculated by summing over many (¹50) sample orientationsusing the angular dependence results discussed below. As mentioned above, thepresently available polarization calculations (such as in ® gure 41) neglect correctionsdue to the inter-tube interactions within a nanotube bundle which lower thesymmetry in-plane normal to the nanotube axis, and therefore these predictionsshould be used with caution when interpreting Raman spectra for nanotube bundles.The calculations in ® gure 41 show that the most intense mode is the low frequencyradial breathing mode with A1g symmetry in the VV geometry. The two lowfrequency E2g modes have lower intensities and ® nally the low frequency E1g mode



is predicted to have even lower intensity in the VV con® guration. However, for theVH geometry, the E1g and E2g modes are all predicted to have a modest reduction inintensity relative to the VV con® guration, but the radial breathing A1g mode isexpected to have a very low intensity in the VH con® guration.

In contrast to the low frequency A1g radial breathing mode, the high frequencyA1g mode, at 1587 cm¡1 which is associated with the tangential band, does not showsuch a large suppression between the VV and VH geometries, because of theconnection of this A1g mode through zone folding to the Raman-active E2g modeof graphite which has intensity only in the VH geometry. It is also reasonable thatthe tangential motions of the higher frequency Raman modes of a nanotube shouldhave a large Raman intensity, because of the relation of these modes to the E2g modeof graphite. However, the directions of the carbon atom motions of the tangentialA1g mode are di� erent for armchair and zigzag nanotubes, since the CˆC bond-stretching motions can be seen in the horizontally and the vertically vibrating CˆCbonds for armchair and zigzag nanotubes, respectively. Thus the curvature of the


0 400 800 1200 1600

E1g

E2g

A1g

17

165

118 368

1585

1587

1591

E2g

A1g

E2g

E1g

VV

(10,10)0 400 800 1200 1600

E1g

E2g

A1g

17

118

165

368

1585

1587

1591

E2g

E2g

E1g

A1g

VH

0 400 800 1200 1600

E2g

E1g

A1g

19

123

173

386

1585

1586

1591

A1g

E2g

E1g

E2g

Ram

an in

tens

ity (

arb.

uni

ts)

Raman shift (cm-1)

(11,8)0 400 800 1200 1600

E1g

E2g

A1g

19

173

123 386

1585

1586

1591

E2gE1g

E2g

A1g

0 400 800 1200 1600

E1g

E2g

A1g

18

168

119 376

1587

1590

1571

E2g

E2g

A1g

E1g

0 400 800 1200 1600

E1g

E2g

A1g

18

119

168376

1587

1590

1571E2g

E1g

A1gE2g

Raman Shift [ cm ]-1

VH

Ram

an In

tens

ity [

arb

units

. ]

r = 6.47 A

r = 6.78 A

(10,10)

(17,0)

(11,8)

r = 6.66 A

(a)

(b)

(c)

VV

Figure 41. The polarization dependence of the Raman scattering intensity of selected lowand high frequency Raman-allowed modes for (a) (10, 10) armchair, (b) (17, 0) zigzagand (c) (11, 8) chiral nanotubes for which the nanotube radii are given on the right.The left column is for the VV scattering con® guration and the right column is for theVH con® guration [2, 43].


nanotube a� ects the frequency of these modes di� erently. Although these higherfrequency modes (! > 1500 cm¡1 ) are di� cult to distinguished from one anotherbecause of their very similar frequencies, it should be possible to identify the di� erentmodes experimentally, once puri® ed aligned samples become available, throughstudies of the angular dependence of the Raman intensities, which are discussedbelow. The calculated results show almost no intensity for the intermediatefrequency Raman modes around 1200± 1500 cm¡1 [42]. The Raman experiments onsingle-wall nanotubes show some weak features [28], which might arise from alowering of the symmetry of the nanotube due to nanotube deformations or due tointer-tube interactions, or may be associated with ® nite nanotube lengths [79].

As discussed below, angular dependent polarization studies of SWNTs areexpected to provide information about the identity of speci® c features in theexperimental Raman spectra with mode symmetries, thereby allowing interpretationof the experimental spectra in terms of theoretical calculations of mode intensities.Theoretical angular dependent calculations of the intensities of the various Raman-active modes have been carried out only for the (10, 10) armchair nanotubes and theresults are plotted in ® gure 42 as a function of nanotube orientation with respect tothe polarization direction. These three rotations of the nanotube are denoted by³i…i ˆ 1 ;2 ;3), as shown on the right hand side of ® gure 42. Here ³1 is the angle thatthe nanotube axis makes with the z axis as the nanotube axis rotates from the z axisto the x axis in the xz plane normal to the y axis. The polarization vectors of the lightlie along the z and x axes, respectively, for the V and H polarizations. Likewise ³2 isthe angle that the nanotube axis makes with respect to the z axis as the nanotube axisrotates in the yz plane from the z axis to the y axis. The angle ³3 denotes the rotationangle of the nanotube around its own axis. The x ;y ;z axes are de® ned so that the twoinequivalent carbon atoms for the armchair nanotubes are both placed along the xaxis when ³3 ˆ 08 (see ® gure 1). In an actual `aligned’ sample of SWNTs, the axes ofthe nanotubes are aligned but the tube orientations with respect to ³3 are random.Figure 42 shows that the angular dependence of the Raman intensities as a functionof ³1 and ³2 are somewhat di� erent from each other for the VV con® guration andvery di� erent from each other for the VH con® guration. We further note that theten-fold symmetry axis (C10) of the (10,10) nanotubes which is not a symmetryoperation that is compatible with the Cartesian axes.

First we show how ® gure 42 can be used to identify mode symmetries in a Ramanpolarization experiment using aligned nanotubes. In ® gure 42 we see that the Ramanintensity for the A1g mode at 1587 cm¡1 as a function of ³1 for the VV con® gurationhas a maximum at ³1 ˆ 0, and ³2 ˆ 0, and is the only tangential mode withappreciable intensity for this polarization geometry. The tangential A1g mode ispredicted to have no intensity in the VH con® guration for ³1 ˆ 0, ³2 ˆ 0. On theother hand, the E1g mode at 1585 cm¡1 is predicted to have a maximum intensity inthe VV con® guration at ³1 ˆ 458, but in the VH con® guration, the E1g mode at1585 cm¡1 is predicted to have a maximum intensity at ³1 ˆ 0, ³2 ˆ 0 and aminimum intensity at ³1 ˆ 458 or ³2 ˆ 908. These polarization properties shouldallow us to distinguish these two modes with close-lying frequencies from each otherexperimentally, if we have an axially aligned nanotube sample. Also the E2g mode at1591 cm¡1 should be distinguishable from the A1g and E1g modes, since the E2g modeis expected to have a maximum intensity at ³1 ˆ 908 and ³2 ˆ 908 in the VVgeometry; the E2g mode is the only tangential mode that has appreciable intensityat ³2 ˆ 908 in the VH con® guration.



The lower frequency Raman-active modes should also be distinguishable inaccordance with their polarization behaviour. In ® gure 42, the two E2g modes at 368and 1591 cm¡1 have almost the same relative intensities in the VV con® guration.However, the ³2 dependence of the VH con® guration clearly distinguishes thedi� erent polarization behaviours between these two E2g modes [43]. It should benoted that the polarization characteristics presented in ® gure 42 are valid only forarmchair nanotubes, and these characteristics are expected to di� er somewhat fordi� erent nanotube chiralities.

Even the nanotube modes belonging to the same irreducible representation donot always have the same angular dependence with regard to the polarization of thelight. For example, the intensity of the A1g mode at 165 cm¡1 is expected, on the basisof ® gure 42, to show a di� erent angular dependence from that of the A1g mode at1587 cm¡1 [43]. Also the symmetry analysis for the lower symmetry E1g and E2g

modes may be di� cult, even if we can have an aligned nanotube sample for whichthe direction of the carbon atoms is ordered, since the 10-fold symmetry of the(10,10) nanotube does not satisfy the symmetry of the triangular nanotube lattice so


0 30 60 90q

3 [ deg. ]

A1g

A1g

1587

165

0 30 60 90q

3 [ deg. ]

E1g

E1g

1585

118

0 30 60 90q

2 [ deg. ]

Ra

ma

n in

ten

sity

(a

rb.u

nits

) A1g

E2g

E1g

1587

118

165

1591E2g

A1g

368

118

E1g15850 30 60 90

q2 [ deg. ]

E1g

E1g

E2g

E2g

E2g

17

368

1591

1585

118

0 30 60 90

1 [ deg. ]

1585

A1g

E1g

E1g

A1g

E2g

E2g

1587

165

1591368

0 30 60 90q

1 [ deg. ]

VV VH

E1g

1591

A1gE1g

E2g

1585

A1g165118

368

1587

E2g

2

z

x

y

z

x

yq 3

q 1

x

z

y

q

q

Figure 42. The polarization dependence of the Raman intensities as a function of theorientation of the nanotube axis for a (10, 10) armchair nanotube in three principalplanes (see text). The left and right hand ® gures give the angular dependence of themode intensities (labelled by symmetry type and frequency) corresponding to the VVand VH polarizations (see text) [43].


that averaging over ³3 is necessary to describe a nanotube bundle. Even for analigned SWNT bundle where the SWNTs make an arbitrary angle with the x and zaxes, since the A1g mode at 1587 cm¡1 is independent of ³3, the signal for the A1g

tangential mode will be clearly seen in the VV con® guration. The (9,9) armchairnanotube is of special interest, since it is one of a few examples where the n-foldsymmetry of the nanotube matches the triangular lattice, so that detailed angle-dependent selection rules can be expected in this case [2].

The only experimental polarized Raman spectra reported to date on SWNTs arefor nanotubes synthesized in the porous channels of aluminophosphate singlecrystals [137]. From the frequency of the Raman feature that was identi® ed with aradial breathing vibration (¹530 cm¡1), nanotube diameters of < 5 AÊ were inferred[81], smaller than previously reported in the literature for any SWNT [1, 2], therebyindicating the need for further characterization studies on these samples. Theintensity of the tangential Raman G-band was found to be sensitive to thepolarization geometry, with the tangential mode intensity being the strongest forthe incident and scattered optical E ® elds along the nanotube axis, in agreement withthe polarization e� ects expected for aligned dipole antennas on the basis ofelectromagnetic theory. Using a lineshape analysis for the tangential band basedon three Lorentzian components (see section 4.2), the following peak frequencies,mode symmetries, and FWHM linewidths were reported: 1585 (E2g ; 36), 1599 (E1g ;56) and 1615 (A1g ; 30) cm¡1. While using the same symmetry assignment argumentsas Kasuya et al. [93], the relative frequency di� erences obtained by Sun et al. [137]were much smaller than would be implied by the results in ® gure 26, for nanotubeswith such small nanotube diameters [137]. Con® rmation of these results on wellcharacterized, aligned SWNTs is needed.

The most detailed experimental polarization results for Raman spectra presentlyavailable are for high purity aligned arrays of multi-wall carbon nanotubes(MWNTs) [136]. These aligned MWNTs were prepared on silica substrates fromthe thermal decomposition of a ferrocene± xylene mixture and had an averagenanotube diameter dt ¹25 nm, with the MWNTs aligned normal to bare silicasubstrates [139]. The nanotube diameters in this MWNT sample are probably largeenough so that the resonance Raman scattering e� ect is substantially weaker than istypical for SWNTs (1 < dt < 2 nm). This is one reason why we might expect the non-resonant bond polarizability calculation discussed above to be more applicable todescribe the polarized Raman spectrum for a MWNT sample than for typical SWNTsamples. The thickness of the aligned nanotube bundle was intentionally kept below1 mm so that the focused laser excitation beam (beam diameter d ¹ 1 mm) over® lledthe aligned MWNTs during the polarized Raman scattering measurements. Thisstudy measured polarization e� ects associated with …k;k† and …k ;?† scattering forthe tangential G-band (1500± 1650 cm¡1 discussed in section 4.2) and for the so-calledD-band (1300± 1400 cm¡1 discussed in section 4.8), as well as providing con® rmationfor a predicted angular dependence of the scattered intensity as a function of theangle ³m between the polarization direction and the nanotube axis [136]. The angle³m is the angle that was measured in the polarization experiment.

The polarized spectra in ® gure 43 for four polarization geometries, labelled …XY †,…Y X†, …Y Y † and …XX†, are shown for the Raman intensities of the D-band and thetangential G-band for these MWNTs in the backscattering con® guration, with thegreatest intensities observed for the …XX† polarization geometry. The geometry andnotation used to describe these polarization studies on MWNTs are shown in the



inset to ® gure 43, where x ;y ;z refer to the coordinates of the MWNTs and X;Y ;Zrefer to the laboratory frame of the light. In this notation, a backscattering Ramanexperiment is described by Z…XY † -Z to indicate light incident along the Z axis withthe electric vector in the X direction for the incident light beam, while the Poyntingvector for the scattered light is along the ¡Z direction, with the electric vector for thescattered light beam in the Y direction. For the polarization results on the MWNTsshown in ® gures 43 and 44, the backscattering geometry was used with light normalto the nanotube axis, so that the abbreviated notation …XY †, giving the directions ofthe incident and scattered electric ® eld polarizations, is used to label the polarizationgeometries shown in these ® gures.

A Lorentzian lineshape analysis for the MWNTs spectra in ® gure 43 shows fourpeaks at 1354, 1576, 1583 and 1624 cm¡1 in almost all the polarization geometries,but with very di� erent polarization-dependent relative intensities [136]. From® gure 43, the experimental polarized integrated intensity ratios for the tangentialband (at 1584 cm¡1 for the XX spectrum) are IXX=IXX :IY Y =IXX :IXY =IXX :IY X=IXX =1.00:0.29:0.19:0.39 for this MWNT sample. These polarization results can in part be


Figure 43. Polarized Raman spectra for MWNTs taken at 514.5 nm (2.41 eV) for 4scattering geometries. The peak frequencies for the various features are in cm¡1 andthe FWHM linewidths in cm¡1 are given in parentheses. The inset de® nes thepolarization geometries for the aligned nanotubes that lie parallel to the x axis butthere is no preferred angular alignment of the nanotubes in the plane normal to thenanotube axis [136].


explained by the nanotube geometry which gives rise to slightly di� erent forceconstants along the nanotube axis relative to the circumferential direction, where thenanotube curvature reduces the force constant. This anisotropy in the force constantaccounts physically for the frequency di� erence between the components at1575 cm¡1 (attributed to vibrations in the circumferential direction) and at1584 cm¡1 (attributed to vibrations along the nanotube axis) for the …XX† polariza-tion (see ® gure 15). Similar frequency di� erences are observed for the otherpolarization geometries, and all the upshifted and downshifted modes in ® gure 43average to 1580:8 § 0:8 cm¡1, in good agreement with the E2g2 graphite modefrequency. Because of the small number of allowed k vectors in the circumferentialdirection, we expect the frequency di� erence between the lowest and middlefrequency components to increase with decreasing nanotube diameter [93]. Forexample, SWNTs with diameters of dt ¹1.4 nm show these two spectral features [28]to be at 1567 cm¡1 and 1593 cm¡1 (see section 4.2), and these two mode frequenciesalso average to ¹1580 cm¡1. The feature near 1620 cm¡1 which is quite pronouncedin the spectra for MWNTs, and not present in SWNTs, is associated with themaximum in the graphene 2D phonon density of states and is called the D 0 band[116].

No theory has yet been developed to describe polarization e� ects for MWNTs.Therefore the experimental polarization results for the tangential band in ® gures 43


Figure 44. Dependence of the measured Raman intensity for VV scattering as a function ofthe angle ³m between the polarization direction and the nanotube axis (see inset to® gure 43). The inset to this ® gure shows the dependence of the experimentaltangential band intensity ratio R ˆ IVV…³m†=IVV…0† (open and ® lled circles) as afunction of ³m in comparison to theory ( ® gure 42) for SWNTs [79, 136].


and 44 have been interpreted in terms of the theory developed for (10, 10) SWNTs[43] as a ® rst approximation. The theory of polarization e� ects in single-wallnanotubes was ® rst used to distinguish between the contributions to the tangentialband from modes with A1g, E1g and E2g symmetry (see ® gure 15) [43], whichaccording to bond polarization model calculations [43, 79] for a (10,10) SWNTpredict Raman-active frequencies at 1587, 1585 and 1591 cm¡1, respectively, alloccurring within a range of 6 cm¡1, which is much smaller than the observedlinewidth of the tangential G-band for Raman scattering, even for semiconductingnanotubes [43, 137] (see section 4.2).

The experimental intensities for the …XX† and …Y Y † geometries in ® gure 43 andin the inset to ® gure 44 correspond, respectively, to ³m ˆ 0 and ³m ˆ 908 for the VVpolarization con® guration, while the …XY † and …Y X† geometries, respectively,correspond to ³m ˆ 0 and ³m ˆ 908 for the VH con® guration, where, V and Hdenote vertical and horizontal, respectively (see ® gure 42). Calculations show thatthe A1g mode is dominant for the tangential band in the …XX† con® guration, while inthe …Y Y † con® guration, both the A1g and E2g modes should contribute to thescattering intensity. The intensity ratios predicted for the A1g mode at ³m ˆ 08, theA1g mode at ³m ˆ 908, and the E2g mode at ³m ˆ 908 are, respectively, 2.91, 0.72 and0.33. Thus the theoretical value for the intensity ratio for IXX=IY Y is 2.91/1.05 =1.00/0.36, which is to be compared to the experimental values of IXX=IY Y = 1.00/0.29. This observation would be consistent with the peak at 1584 cm¡1 containing theunresolved E2g and A1g modes of the tangential band.

Calculations for the expected Raman intensity for VH scattering [79] requires anaverage to be done over ³3 (see ® gure 42). After carrying out the appropriate angularaverages, a theoretical value for the intensity ratio for the …XY † to …Y X† geometriesof IXY =IY X = 1.00/0.27 was obtained, which is to be compared with the experimentalvalue of 0.49/1.00. In summary, the theoretical predictions for the relative polariza-tion intensities for the tangential band for SWNTs are IXX=IXX :IY Y =IXX :IXY =

IXX :IY X=IXX ˆ 1:00:0.36 :0.13:0.03 for the …XX†:…Y Y †:…XY †:…Y X† polarization geo-metries [43], which are in rough agreement with the corresponding experimentalvalues for aligned MWNTs IXX=IXX :IY Y =IXX :IXY =IXX :IY X=IXX ˆ 1:00:0.29:0.19:0.39[136]. The …XX† intensity depends only on the A1g mode, while the …Y Y † intensityhas contributions from both the A1g and E2g modes. Comparison between theexperimental …XX† and …Y Y † polarization data for MWNTs does not seemconsistent with the mode identi® cations made in ® gure 26 for SWNTs. Furtherwork is necessary to understand the polarization e� ects and to make de® nitivesymmetry assignments of phonon modes in carbon nanotubes.

Based on theoretical considerations, the intensity of the tangential A1g mode as afunction of ³m is expected to exhibit a minimum at ³m ˆ cos¡1 …1=31=2† ˆ 54:78 [43].Figure 44 shows a collection of polarized Raman spectra that were obtained onaligned MWNTs in the VV con® guration as a function of ³m . Clearly the experi-mental polarized tangential band intensity ratio R ˆ IVV…³m†=IVV…0† exhibits aminimum near ³m ˆ 558, in good agreement with the theoretical curve for theintensity of the VV signal versus ³m shown in the inset to ® gure 44. It should benoted that the intensity ratio R denoted by the open circle in the inset to ® gure 44was obtained from the XX and Y Y data depicted in ® gure 43. The data set in® gure 44 also suggests that the tangential band intensity is dominated by theintensity of the A1g symmetry mode at 1584 cm¡1. It is interesting that the experi-



mental D-band intensity is also a minimum near ³m ˆ 558. No theory is yet availablefor the angular dependence of the D-band intensity.

It would also be interesting to measure the polarization e� ects associated with theA1g radial breathing mode in SWNTs, which was already considered in a preliminaryway [137]. Since the A and B carbon atoms within the unit cell of the graphene 2Dhoneycomb lattice move similarly under breathing mode displacements, there shouldbe no intensity for the …XY † or …Y X† polarization geometries for isolated single-wallnanotubes (see ® gure 42). Theory for (10, 10) SWNTs further predicts that the…Y Y † intensity should be stronger than that for the …XX† intensity, with apolarization intensity ratio …XX†:…Y Y † = 0.20:1.00 for the radial breathing mode.Since the intensity of the radial breathing mode drops rapidly with increasing tubediameter (where we note that the experimental intensity of this mode for a (20,20)nanotube was observed to be vanishingly small) [82], polarization studies of theradial breathing mode should be done on SWNTs with dt < 2 nm. In the actualexperiments, inter-tube coupling within a nanotube bundle (see section 3.3) and theresulting lowering of the symmetry of the nanotube cross-sections may result in somerelaxation of the polarization e� ects predicted for isolated SWNTs.

Returning to ® gure 43 we note that the optical absorption spectra of a single-wallcarbon nanotube, calculated for polarized light by Ajiki and Ando [140, 141], showthat the selection rules for optical absorption between the valence and conduction psubbands with subband index n are D n ˆ 0 and D n ˆ 1 for polarization parallel andperpendicular to the nanotube axis, respectively. The optical absorption for thepolarization perpendicular to the nanotube axis is predicted to be suppressed almostcompletely when the depolarization e� ect is taken into account. Thus the opticalabsorption occurs only when the polarization is parallel to the nanotube axis forinter-subband transitions, with D n ˆ 0 for single-wall carbon nanotubes. Thedepolarization e� ect should, however, be relaxed in MWNTs, especially as thediameter increases. On the basis of the depolarization e� ect, we expect more e� ectiveexcitation of resonant Raman scattering for the …XX† scattering geometry, and formetallic nanotubes, for which the nanotube acts as a radiation pipe for the laserexcitation. The polarized VV Raman spectra for aligned MWNTs in ® gure 43 appearto be consistent with the optical anisotropy present in the MWNTs. The discrepancybetween the experimental polarized Raman intensity ratios (regarding ® gure 43) andtheoretical predictions (® gure 42) may be due to a depolarization (antenna) e� ectoperative also on the scattered light which tends to enhance the experimental Y X:XYintensity ratio relative to the calculations.

4.8. D-band and G 0-band spectraOne of the interesting features of the Raman spectra in sp2 carbon materials is

the laser energy dependence of the frequency of the disorder-induced D-band whichis observed between 1250 and 1450 cm¡1. The D-band is activated in the ® rst-orderscattering process by the presence of in-plane substitutional hetero-atoms, vacancies,grain boundaries or other defects and by ® nite size e� ects, all of which lower thecrystalline symmetry of the quasi-in® nite lattice. The association of the D-band withsymmetry-breaking phenomena results in a D-band intensity that is proportional tothe phonon density of states shown in ® gure 12 (c) for graphite, which should also beapplicable for (10, 10) SWNTs. A dominant feature of the second-order Ramanspectrum is the G 0-band which is the overtone of the D-band and appears in therange 2500± 2900 cm¡1 when the laser excitation energy Elaser is varied from 1 to



4.5 eV. Furthermore, the second-order G 0-band is observed even in the case ofcrystalline graphite, where the disorder-induced D-band is absent, so that the G 0-band is an intrinsic property of the 2D graphene lattice. It has been known for abouttwo decades [142], that there is a strong dependence of the peak frequency for the D-band and for the G 0-band on Elaser, and this phenomenon occurs in a similar way inall kinds of sp2 carbon materials, such as, graphon carbon black [143], hydrogenatedamorphous carbon [144], glassy carbon and crystalline graphite [145± 147], andmulti-component carbon ® lms [148]. It is believed that the D-band feature observedin bundles of SWNTs has contributions from the SWNTs themselves, perhaps due toa ® nite length e� ect [79], as well as from other carbonaceous materials (e.g.amorphous sp2 carbon coatings) present in the imperfectly puri® ed SWNT samplesthat are now available.

Figure 45 illustrates the basic mechanism responsible for the laser excitationdependence of the D-band and the G 0-band frequencies in sp2 carbons [117, 146,147]. In the upper part of ® gure 45, we see that electronic transitions between the pand p¤ electronic states of 2D graphite with energies corresponding to visiblephotons only occur in the vicinity of the K point in the Brillouin zone (BZ) throughwhich the Fermi level EF passes [2, 41]. The phonons associated with the D-band andthe G 0-band have the same wavevectors D q as the electronic states D k which are inresonance with the laser. Moreover, it is argued that these phonons belong to theoptic branch that contains the zone centre E2g2

graphitic mode, represented by aheavy curve in ® gure 45 [2, 41]. The reason why the phonons belonging to thisparticular optic branch exhibit an especially large Raman cross-section compared toother phonons with the same wavevector D q is attributed to their breathing-modedisplacements (see ® gure 46) which would be expected to show strong deformationpotential coupling to the electronic states [117]. Referring to ® gure 46, we see that forsp2 carbons all the carbon atoms about the points labelled by £ vibrate throughbreathing mode atom displacements with respect to point £, while exhibiting typicaloptical mode displacements with respect to the centres of the other two hexagons inthe honeycomb lattice [117].

The second-order G 0-band in the Raman spectra of various sp2 carbon materialsis generally much more intense than the disorder-induced D-band. This is due to thefact that the second-order G 0-band is symmetry-allowed by momentum conservationrequirements, whereas the disorder-induced D-band only appears when there is abreakdown in the in-plane translational symmetry [117]. For a given sp2 sample, theintensity of the D-band for sp2 carbons increases smoothly as Elaser decreases due tostructural defects or to ® nite size e� ects [145, 149]. In contrast, the G 0-band issymmetry-allowed and is an intrinsic feature of all sp2 carbons. Therefore, a Ramaninvestigation of the second-order G 0-band by varying Elaser for the incident photonprovides an experimental way to probe the particular optical phonon branchrepresented by the heavy curve in ® gure 45.

Raman spectra showing the D-band and the G 0-band in SWNTs are shown in® gure 47 at laser excitation 632.8 nm (1.96 eV) and 514.5 nm (2.41 eV). These spectrashow features for SWNTs with almost the same peak frequencies as the correspond-ing D-band and G 0-band in sp2 carbons, and these features in the SWNT spectrashow a qualitatively similar dependence of their peak frequencies on Elaser as is foundin other sp2 carbons. The measurements are made on a SWNT sample with a broadnanotube diameter distribution, as determined by transmission electron microscopy(TEM) measurements and shown in the lower right inset to ® gure 47. The most



prominent feature in the second-order spectra in ® gure 47 is the G 0-band feature inthe 2600± 2700 cm¡1 range, and we see that the peak frequency of the G 0-band featureis strongly dependent on Elaser with !G 0band ˆ 2633 cm¡1 and 2682 cm¡1 at 1.96 eVand 2.41 eV, respectively. The spectrum in ® gure 47 shows a well-de® ned feature at1323 cm¡1 for the Elaser ˆ 1:96 eV trace and at 1345 cm¡1 for the Elaser ˆ 2:41 eVtrace, which can be identi® ed with the so-called `D-band’ , as discussed above.Figure 47 furthermore shows higher-order features for the D-band and the G 0-bandover a very broad range of phonon frequencies [80]. Whereas the frequencies of theD-band (!D band ) and the G 0-band (!G 0 band ) depend strongly on the laser energy, theintensities of the D-band and G 0-band features for a given nanotube sample exhibit


Figure 45. Electronic energy bands E…k† (top) and phonon dispersion curves !…q† of 2Dgraphite (bottom) [2, 41]. Both the phonon branch that is strongly coupled toelectronic bands in the optical excitation, and the electronic bands near the Fermilevel (E ˆ 0) that have a dispersion relation that is linear in k are indicated by heavylines. The initial slope for the low frequency TA phonon branch (which is initially thesame along GM and along GK) is also indicated by heavy lines. The strong couplingbetween the electrons of wave vector D k, measured from the K-point in the Brillouinzone, to phonons of wave vector D q ˆ D k is responsible for the frequencydependence of the D-band and the G 0-band features in the Raman spectra of sp2

carbons and carbon nanotubes [117].


only a weak dependence on Elaser, and the data in ® gure 47 are consistent with!G 0band ’ 2!Dband for SWNTs.

Figure 48 compares the second-order spectra in the 2400± 3500 cm¡1 range forSWNTs and graphite. Whereas the G 0-band for SWNTs and turbostratic 2Dgraphene layers is well described by a Lorentzian lineshape, the G 0-band for graphiteshows a doublet structure (see ® gure 48 and the upper inset to ® gure 47) whichphysically is related to the two crystallographically inequivalent graphene layers in3D graphite [116, 151]. Whereas the wavevector for graphite is essentially acontinuous variable, the wavevector in the circumferential direction for nanotubeshas only a few (¹40 for a (10, 10) nanotube) allowed values. Furthermore, incontrast to the radial breathing mode (section 4.1) and the tangential modes(section 4.2) for which only a few nanotubes of typical SWNT samples are inresonance with Elaser and therefore contribute resonantly to a given Ramanspectrum, the mechanism described by ® gure 45 indicates that all nanotubes in thesample contribute to the D-band and G 0-band spectrum at each Elaser value. Inaddition, the zone folding of the K-point phonons in the large 2D Brillouin zone(BZ) will give rise to Raman-allowed k ˆ 0 phonons in the small 1D BZ of thenanotubes and these activated phonons will be subject to the same nanotube-selective resonant Raman scattering process discussed in sections 4.1 and 4.2 forthe radial breathing mode and the tangential modes, respectively.

The strong upshift in the frequency of the G 0-band as Elaser increases isdemonstrated in the Raman spectra shown in ® gure 49 (b) and in the plot of thepeak frequencies of the G 0-band versus Elaser given in ® gure 49 (a). We explain thestrong linear dependence of the G 0-band peak frequency on Elaser in single-wallcarbon nanotubes by the resonance of Elaser with electronic interband transitionsnear the K-point in the 2D Brillouin zone as described in ® gure 45 for sp2 carbons.The argument for the linear dependence of !Dband on Elaser is as follows (see® gure 45). From the linear k dependence of E…k† near the K-point in the Brillouinzone, the energy for an interband transition at a point D k from the K-point is

D E ˆ ¬D k ; …30†


Figure 46. Schematic diagram of the atomic displacements (arrows) in the graphene planefor the E2g2 mode at the G-point, which can be viewed with respect to the unmarkedcentres of the hexagons. For the atomic motions of the six atoms about the hexagoncentres denoted by £, breathing-type displacements are obtained corresponding tonormal modes associated with the K-point in the Brillouin zone [117].


which at resonance with Elaser yields D E ˆ Elaser. Strong electron± phonon couplingoccurs when the phonon and electron wave vectors are equal D q ˆ D k. The D-bandfrequency !Dband relative to !K at the K-point is

!Dband ˆ !K ‡ D q ; …31†yielding

Elaser

¬ˆ !Dband ¡ !K

; …32†

so that

D Elaser ˆ ¬

D !Dband: …33†

Since the G 0-band is the overtone of the D-band, we can write !G 0band ’ 2!Dband forSWNTs. Because of the higher intensity and much better data quality for the G 0-band relative to the D-band, we discuss the G 0-band ® rst.

From ® gure 49 (a), we ® nd a value for the slope @!G 0band=@Elaser ˆ 106 cm¡1/ eVfor SWNTs as compared to @!G 0band=@Elaser ˆ 101 cm¡1=eV for 3D graphite [117,118]. For all the Raman spectra shown in ® gure 49 (b), the lineshape for the G 0-bandfeatures is ® t by a single Lorentzian component, with a linewidth that has a veryweak dependence on Elaser. The data in ® gure 49 (a) for SWNTs extrapolate to


Figure 47. Raman spectra of SWNTs over a very broad phonon frequency range excitedwith 514.5 nm (2.41 eV) and 632.8 nm (1.96 eV) excitation. The upper inset illustratesin more detail the C± C tangential stretching G-band modes and the G 0-band ofSWNTs (solid lines) and a comparison is made to these features in highly orientedpyrolytic graphite (HOPG) (dotted lines). The lower inset shows the diameterdistribution of isolated SWNTs obtained from HRTEM images [80].



Figure 48. Second-order Raman spectra of single-wall carbon nanotubes and of 3D graphitefor three di� erent laser excitation energies [150].

Energy (eV)2.01.6 2.4 2.8

Ram

an s

hift

(cm

)-1

2600

2620

2640

2660

2680

2700

2400 2500 2600 2700

2.71

2.41

1.96

2.19

1.83 eV

2400 26002500 2700


Raman shift (cm-1)

(a) (b)

Figure 49. The Raman spectra for the intense G 0-band for ® ve values of Elaser are shown in(b), while (a) shows a plot of the peak Raman frequency for the G 0-band featuresversus laser excitation energy Elaser [114].


2429 cm¡1 at Elaser ˆ 0, and this phonon frequency is approximately twice the K-point phonon frequency in the 2D Brillouin zone of sp2 carbons. This extrapolationalso agrees quite well with the direct measurement of this same phonon frequency(2440± 2445 cm¡1) for SWNTs, as shown in ® gure 50. The low intensity of the K-point second-order feature is attributed to its non-resonant origin, since the K-pointfeature corresponds to the Elaser ˆ 0 limit in ® gure 45. The extrapolation of theG 0-band peak frequency to Elaser ˆ 0 yields 2429 cm¡1 which is quite consistent withthe corresponding extrapolation of the D-band peak frequency to 1415 cm¡1 atE laser ˆ 0 as discussed below.

A series of spectra showing the D-band in single-wall carbon nanotubesare presented in ® gure 51 for several values of E laser . The strong dependenceof the D-band peak frequency as a function of E laser is apparent from thespectra in ® gure 51, and a summary of the collected D-band frequencies as afunction of E laser is shown in ® gure 52 from which we obtain a value of@!Dband=@E laser ˆ 51:2 cm¡1= eV and an intercept at Elaser ˆ 0 of 1215 cm¡1. Meas-urements on a number of SWNT samples give an average value of@!Dband=@E laser ˆ 52 § 1 cm¡1/ eV which is to be compared with an average valuefor graphite of 48 § 3 cm¡1/ eV [117]. As mentioned above, the extrapolated value of1215 cm¡1 for the D-band frequency at the K-point is in good agreement with halfthe extrapolated value of !G 0band ˆ 2429 cm¡1 to E laser ˆ 0.

We now summarize a number of characteristics for the D-band and G 0-band forcarbon nanotubes. The ® rst relates to polarization e� ects, which have been reportedonly for MWNTs thus far, and only for the D-band (not for the G 0-band). In thisconnection, ® gure 43 shows that the D-band spectra for multi-wall nanotubes(¹25 nm diameter) have a strong polarization dependence that is di� erent from thatof the Lorentzian components of the tangential G-band, and also very di� erent fromthe weak polarization dependence observed for disordered carbons [136] (see


2360 2400 2440 2480 2520Raman shift (cm

?1)

0

Ra

ma

n I

nte

ns

ity

(a

.u.)

1.83 eV

1.92

1.96

2.41

2.54

2.71

2400 2440 2480 25202360Raman shift (cm )-1


Figure 50. The weak non-resonant Raman feature associated with the second harmonic(overtone) of the K-point phonon in the 2D Brillouin zone for six values of Elasertaken on a nanotube sample with dt ˆ 1:49 § 0:20 nm [114].



1190 1240 1290 1340 1390


Raman Shift (cm )-1

2.71

2.54

2.41

2.19

1.96

1.92

1.83

1.58

1.49 eV

Figure 51. Raman spectra in the range 1190± 1390 cm¡1 for SWNTs showing the upshift ofthe D-band peak frequency with increasing Elaser [101, 102].

Figure 52. Frequency of the `D-band’ for single-wall carbon nanotubes as a function oflaser excitation energy. The line is a least squares ® t to the data points for one SWNTsample. The ® t to the data points yields a slope of 51.2 cm¡1/ eV and an intercept of1215 cm¡1 at Elaser ˆ 0 [150].


section 4.7). Further study of the polarization e� ects associated with the D-band incarbon nanotubes both experimentally and theoretically is needed.

Di� erences between the Stokes and anti-Stokes spectra are also found in ® gure 33at Elaser ˆ 1:58 eV and 1.96 eV for the D-band and G 0-band with regard to their peakfrequencies and the relative intensities. No second-order features are observed in theanti-Stokes spectra for Elaser ˆ 1:96 eV, which is consistent with a Boltzmann factorreduction in relative intensity. However, the observation of Raman intensity in theG 0-band in the anti-Stokes spectrum at 1.58 eV cannot simply be explained by aBoltzmann factor exp…¡Eph=kBT † e� ect, but rather may be due to an enhancedelectron± phonon coupling mechanism for metallic nanotubes. Additional enhance-ment of the anti-Stokes process for metallic SWNTs relative to semiconductingSWNTs is observed experimentally, implying a stronger electron± phonon couplingfor metallic nanotubes.

The di� erent frequency shifts of the G 0-band features in ® gure 33 can beexplained by the di� ering energies of the scattered photons which are involved withthe 2D resonance enhancement e� ects associated with the K-point in the Brillouinzone [117]. For example, an expected di� erence of 34 cm¡1 between the anti-Stokesand Stokes G 0-band frequencies is predicted from the measured …@!G 0band=@Elaser†slope of 106 cm¡1/ eV for the G 0-band dispersion [118], which is in good agreementwith the observed frequency di� erence of 36 cm¡1 shown in ® gure 33.

Next we review present knowledge about the D-band and G 0-band features asthey are revealed in the surface-enhanced Raman spectra SERS discussed insection 4.6. In ® gure 37, we see that the D-band feature (1307± 1315 cm¡1 ) in the® rst-order Stokes spectrum at 1.96 eV which lies within the metallic window for thissample is upshifted by 4± 8 cm¡1 in the surface enhanced resonant Raman spectrum(SERRS) relative to the normal resonant Raman spectrum (RRS), while thecorresponding second-order G 0-band is upshifted by 24 cm¡1 upon interaction withthe metal substrate. It is interesting to note that the upshift of the G 0-band associatedwith semiconducting nanotubes (observed at Elaser ˆ 2:41 eV) is only 13 cm¡1,showing that the SERRS-induced shift in the G 0-band is much larger for Elaser

values where metallic nanotubes contribute strongly to the Raman spectra than forElaser in the semiconducting regime. Furthermore at Elaser ˆ 1:96 eV (within themetallic window), the intensity of the G 0-band relative to that of the tangential mode,denoted simply by …IG 0 =Itang†, has a value of 0.4 for the normal resonant Ramanspectra (RRS) as compared to a signi® cantly larger value of 0.6 for the SERRSspectra. In contrast, for Elaser ˆ 2:41 eV, where only the semiconducting nanotubesare in resonance with Elaser , the ratio …IG 0 =Itang† has the same smaller value of 0.2 forboth the RRS and SERRS spectra.

An even larger upshift of 36 cm¡1 is observed in the frequency of the second-order G 0-band for Elaser ˆ 1:49 eV for the nanotubes adsorbed to a Ag nano-structural substrate (from 2600 cm¡1 for the normal RRS spectrum to 2636 cm¡1

for the SERRS spectrum) at the anti-Stokes side of the Raman spectra. This largerupshift of the G 0-band is attributed to the higher charge transfer that is produced bythe interaction of the metallic nanotubes with the nanoscale Ag particles in thecolloidal Ag substrate relative to the ® lm Ag substrate where the upshift was only¹24 cm¡1. Since the anti-Stokes spectrum at Elaser ˆ 1:49 eV is dominated byresonance with metallic nanotubes, we can explain the large intensity of the G 0-band relative to the tangential band, which seems to be a factor of ¹2 larger for theRRS spectrum relative to the SERRS spectrum. This observation is explained by the



lower frequency of the G 0-band in the RRS spectrum relative to the SERRSspectrum. Heating e� ects are less important for SERRS than for RRS because ofthe good thermal contact between the metallic nanotubes and the metal substrates inSERRS experiments. It would be expected that a phonon with a high vibrationalfrequency (¹2600 cm¡1 ) would be very weak in the room temperature anti-Stokesspectrum because of the highly unfavourable Boltzmann factor. The fact that the G 0-band can be observed in the SERRS anti-Stokes spectrum at 1.49 eV is attributed toan optical pumping e� ect which is operative for metallic nanotubes (see section 4.6)[130].

The large upshifts in the G 0-band frequencies in the SERRS spectra forall nanotubes relative to the corresponding RRS spectra, and the relative increasein the intensities of the G 0 Raman band in the SERRS spectra for the metallicnanotubes indicates that for all k values the electronic structure E…k† of the SWNTsis signi® cantly perturbed by the metal substrates to which the SWNTs are adsorbed.Following this line of reasoning, it can be argued that the interaction betweenthe nanotubes and the metal substrate also perturbs the electronic structure nearthe K-point of the 2D Brillouin zone, so that the D-band and the G 0-bandvibrational spectra, which are resonantly excited by a given laser energy Elaser,now correspond to a slightly larger D k relative to the K-point in the 2D Brillouinzone. But since the D-band (and the G 0-band) phonon that is resonantly enhancedhas the same wave vector as that of the resonant electronic transition D q ˆ D k [117],the upshift in the SERRS spectrum relative to the RRS spectrum is a measure of thestrength of the electronic interaction between the nanotube and the metal substrate.Because of the much larger experimentally observed upshift of the G 0-bandfrequency for the metallic nanotubes, we conclude that the interaction of the metallicnanotubes with the metal substrate is much stronger than for the semiconductingnanotubes. This conclusion is also consistent with the increased intensity of certaincomponents associated with the tangential band of metallic nanotubes in the SERRSspectra relative to the intensity of the same components in the RRS spectra. Thesearguments also support the conclusion that electron± phonon interaction e� ects arestronger for the metallic nanotubes than for the semiconducting nanotubes,consistent with the large carrier density in metallic single-wall nanotubes.

4.9. Overtone and combination modesIn molecular (0D) Raman spectroscopy, the combination and overtone modes

are generally more important for the interpretation of the phonon data than is thecase for 3D solids. This is related to the fact that in 3D solids the inclusion of two ormore phonon modes in a scattering process relaxes the wave vector selection rule,and thereby gives relatively broad structures in the Raman spectra. However, in 1Dmaterials the k-space integration is only in one direction and thus the higher-orderRaman features remain relatively sharp.

We discuss below the various features in the second-order spectrum associatedwith the harmonics (overtones) and combination modes of the two dominantfeatures of the ® rst-order spectrum, which are the low frequency radial breathingmode !RBM and the high frequency tangential band !tang. We also discuss harmonicsand combination modes for the D-band and the G 0-band discussed in section 4.8.

In the case of fullerenes, a more complete analysis of the combination andovertone modes has been made [152, 153] and a detailed tentative identi® cation of all



the phonons has been given. At the present time the available data for carbonnanotubes does not support such a de® nitive identi® cation.

4.9.1. OvertonesIn ® gure 53, we see (on a scale amjli® cation of 20£) spectral features identi® ed

with the second harmonic (overtone) of the radial breathing mode 2!RBM (seesection 4.1) at two di� erent laser excitation energies. The spectra at Elaser ˆ 1:58 eV(785 nm) show features in the ® rst-order spectrum at !RBM ˆ 150 cm¡1 and 162 cm¡1

as well as second-order lines at approximately 2!RBM ˆ 301 cm¡1 and 330 cm¡1. Achange in Elaser excites di� erent nanotubes, so that for Elaser ˆ 2:54 eV (488 nm), the® rst-order Raman spectrum shows a strong feature at !RBM ˆ 159 cm¡1 and a muchweaker second harmonic at 2!RBM ˆ 320 cm¡1 (see ® gure 53). Similar trends areobserved at other values of Elaser [114]. The intensity of the overtone of the radialbreathing mode reaches a maximum for Elaser values where contributions frommetallic nanotubes dominate the spectra for the tangential band (see section 4.2),suggesting that the 2!RBM feature also is stronger for metallic nanotubes relative tosemiconducting nanotubes.


Figure 53. The Raman spectra for the radial breathing mode band (!RBM) and its secondharmonic at 2!RBM for two laser excitation energies 1.58 eV (785 nm) and 2.54 eV(488 nm) for an SWNT sample with dt ˆ 1:49 § 0:20 nm [114].


In contrast, the second harmonic of the tangential band 2!tang which occurs inthe range 3100± 3250 cm¡1 (see ® gure 54) has very di� erent characteristics from thesecond harmonic of the radial breathing mode 2!RBM shown in ® gure 53. We see in® gure 54 the evolution of the second harmonic of the tangential band at 2!tang for® ve laser energies in the range 1.96± 2.71 eV. The central frequency and linewidth ofthis second-order band exhibits a relatively weak dependence on Elaser for2:19 < Elaser µ 2:71 eV, where the semiconducting nanotubes dominate the ® rst-order spectra for this sample. However, for Elaser ˆ 1:96 eV, where the dominantcontribution to the ® rst-order spectrum comes from metallic nanotubes, the second-order spectrum is downshifted and is much broader than for the higher Elaser values,consistent with the behaviour of the ® rst-order tangential band discussed insection 4.2.

The feature in the second-order Raman spectrum for 3D graphite near 3240 cm¡1

(see ® gure 48) is strongly a� ected by the shape of the uppermost optical branch ofthe !…q† phonon dispersion curves for graphite [116, 154], which exhibits a peak inthe phonon density of states near 1620 cm¡1, associated with intermediate wavevector (non-zone centre) phonons. This peak in the phonon density of states isresponsible for the feature in the second-order spectrum of graphite near 3240 cm¡1,and the frequency is upshifted by 76 cm¡1 from twice the zone-centre phonon modein graphite at 2 £ 1582 cm¡1 ˆ 3164 cm¡1. Calculations of the phonon dispersion


Figure 54. Raman spectra for a SWNT sample with dt ˆ 1:49 § 0:20 nm for the secondharmonic of the tangential G-band mode, at ® ve laser excitation energies. The ® rst-order spectra for the tangential G-band mode are displayed in ® gure 27 [114].


curves in SWNTs [2, 60] show no corresponding peak in the density of states awayfrom the Brillouin zone centre, and therefore we only expect a Raman feature closeto twice that for the ® rst-order tangential band, consistent with experiments forSWNTs [114].

A Lorentzian lineshape analysis of the second-order spectrum associated with thesecond harmonic of the tangential band shows qualitatively di� erent behaviour forsemiconducting nanotubes in comparison to metallic nanotubes (see ® gure 55). ForElaser ˆ 2:71 eV where semiconducting nanotubes mainly contribute resonantly tothe ® rst and second order spectra [94], the dominant feature in the second-orderspectrum is at 3181 cm¡1, which is close to twice the frequency of the dominant modein the ® rst-order spectrum 2…1592† ˆ 3184 cm¡1, and the second-order feature is onlyslightly broader than twice the FWHM linewidth of the ® rst-order feature. Thefrequencies of the two weaker features in the second-order spectrum at 2.71 eVcorrespond approximately to twice the frequencies of the ® rst-order features. AsElaser decreases from 2.71 eV, the peak frequency of the entire second-order band (see® gure 54) downshifts, especially for the lowest value of Elaser, because new tangentialpeaks associated with metallic nanotubes are resonantly enhanced. In contrast, thesecond-order spectrum at Elaser ˆ 1:96 eV in ® gure 54 shows a broad, asymmetricband with more scattering intensity at low phonon frequencies. Analysis of thelineshape of this Raman band in ® gure 55 shows a feature at 3082 cm¡1, which is


33003000 3100 32001700160015001400

1400 1500 1600 1700 3000 3100 3200 3300

1.96 eV 1.96 eV

2.71 eV 2.71 eV

1567

3153

1515

15401581

1599

15923181

3216

3082

3122

3153

31783203

1601

1590



Raman shift (cm -1) Raman shift (cm -1)

Figure 55. A line shape analysis of the spectral features in the ® rst-order spectra (left) andin the second-order spectra (right) for the tangential bands taken for Elaser ˆ 1:96 eV(632.8 nm) (top) and 2.71 eV (457.9 nm) (bottom) for a SWNT sample withdt ˆ 1:49 § 0:20 nm [114].


close to 2 £ 1540 cm¡1, thereby providing support for the interpretation that metallicnanotubes are contributing to this second-order Raman band at Elaser ˆ 1:96 eV.The second-order tangential band at Elaser ˆ 1:96 eV in ® gure 55 is so broad andfeatureless, that it is not meaningful to attach signi® cance to the Lorentziancomponents obtained from a Lorentzian lineshape analysis. The spectra in® gure 47 show that the overtones for the tangential band are strong enough toobserve well-resolved spectral features to third order (3G) (see section 4.9.3) [80].

The largest linewidths are observed when both the incident and scattered photonsare in resonance with metallic nanotubes for both the ® rst and second orderscattering processes. Intermediate linewidths are observed when both semiconduct-ing and metallic nanotubes contribute to the scattering process.

4.9.2. Combination modesResonant Raman e� ects associated with the 1D electron density of states

singularities also give rise to resonant Raman e� ects in the combination modes.One example of a combination mode occurs at the sum frequency between atangential and one or two radial breathing mode phonons !tang ‡ !RBM and!tang ‡ 2!RBM, as shown in the spectra in ® gure 56 taken at Elaser ˆ 1:58 eV and1.96 eV. Taking the radial breathing mode at !RBM ˆ 150 cm¡1 for Elaser ˆ 1:58 eV(see ® gure 53), and the most intense line at 1591 cm¡1 for the tangential band !tang

yields a sum of !tang ‡ !RBM ˆ 1741 cm¡1, in good agreement with the combinationmode at 1742 cm¡1 [114].

The other higher frequency feature in ® gure 56 is tentatively assigned to a secondcombination band associated with !tang ‡ 2!RBM. Taking !tang ˆ 1540 cm¡1 and!RBM ˆ 160 cm¡1 at Elaser ˆ 1:58 eV (see ® gure 56 and [114]), we obtain(!tang ‡ 2!RBM† ˆ 1860 cm¡1 for the metallic regime, in good agreement with theexperimental peak at 1871 cm¡1. For the spectrum at Elaser ˆ 1:96 eV, we take!tang ˆ 1590 cm¡1 appropriate for semiconducting nanotubes and !RBM ˆ 165 cm¡1,then we get !tang ‡ 2!RBM ˆ 1920 cm¡1, which accounts for the observed frequencyin ® gure 56 at 1925 cm¡1 [114].


Figure 56. Spectral features tentatively associated with combination bands for !tang ‡ !RBMand !tang ‡ 2!RBM in the second-order Raman spectra of carbon nanotubes atElaser ˆ 1:58 eV (785 nm) and 1.96 eV (632.8 nm) [114].


Figure 37 shows that the SERS e� ect upshifts the !tang ‡ !RBM combinationmode by ¹10 cm¡1, but upshifts the !tang ‡ 2!RBM combination mode by ¹31 cm¡1.The observed combination mode frequencies suggest that the (!tang ‡ !RBM)combination mode might be more enhanced by the SERS process for metallicnanotubes. The larger upshift for the (!tang ‡ 2!RBM) combination mode iscorrelated with a larger variation of this mode frequency with Elaser in the normalresonant Raman spectra [114].

4.9.3. Overtones and combination modes for the D-band and G 0-bandSince the D-band and G 0-band involve a di� erent resonance process (see

section 4.8) than the radial breathing mode or the tangential band (also called theG-band) , the harmonics and overtones of the D-band and G 0-band show di� erentproperties, as discussed below. The overtones and combination modes for the D-band and G 0-band at Elaser ˆ 1:92 eV (632.8 nm) and 2.41 eV (514.5 nm) are shown in® gure 47. A listing of the frequencies and dispersion of the D-band, the G 0-band, thetangential G-band and of their overtones and combination modes is given in table 4for Elaser ˆ 2:54 eV (488 nm) excitation [80]. The table includes their assignments,which are made on the basis of their frequencies and their dispersion …@!=@Elaser†.Their observed dispersion …@!=@Elaser†expt is determined from the spectral informa-tion provided for Elaser ˆ 1:92 eV, 2.41 eV and 2.54 eV [80] and the characteristicdispersion of the D-band and G 0-band discussed in section 4.8. The theoreticalestimates, !th, given in table 47 are obtained from the assignments listed for eachspectral feature and the values of the G 0-band and G-band (tangential band) modefrequencies. Also listed in the table is the predicted dispersion …@!=@Elaser†th for eachentry based on …@!G 0 band=@Elaser† ˆ 106 cm¡1/ eV and …@!G band=@Elaser† ˆ 0. Goodagreement is obtained between the observed mode frequencies and their dispersionsand the values predicted from the mode assignments and from the well-establishedfrequencies and dispersion for the D-band, G-band and G 0-band. The largedispersion of the D-band and the G 0-band and the absence of dispersion for theG-band are key factors in the identi® cation of the overtones and combination modesin table 4. The strong electron± phonon coupling in the SWNTs makes it possible to


Table 4. Experimental (expt) peak positions ! (cm¡1) and the theoretical assignments (th) ofselected ® rst-order and higher-order Raman features of SWNTs excited by 488 nm(2.54 eV) excitation [80, 101].

!expt !th …@!=@E†expt …@!=@E†th

(cm¡1) Assignment (cm¡1) (cm¡1/eV) (cm¡1=eV)

1352 D 1348 50 531582 G 1582 0 02696 G 0 2696 109 1062946 D + G 2930 50 533184 2G 3164 5 04273 G 0 ‡ G 4278 129 1064800 3G 4746 6 05330 2G 0 5392 260 2125848 G 0 ‡ 2G 5860 133 1066885 2G 0 ‡ G 6974 194 212


observe overtones beyond the fourth harmonic. It should be noted that theexperimental dispersion of the G 0-band appears to be greater than 106 cm¡1/ eVfor the higher-order modes.

4.10. Raman studies of doped carbon nanotubesRaman spectra have also been taken on doped single-wall carbon nanotube ropes

[39, 40, 155± 159]. These spectra show e� ects similar to the e� ect of alkali metal andhalogen intercalation into graphite [160], which exhibits characteristic upshifts in theE2g2 mode frequency associated with the donation of electrons from graphite to thehalogens in the case of acceptors, and characteristic downshifts in mode frequenciesassociated with donor charge transfer to graphite in the case of alkali metalintercalation (see ® gure 57 for the upshift in the tangential G-band frequency foriodine intercalated single-wall carbon nanotubes) [155]. To date, Raman scatteringstudies have been carried out mainly on as-prepared (and unpuri® ed) material takendirectly from the synthesis chamber and exposed to the following reactants: (donors)Li [158], K and Rb [161], and Cs [39, 40]; (acceptors) sulphuric acid (H2SO4 ) [159],Br2 [39, 40, 161], iodine (I2) vapour [161], and molten iodine [155]. It is of interest tomention that iodine does not form a graphite intercalation compound (GIC) butdoes e� ectively dope SWNTs. Many authors have assumed that for charge transferprocesses in single-wall carbon nanotubes the dopant resides as ions (and alsopossibly as neutral atoms) in the interstitial channels between the nanotubes in thetriangular nanotube lattice. While this presumption has yet to be con® rmed by X-raydi� raction, or other structural probes, the presumption implies that isolated SWNTswhich have no interstices may behave di� erently from SWNT bundles which do haveinterstices regarding the stability of the adsorbed dopants in charge transferprocesses.


Figure 57. Raman spectra of pristine, moderately I-doped and saturation I-doped single-wall nanotube samples (T ˆ 300 K, 514.5 nm laser excitation) where molten I2 is thedopant. The insert shows the photoluminescence spectrum due to the intercalatedpolyiodide chains in the moderately doped sample where sharp Raman lines aresuperimposed on the broad PL spectrum [155].


Measurements of the RBM spectra for Br2 doped SWNTs with diameters in therange 1:24 < dt < 1:58 nm show that in addition to the RBM features near 190 cm¡1,a new feature appears near 240 cm¡1 for Elaser > 1:8 eV, as shown in ® gure 58 (a) [39,40]. The Elaser dependence of the ratio of the intensity of the 240 cm¡1 feature to thatof the 180 cm¡1 is shown in more detail using the right hand scale of ® gure 58 (b),and these data are to be compared with the optical absorption spectra of the sameSWNT sample, shown on the left scale of the same ® gure, before Br2 doping. Figure58(b) shows absorption peaks for the ES

11…dt†, ES22…dt† and EM

11…dt† interbandtransitions between van Hove singularities, as discussed in section 2.2 [29]. Dopingwith bromine to the saturation limit completely suppresses the three opticalabsorption features [37], as EF is lowered by about 1 eV through the charge transferprocess, giving rise to a suppression of interband transitions by 2 D EF ¹ 2 eV, asshown in ® gure 58 (b). The authors [39, 40] argue that upon evacuation of the samplechamber, the bromine dopant is completely removed from the isolated SWNTspresent in the sample, but not from the bundles which retain some of the adsorbedbromine even after evacuation of the chamber at 300 K. The authors [39, 40] relatethese arguments to the observation of the same tangential mode spectrum beforebromine addition and after evacuation of the sample chamber, as shown in ® gure 59.Because of the lowering of the Fermi level for the Br-doped nanotubes, no opticalabsorption and therefore no resonant Raman scattering occurs within the EM

11…dt†


Figure 58 (a). Resonance Raman spectra at many values of Elaser for a fully doped bromineSWNT sample prepared using a NiY catalyst (1:24 < dt < 1:58 nm). An additionalpeak around 240 cm¡1 can be seen for laser excitation energies Elaser ¶ 1:96 eV. Aftertaking these spectra, the sample chamber was evacuated at room temperature. (b)(left scale) The dependence on photon energy of the optical density (absorptionspectra) for a pristine (undoped) SWNT sample, and (right scale) the Ramanintensity ratio of the features at ¹240 cm¡1 appearing only in the doped sample tothe RBM feature at ¹180 cm¡1. The additional Raman peak near 240 cm¡1 onlyappears when the metallic window is satis® ed by the laser excitation energy [39, 40].


metallic window for the Br-doped SWNTs. The authors therefore argue [39, 40] thatonly the isolated SWNTs, from which the adsorbed Br is readily removed byevacuation of the sample chamber, can be resonantly excited by the Raman processat Elaser ˆ 1:78 eV for the evacuated sample. In this connection ® gure 59 shows thatthe Raman peak at 1540 cm¡1 identi® ed with metallic nanotubes is not present in theRaman spectrum at 1.78 eV for the evacuated sample. Their interpretation of thisobservation [39, 40] is that the Br-doped nanotubes in the nanotube bundles do notcontribute to the 1540 cm¡1 feature because light in the EM

11…dt† metallic window isnot absorbed when bromine is adsorbed to the SWNT bundles, and that the isolatedSWNTs also do not contribute to the 1540 cm¡1 feature, suggesting that theexcitation of the 1540 cm¡1 structure requires the nanotubes to be within a SWNTbundle [39, 40]. This conclusion needs to be con® rmed by other independentexperiments and understood in terms of a physical mechanism for the observationof the 1540 cm¡1 feature associated with metallic nanotubes in the Raman spectra. Itcan also be argued that only the metallic nanotubes which have mobile charge retain


Figure 59. The Raman spectra for the tangential band for a pristine sample (before dopingthe SWNT sample with bromine) (upper ® gure) and for the same SWNT samplewhich was evacuated after Br2 doping to saturation at room temperature (lower® gure). The spectral feature at 1540 cm¡1 observed in the sample before Br2 doping(upper ® gure) is missing in the spectrum for the evacuated sample (lower ® gure). Thelaser excitation energy Elaser ˆ 1:78 eV is within the metallic window for this SWNTsample before doping, but the downshift in Fermi level by ¹1 eV due to Br2 doping[37] prevents optical absorption at Elaser ˆ 1:78 eV associated with the EM

11…dt† inter-subband transition for the Br2-doped sample (see text) [39, 40].


adsorbed bromine after evacuation of the sample chamber, so that only semicon-ducting nanotubes can contribute resonantly to the tangential band Raman spectraat Elaser ˆ 1:78 eV for the evacuated sample chamber, since the metallic nanotubes inthis case do not absorb light at 1.78 eV [37].

Recently, the high resolution Z-contrast STEM technique, which is a sub-nanoscale probe capable of imaging individual atoms, was used to produce imagesof individual SWNTs that had been doped through contact with molten iodine [38].The images were taken on nanotubes which protruded beyond the ends of the iodinedoped bundles, showing chains of iodine atoms located inside the nanotube, but notin the interstitial channels. Supporting theoretical calculations con® rmed that chargetransfer actually occurs between iodine and the nanotube wall in the case of linear I¡

5and I¡

3 intercalated molecules. From these theoretical results, a strong interactionbetween the iodine molecule and the nanotube substrate was inferred. Several suchZ-contrast images on di� erent SWNTs were obtained and the images wereinterpreted to indicate that the iodine enters within the nanotube core region andthat charged polyiodide chains are intertwined inside an individual nanotube in theform of a double helix. Figure 60 [38] shows a high resolution Z-contrast image of aSWNT containing strands of iodine. The image shows a specially processed image toreduce the noise. The period of the helix is ¹5 nm, and the maximum separation ofthe strands is ¹0.65 nm, consistent with the inside diameter of a (10, 10) armchairnanotube. Other iodine helicities were also observed in other images. It should benoted that no post-synthesis treatment to open the nanotube ends was carried out, soit is not known if some nanotube ends were naturally open or whether the iodineitself opened the nanotube ends to receive the dopant, or if the dopant enteredthrough defects in the nanotube wall. Because most of the nanotube ends werepresumed to be closed, at least at the start of the iodine doping reaction, the bulk ofthe iodine dopant was viewed as residing in the interstitial channels [38]. Calculations


Figure 60. Z-contrast image of iodine atoms inside a single-wall carbon nanotube (upperleft). The inset in this image is a maximum-entropy processed image to reduce thenoise, and the schematic diagram shows the proposed structure for the iodine atomsas a double helix structure. The schematics (right and below) show how the proposediodine chains are located within a (10,10) armchair nanotube (from Fan et al. [38]).


of the Raman mode frequencies for a charged helical chain of iodine atoms have yetto be carried out.

Charge transfer reactions of as-prepared bundles of single-wall carbon nanotubeswith molten iodine (see ® gure 57) produced a much more dramatic e� ect on theRaman spectrum than was observed when the reaction was carried out in iodinevapour [155]. The Raman spectra clearly showed the e� ects of a charge transferreaction between the iodine and the single-wall nanotube bundles to be reversibleand uniform, based on the upshift of the tangential band frequencies with increasingexposure of the nanotube bundles to iodine, and the downshift of !tang when theiodine was de-intercalated. This reaction produces an air stable compound, which byweight uptake measurements is found to have an approximate stoichiometry C12I.The Raman spectra in ® gure 57 for undoped (pristine), moderately I-doped andsaturation I-doped single-wall nanotube bundles [155, 157] show a low frequencyregion which is rich in structure, identi® ed with the presence of intercalated chargedpolyiodide chains (I¡

3 and I¡5 ). No Raman evidence for neutral I2 (215 cm¡1 ) was

found in the nanotube samples. Saturation doping was found to convert the I¡3

observed in the moderately doped material into I¡5 . Analysis of these data suggested

that intercalation downshif ts the radial breathing mode band from 186 to 175 cm¡1,while the tangential band is upshif ted by 8 cm¡1 from 1593 to 1601 cm¡1. The radialmode downshift may be due to a coupling of the nanotube wall to the heavy iodinechains, while the small (8 cm¡1 ) tangential mode upshift (compared to the 24 cm¡1

upshift of the tangential mode upon Br2 intercalation) is attributed to the iodinechains being only singly ionized. If all the iodine were in the form of I¡

5 in the C12Icompound, we might write the formula as C‡

60 (I5)¡, which is equivalent to one holeper 60 C-atoms, and we therefore ® nd an upshift of 8 cm¡1 per additional hole per 60C-atoms in the single-wall nanotubes (or 480 cm¡1 per hole per C-atom). This shiftcan be compared to that of 6 cm¡1 per added electron in M-doped C60 (or 360 cm¡1

per electron per C-atom) [1].Both the radial and tangential Raman-active nanotube modes have been

observed to upshift or downshift signi® cantly with doping. Since the sign of themode frequency shift in nanotubes is consistent with earlier studies of intercalationin GICs and C60 , the shift has therefore been interpreted [161] in terms of C± C bondexpansion or contraction. For example, Br2 intercalation makes the tangentialmodes upshif t by 24 cm¡1. In the case of the donor dopants (K, Rb), the tangentialvibrational bands downshif t, and the spectra are remarkably similar, suggesting thatunder the experimental conditions used, both reactions proceed to the same endpointstoichiometry. The highest frequency tangential modes in the K and Rb intercalatednanotube Raman spectra were ® t to a Breit± Wigner± Fano lineshape, similar to thatfound for the ® rst stage MC8 GICs (M ˆ K, Rb, Cs), although the coupling constant…1=qBWF† is a factor of three lower than the …1=qBWF† observed in GICs [162],thereby leading to a more symmetric (Lorentzian-like) lineshape for the dopednanotubes. It should be noted that the spectra in ® gure 57 were taken for Elaser

outside the metallic window, while the spectra in ® gure 59 were taken for Elaser withinthe metallic window for their respective SWNT samples. While there is noinconsistency between the reported Raman results in the two cases, further workas a function of Elaser is needed to gain a detailed picture of the mechanismresponsible for the Raman feature at 1540 cm¡1 identi® ed with metallic nanotubeswith dt ¹ 1:4 nm. A variety of other Raman features in the region 900± 1400 cm¡1

were also observed for the donor intercalated nanotubes, but no speci® c assignments



of the mode features were made, since the corresponding modes in the undopedSWNT sample were not observed.

In situ Raman scattering studies, performed during the electrochemical anodicoxidation of single-wall carbon nanotube bundles in sulphuric acid [159], are ofspecial interest, and can be directly compared with similar studies on H2SO4 GICs[163]. In the case of the nanotubes, a rapid spontaneous shift of ¹15 cm¡1 in thetangential Raman modes was observed under open circuit conditions which was notobserved in the GIC system. In the H2SO4 single-wall nanotube study, a directmeasure was obtained for the charge transfer e� ect on the tangential modefrequency, i.e. 320 cm¡1 per hole per C-atom, in reasonable agreement with valuesdiscussed above for M-doped C60 [1].

5. Thermal properties

The thermal properties of carbon nanotubes, including their speci® c heat,thermal conductivity and thermopower, strongly depend on the phonon dispersionrelations and the phonon density of states. Although our present knowledge of thesethermal properties is very limited [54], we expect these properties to be unique tocarbon nanotubes and to display features characteristic of the low dimensionality ofthe carbon nanotubes.

5.1. Speci® c heatSince almost all experiments on thermal properties are done on collections of

carbon nanotubes, which contain both metallic and semiconducting nanotubes, theheat capacity (and the speci® c heat which is the heat capacity per gram of sample), inprinciple, consists of contributions from both phonons and electrons. For 3Dcrystalline graphite and 2D graphene layers, from which carbon nanotubes arederived, the dominant contribution to the heat capacity comes from the phonons,while the electronic contribution is so small that it can be essentially neglected, evenat low temperatures (i.e. below 4 K). Thus, we expect the heat capacity for SWNTsto be dominated by phonons. The phonon contribution to the heat capacity can ingeneral be written as:

Cph ˆ…1

0

kB

-h!

kBT… †2

» !… † exp …-h!=kBT †d!

exp …-h!=kBT † ¡ 1‰ Š2; …34†

where »…!† is the phonon density of states determined by the phonon dispersionrelations !…q† for the various phonon modes, which have been calculated for SWNTs(see section 3.1). In the low T regime, Cph for isolated SWNTs is expected to bedominated by the four acoustic phonon modes discussed in section 3.1, including thenon-degenerate LA mode, the doubly-degenerat e TA modes, and the non-degeneratetwist mode (TW) of the nanotube. All of these modes, which are populated at low T ,exhibit a linear ! ˆ vk dependence, where v and ! are the velocity of sound and thefrequency for each of the acoustic modes. From equation (34) we see that the low-temperature speci® c heat contains information about the dimensionality of thesystem, primarily through the density of phonon states »…!†, which is sensitive todimensionality. Because the low frequency phonons for an isolated single-wall



carbon nanotube have di� erent characteristics from 2D graphite (see section 3.1) andespecially di� erences in the low frequency phonon density of states (see ® gure 12 (c)),we expect the speci® c heat for the SWNTs to have a di� erent temperaturedependence than that for a 2D graphene sheet.

For an isolated graphene sheet, two of the (! ˆ vk) acoustic modes have a veryhigh sound velocity with vLA ˆ 24 km s¡1 for the LA mode and vTA ˆ 15 km s¡1 forthe TA mode, while the third out-of-plane transverse (ZAP) mode is described by aparabolic dispersion relation, ! ˆ ¬k2, with ¬ ¹ 6 £ 10¡7m2 s¡1 [151]. Thus the low-temperature speci® c heat of a graphene sheet has a contribution from the in-planemodes that is proportional to T 2, and a smaller contribution from the ZAP modethat goes as T 3.

The e� ects of combining weakly interacting graphene sheets in a correlatedstacking arrangement to form crystalline 3D graphite introduces dispersion alongthe c axis, which raises the dimensionality of the system, and the speci® c heatconsequently exhibits a temperature dependence Cph ¹ T 2:2 at low temperatures.Since the c-axis phonons have very low frequencies, thermal energies of ¹50 K aresu� cient to occupy all the ZAP phonon states, so that for T > 50 K, the speci® c heatof 3D graphite shows a T 2 dependence and basically exhibits 2D behaviour. Thiscrossover between 2D and quasi-3D behaviour, below which the interplanar coupledbehaviour becomes important, is denoted by a maximum in a plot of Cph versus T 2

[164± 166].Now let us consider the speci® c heat for isolated single-wall carbon nanotubes

(i.e. non-interacting nanotubes) . Our discussion of !…k† for SWNTs (see section 3.1)shows that SWNTs have 4 acoustic branches, which for small k have a lineardispersion relation ! ˆ vk. In the low T regime, only the acoustic branches of theSWNTs will be populated. The LA mode in the SWNTs is exactly analogous to theLA mode in a graphene sheet. The TA modes in a SWNT, on the other hand, are acombination of the in-plane and out-of-plane TA modes in a graphene sheet, whilethe twist mode is special to carbon nanotubes and is related to the in-plane TA mode(see section 3.1). These modes all show a linear ! ˆ vk relation at low k, but as® gure 14 shows, the various acoustic branches ¯ atten at rather di� erent k vectors.Furthermore there is no mode for the isolated single-wall nanotube that is analogousto the ZAP mode of a graphene layer which has a quadratic dependence on phononwavevector. All four acoustic phonon modes for the nanotube have high phononvelocities: vLA ˆ 20 km s¡1, vTA ˆ 9 km s¡1, and vTW ˆ 15 km s¡1 for a (10, 10)nanotube. The lowest lying optic mode for a (10, 10) nanotube (® gure 14) is thesquash mode, which is calculated to peak at 17 cm¡1 for an isolated nanotube, whichcorresponds roughly to a thermal energy of 25 K.

Since the phonon dispersion relations for all four acoustic branches are of theform ! ˆ vk, at least at low wave vectors, and the phonon density of states isexpected to be constant, independent of !, we can then expect Cph to be linear intemperature at low T [59]

Cph ˆ constk2

BT-hv

; T ½-hv

kBdt; …35†

where v is an appropriately averaged velocity of sound and dt is the nanotubediameter. As the temperature is raised, the lower lying s̀ubbands’ will becomeoccupied (see ® gure 14), and the power-law dependence T p of the speci® c heat willshow p increasing from 1 toward 2 approaching the power-law dependence p ˆ 2



which is valid for an isolated graphene layer. Preliminary inelastic neutron scatteringstudies suggest that the phonon density of states of SWNTs above ¹400 cm¡1 arevery close to that of a graphene layer, suggesting that the transition to T 2 behaviourfor the speci® c heat is not complete until well above room temperature [167].Benedict et al. [59] have shown theoretically that Cph / T , provided that thenanotube diameter dt and the temperature T are su� ciently small, i.e.T ½ 2-hv=…kBdt†, and we note that the temperature of validity of this relationshipdepends on the nanotube diameter. The crossover from a linear to a quadratic Tdependence is estimated to occur at T ’ 2-hv=…kBdt† or at ¹ 100 K for a (10,10)armchair nanotube.

The electron contribution to the speci® c heat is also expected to be linear in T formetallic nanotubes, especially at low T , while the contribution of the semiconductingnanotubes is much smaller at low T and is expected to exhibit an exponentialdependence since the density of states at EF in this case vanishes. Therefore, the ratioCel=Cph is expected to be independent of T at low T where Cel has the bestopportunity to contribute signi® cantly to the speci® c heat. But since v=vF ’ 10¡2,where v is the velocity of sound and vF is the Fermi velocity, and the carrier densityfor SWNTs is lower than in graphite, we expect phonons to dominate the speci® cheat of isolated SWNTs over the entire temperature range of measurement. We willsee below that dimensional crossover e� ects should also be present in bundles ofSWNTs because of the weak force constants between carbon atoms on adjacentnanotubes.

Figure 14 shows the low-energy phonon dispersion relations for a single (10, 10)nanotube. Rolling a graphene sheet into a nanotube has two major e� ects on thephonon dispersion relations as discussed in section 3.1. First, the two-dimensionalphonon dispersion relations of the sheet is collapsed onto one dimension because ofthe periodic boundary conditions in the circumferential direction of the nanotube,leading to the development of discrete phonon s̀ubbands’ . At the G-point, thesplitting between the subbands is of order [59]

D E ˆ 2-hvdt

: …36†

The second e� ect of rolling the graphene sheet to form a nanotube is to create a newacoustic mode (the twist mode) and to create low-energy subbands by zone foldingthe acoustic phonon branches, as discussed in section 3.1.

Experimental measurements by Yi et al. [168] of the speci® c heat for MWNTs(20 < dt < 40 nm) in the temperature range 10 < T < 300 K show a linear depen-dence of Cp on T (see ® gure 61). The inset to ® gure 61 compares Cp…T † for MWNTsand graphite, emphasizing the di� erence in behaviour in the low temperature speci® cheat. The extraordinarily large range of this linear T dependence (10 < T < 300 K)is somewhat unexpected considering the large diameters of these MWNTs. Theauthors attributed this linear T behaviour out to relatively high T values to theessentially isolated nanotube shell constituents of these MWNTs [168]. Thusthe authors conclude that the interaction between carbon atoms on adjacent wallsis very weak because of the turbostratic stacking of sequential layers.

The speci® c heat measurements of Mizel et al. [164] on MWNTs and on ropes ofSWNTs (see ® gure 62) give similar results to Yi et al. [168] in the temperature range100 < T < 200 K, regarding both the absolute magnitude of the speci® c heat and thelinear temperature dependence of the speci® c heat. The roughly quadratic depen-



dence of the speci® c heat at low temperature in ® gure 62 for ropes of SWNTs,MWNTs and graphite is clearly di� erent from the low temperature behaviour in® gure 61. More detailed plots of the speci® c heat data in ® gure 62 as Cp=T versus T(not shown) [164] reveal some di� erences in behaviour between the SWNT ropes, the


Figure 61. Measured speci® c heat of MWNTs [164, 168], compared to the calculatedphonon speci® c heat of graphene, graphite and isolated nanotubes [54].

Figure 62. Measured speci® c heat of MWNTs [164, 168], compared to the calculatedphonon speci® c heat of graphite and ropes of SWNTs.


MWNTs and graphite below 50 K, and especially below 5 K. Much smallerdi� erences are found between the behaviour of the MWNTs and graphite at lowtemperature [164]. Further work on better characterized samples is necessary todetermine the speci® c heat for nanotubes at low temperature and to determine thedi� erences in Cp…T † between SWNTs, SWNT bundles, MWNTs and graphite.

Calculations have been carried out to consider the e� ect of inter-tube interactionsin a SWNT rope on the phonon modes at low frequencies [59]. These calculationsshow that inter-tube interactions introduce a weak dispersion in the transversedirection, just as occurs in 3D graphite relative to a graphene sheet. In addition,some calculations indicate that the twist mode becomes an optical mode because ofthe presence of a non-zero shear modulus between neighbouring nanotubes [59].More detailed experiments on highly puri® ed and well characterized samples areneeded to evaluate the importance of these inter-tube interactions on the lowtemperature speci® c heat.

Also, the magnetic ® eld dependence of the speci® c heat might be measurablebecause of the large e� ect that the magnetic ® eld could have on the density ofelectronic states and therefore also on the interband transitions. Such e� ects havebeen considered theoretically [169], but no experimental observations of such e� ectshave been reported thus far.

5.2. T hermal conductivityDiamond and graphite (in-plane) display the highest measured thermal con-

ductivity of any known material, and, of course, the thermal conduction is entirelyby phonons in the case of diamond, and almost totally by phonons in the case of thebasal plane of graphite. The apparent long-range crystallinity of nanotubes suggeststhat the thermal conductivity of nanotubes along the nanotube axis should also behigh. The thermal conductivity therefore provides an important tool for probing theinteresting low-energy phonon structure of nanotubes.

In the one-dimensional limit, the phonon thermal conductivity can be written as:

µph ˆX

Cv2½ ; …37†where C, v and ½ are the heat capacity per unit volume, phonon group velocity alongthe nanotube axis and the relaxation time of a given phonon mode, and the sum inequation (37) is over all phonon modes. While the phonon thermal conductivitycannot be measured directly, the electronic contribution µel can generally bedetermined from the electrical conductivity ¼ by the Wiedemann± Franz law.

µel

¼Tº L 0 ; …38†

where the Lorenz number for a free electron gas is L 0 = 2:45 £ 10¡8…V =K†2 and thetotal thermal conductivity µ is given by µ ˆ µph ‡ µel. Thermal conductivity meas-urements to date [54, 168, 170] indicate that the phonon contribution µph isdominant µph ¾ µel in MWNTs and SWNTs at all temperatures. Therefore thestudy of µ…T † is important for describing phonon properties of carbon nanotubes,the subject of this review article.

In general, at high temperatures, the dominant contribution to the inelasticphonon scattering processes is phonon± phonon Umklapp scattering, while at lowtemperatures inelastic phonon scattering is generally by boundary or defectscattering. Thus at low temperature (T ½ YD), the relaxation time ½ is expected



to be independent of T so that the temperature dependence of the phonon thermalconductivity is the same as that of the speci® c heat discussed in section 5.1. Since theDebye temperature YD is large (>2000 K) for graphite and presumably also for the1D nanotubes, it is easy to satisfy T ½ YD even at room temperature. However, inan anisotropic material like graphite or 1D carbon nanotubes, the weighting of eachphonon mode by the factor v2½ becomes especially important, since for an isotropicmaterial the thermal conductivity is most sensitive to those phonon modes with thehighest group velocities and the longest scattering times. In analogy to graphite,where the in-plane thermal conductivity can be closely approximated by neglectingthe inter-planar coupling [151], it is expected that for SWNTs, µ…T † along thenanotube axis should be insensitive to inter-tube phonon modes, which havesigni® cantly smaller v than for on-tube phonons, and, in a ® nite bundle, asigni® cantly reduced ½ is expected for inter-tube modes as compared to the on-tubemodes. In this respect, µ…T † in SWNTs is expected to display 1D properties moreclearly than the speci® c heat.

The expected linear T dependence of µ…T † at low temperature 10 < T < 25 K isshown in ® gure 63 for a SWNT mat sample (dt ¹ 1:4 nm), and for highertemperatures a more rapid increase in µ…T † was found [54, 170]. Above 40 K,µ…T † in ® gure 63 can be approximated by a T 2 dependence until about 100 K, abovewhich the slope of µ…T † begins to decrease, indicative of the onset of phonon±phonon Umklapp scattering. From an estimation of the volume density ofnanotubes in this mat sample, the room-temperature magnitude of the thermalconductivity (µmat ) was estimated to be ¹35 W m¡1 K¡1. Simultaneous measurementof the electrical and thermal conductance of a given sample yields a ratio µ=¼Twhich is more than a factor of 102 at all temperatures, con® rming the dominance ofphonons in the thermal conductivity of SWNTs. The linear T dependence of µ…T †for T < 25 K in ® gure 63 implies a linear T low-temperature dependence for thespeci® c heat of SWNTs, which was not seen experimentally in ® gure 62 by this group(see section 5.1) [164].


Figure 63. Thermal conductivity of a SWNT mat sample. The linear T dependence of µ…T †at low T is emphasized in the inset [54, 170].


Measurements of the thermal conductivity for aligned multi-wall nanotubes havealso been reported [168] using the 3! measurement technique. These results for µ…T †on multi-wall nanotubes (see ® gure 64) over the temperature range 10 < T < 300 Klook qualitatively similar to those in ® gure 63 for SWNTs, except perhaps at lowtemperature, where there is no obvious region of linear T dependence for theMWNTs, and near room temperature, where the multi-wall µ…T † in ® gure 64 showsless saturation than the µ…T † for the SWNT in ® gure 63. It is curious that the µ…T †for the MWNT in ® gure 64 show a T 2 dependence at low T while a linear C…T † ¹ Tdependence is clearly seen for the same sample in ® gure 61. One possible explanationfor these di� erences in behaviour may be that all the walls of the MWNTs contributeto the speci® c heat, but that thermal contact in the thermal conductivity meas-urements is only made to the outermost wall of the MWNTs. The inner walls, havingsmaller diameters, would be expected to behave more like SWNTs, but the outer-most wall with an average diameter of 30 nm would be expected to be the only partof the MWNT that contributes signi® cantly to the thermal conductivity. Further-more, the outermost wall with a diameter of ¹30 nm is expected to be more like 2Dgraphite than like a SWNT. Since the MWNTs of this sample had 10± 30 walls [168],this interpretation would imply that the measured µ…T † is perhaps one order ofmagnitude lower than the ideal µ…T † because thermal contact is only made to one ofthe walls of each nanotube. If this interpretation is correct, the magnitude of µ…T †for SWNTs could be more comparable to that of graphite (¹2000 W m¡1 K¡1 at300 K) than would be implied by the value (¹27 W m¡1 K¡1 at 300 K) given in® gure 64. Measurements of µ…T † above 200 K are subject to radiation losses whichmust be carefully taken into account.

In contrast, measurements of µ…T † for carbon ® bres [116] show µ…T † to follow aT 2:3 temperature dependence from low T to ¹100 K and which then increases moreslowly until a peak is reached at ¹150 K, above which the thermal conductivitybegins to decrease with increasing temperature, from ¹6000 W m¡1 K¡1 at the peakto ¹2000 W m¡1 K¡1 at room temperature for the most crystalline vapour growncarbon ® bres [116]. This decrease in thermal conductivity is due to the onset ofstrong phonon± phonon Umklapp scattering, which becomes more e� ective withincreasing temperature as higher-energy phonons are thermally populated. For lessstructurally ordered carbon ® bres of various types (see ® gure 65), µ…T † can be ® t to a


Figure 64. Thermal conductivity of a bulk sample of aligned MWNTs and the data forµ…T † is ® t by a T 2 dependence (curve) up to about 120 K [54, 168].


T 2 dependence, indicative of the two-dimensional nature of these disordered carbons(called turbostratic carbons because they lack interplanar stacking order).

We now discuss the experimental low-temperature thermal conductivity of theSWNTs in more detail [170, 173]. At low temperature, some carbon nanotubesamples have been found to have a linear temperature dependence for µ…T † anddeviations from this linear behaviour begin to appear between 25± 35 K, as shown in® gures 63 and 66 [170]. At low temperature, only the four acoustic modes are


Figure 65. Log± log plot of the temperature dependence of the thermal conductivity ofvarious carbon ® bres for a variety of precursor materials and heat treatmenttemperatures. The ® bre with the highest thermal conductivity is a vapour growncarbon ® bre heat treated to ¹28008C and shows a temperature dependence of µ…T †close to that of single crystal graphite [116, 171, 172].


populated (see ® gure 14), while at slightly higher temperatures the lowest zone-folded phonon `subband’ (at 17 cm¡1 or an equivalent thermal temperature of 25 K)should begin to contribute. This behaviour can be accounted for by using asimpli® ed model (see ® gure 66), considering acoustic phonon contributions as wellas optical excitations. In a simple zone-folding picture, the acoustic band has adispersion ! ˆ vk, where k is the phonon wave vector, and the (doubly degenerate)lowest energy phonon subband has been modelled by a dispersion relation

!2 ˆ v2k2 ‡ !20 ; …39†

where the subband energy, as approximated by equation (36), was written as-h!0 ˆ 2-hv=dt and for simplicity the same velocity of sound was used as for theacoustic branch [54]. This approximation is equivalent to taking ! ˆ vk in the limit!0 ! 0. The thermal conductivity from each branch in ® gure 66 can then beestimated using equation (37) and assuming a constant scattering time ½ , andestimating the frequency !0 for the lowest subband edge from the calculateddispersion relations, or from the measured Raman spectra.

Figure 66 shows a comparison between the measured µ…T † of SWNTs, comparedto calculations based on the model discussed above. In making the ® ts, v was chosento be 2 km s¡1, which is a very low value for sp2 carbon-based materials. The topdashed line represents µ…T † of the acoustic band which is linear in T , as expected fora 1D linear ! ˆ vk dispersion relation and a constant ½ . The lower dashed line in® gure 66 represents the contribution from the lowest optical phonon subband, whichis zero at low temperatures, and begins to contribute near 35 K (24 cm¡1). The solidcurve, which is the sum of the two contributions , ® ts the experimental data quite wellwith this choice of parameters. The measured linear slope can be used to calculatethe scattering time, or, equivalently, a scattering length [54].

This model o� ers some support to the interpretation that the observed linear Tdependence of the low-T thermal conductivity is due to one-dimensional behaviour,


Figure 66. Points denote the measured low-temperature thermal conductivity µ…T † ofSWNTs, compared to a model based on ! ˆ vk for only acoustic branches (longdashed curve) and to a model (solid curve) that includes contributions from both theacoustic branch and the lowest optical phonon !2 ˆ !2

0 ‡ …vk†2 branch which isindicated by the dotted curve (see text) [54].


making nanotubes the only material which is of small enough size for this type of 1Dphonon quantization to be observable. The restricted geometry of the SWNTs mayalso a� ect the thermal conductivity at high temperature where Umklapp scatteringin general is the dominant scattering mechanism, but Umklapp scattering is expectedto be suppressed in one dimension because of the low availability of appropriatephonons for conservation of energy and wave vector [174]. In this connection it isinteresting to note that the Umklapp process above 100 K seems to be stronger forthe MWNTs (® gure 64) than for the SWNTs (® gure 63). Measurements on wellcharacterized and puri® ed nanotubes with di� erent diameters, as well as additionaltheoretical modelling, should prove interesting [169, 175].

Calculations of the magnetic ® eld dependence of the thermal conductivity havebeen carried out [175], indicating step structures associated with the Zeemansplitting. These e� ects may be observable below 1 K, and could a� ect thermalproperties through a magnetic ® eld dependent electron± phonon interaction.

5.3. T hermopowerThe thermoelectric power (TEP) or Seebeck coe� cient S gives the voltage

developed across a sample exposed to a temperature gradient. The dominantcontribution to the TEP is usually purely electronic (the `drift’ contribution), thoughin some materials, such as graphite, an electron± phonon scattering contribution (the`phonon drag’ e� ect) also becomes signi® cant in certain temperature ranges,especially at low temperature. Thus even though the thermal properties of nanotubesare dominated by phonons, we expect that the TEP will be primarily governed by theelectronic drift contribution, the electronic band-structur e of the nanotubes, andelectron scattering mechanisms.

We ® rst consider the expected TEP of a metallic SWNT by consideringcalculations for the nanotube electronic band-structure near the Fermi level (see® gure 5). For a conventional metal, the electronic contribution to the thermopowerS…T † is linear in T at low temperature and the magnitude of S is given by the Mottexpression [176], which reduces in one dimension to

S1¡D ˆ ¡ p2k2BT

3ev 0

v‡ ½ 0

½; …40†

where v is the electronic band velocity, ½ is the electronic relaxation time, the primesuperscript indicates that the derivatives of v and ½ are with respect to energy, andthe expression in brackets is evaluated at the Fermi level. In general, S < 0 forelectron-like systems, where v 0 > 0, while S > 0 for hole-like materials. The simplemodels thus far developed for metallic SWNTs have assumed that the overlapintegral in the tight binding approximation for a graphene sheet vanishes, yieldingsymmetric electron and hole bands (mirror bands) . For graphite itself, the C± Cnearest neighbour band overlap integral is s º 0:13 leading to substantial asymmetryin the electronic structure of the valence and conduction bands [2]. The electronicenergy bands for SWNTs near EF (see ® gure 5) are highly linear at EF. Therefore, the® rst term in equation (40) should be close to zero under the assumption of mirrorbands. The second term should also be small, because ½ should be approximatelyconstant when the electronic density of states is constant (as occurs for SWNTs inaccordance with ® gure 6 (b)). Thus an isolated metallic nanotube should have a TEPwhich is close to zero. This is a general result based on the linearity and the electron±



hole symmetry of the simple band-structure shown in ® gure 5. Other mechanismsgiving rise to thermopower (such as phonon drag, or carrier localization) also areexpected to yield S ˆ 0 for systems with electron± hole symmetry. Away from theFermi level, higher subbands display signi® cant curvature. The subbands above EF

are electron-like, while the subbands below EF are hole-like. Therefore, a hole-dopedmetallic nanotube should display a positive TEP, while an electron-doped nanotubeshould display a negative TEP. In view of the positive contributions to S from holesand negative contributions to S from electrons, it can be expected that measurementsof S…T † for well characterized electron and hole concentrations could be used tostudy the departures from electron± hole symmetry and to give information of theappropriate value for the overlap integral that should be used to model SWNTs.

We next consider the expected S…T † of a semiconducting nanotube, based on atypical nanotube band structure (see ® gure 5). When the Fermi level is in the gap, thecontribution from the conduction or valence band is given by

Ssemi…T † ˆ kB

e±

kBT‡ ¯ ln ½

¯ ln "‡ 5

2… †; …41†

where ± is the electrochemical potential and represents the energy di� erence betweenthe band edge and the Fermi level. Thus, when there is a zero density of states at theFermi level, the TEP varies as (1=T ) for a semiconductor, rather than as T for ametal. The total TEP will be the sum of the contributions from both the conductionand valence bands, weighted by their respective conductivities. In an intrinsicsemiconducting nanotube, the Fermi level lies exactly between the conduction andvalence bands where the density of states vanishes. Assuming mirror bands, thecontributions from the conduction and valence bands should be equal in magnitudeand opposite in sign, and the total thermopower from an intrinsic semiconductingnanotube should be zero, just as in the case of metal nanotubes under theassumption of mirror bands. Movement of the Fermi level toward one of the bandswill cause that band to dominate the TEP, producing a semiconducting-like TEPwhich is positive for hole-doping and negative for electron-doping. Higher dopinglevels will move the Fermi level into the conduction or valence band, producing adegenerate semiconductor with a metallic (i.e. linear in T ) electron-like or hole-likethermopower, respectively.

Figure 67 shows the temperature dependence of the thermopower reported for abulk `mat’ sample of SWNT ropes [173]. A linear T dependence of S…T † is observedat low T suggesting that metallic nanotubes make the dominant contribution in thislimit. In light of the expected small TEP of intrinsic metallic and semiconductingnanotubes, the observed magnitude (¹70 mV K¡1 at 300 K) is surprisingly large. Themeasured thermopower (® gure 67) decreases with decreasing T and at low Tapproaches zero with a linear slope, consistent with a non-zero density of states atthe Fermi level.

Other measurements of S…T † on as-grown mat samples prepared by either thepulsed laser deposition (PLD) or arc method yield similar results for S…T †, which arealso similar to the results in ® gure 67 [155]. However, S…T † was strongly reduced (byas much as a factor of 4 by doping with iodine) but S…T † was still found to bepositive after iodine doping and S…T † still showed a linear T dependence at low T .Temperature dependent resistance R…T † measurements on the same samples showeda decrease in R…T † by about a factor of 30, indicating that the e� ect of saturationiodine doping is to introduce a large concentration of holes into the p-electron bands



of the SWNTs [155]. Adsorbed gases are also found to sensitively a� ect themagnitude of S [177].

Attempts have been made to model the observed S…T † of SWNTs in ® gure 67 interms of metallic and semiconducting nanotubes in parallel, and S…T † for these twotypes of nanotubes in parallel has been calculated by weighting S for each nanotubetype by its conductance and dividing by the sum of the conductances [173]. Thismodel, however, fails to ® t the experimental data for reasonable values of the ® ttingparameters. Another approach to ® tting the TEP measurements for SWNTs wasmade by assuming that residual amounts of magnetic impurities from the catalystused to promote the SWNT growth process remains in the nanotube sample aftergrowth is completed [178]. The preparation of a SWNT sample free from transitionmetal catalyst would be interesting for S…T † measurements to determine asymmetriesbetween the electronic structure and occupation of the conduction and valencebands. At present, the connection between thermopower measurements and thephonons in SWNTs remains to be clari® ed.

6. Concluding remarks

The ® eld of carbon nanotube research is remarkable in terms of the uniquephysical properties of the carbon nanotubes, some of which are reviewed in thisarticle. In most sub® elds of condensed matter physics, experimental results have ledthe way and theoretical explanations have followed to give the sub® eld a ® rmfoundation. However, for the case of research on carbon nanotubes, theoreticalpredictions have often led experimental investigations. This situation is directlyrelated both to the di� culty in synthesis of su� cient quantities of pure and well-characterized materials for experimental investigations and to the fundamentalnature of SWNTs as an attractive prototype system for the theoretical investigationof 1D phenomena.

Although extensive experimental study of the phonons in carbon nanotubes isquite recent [28], progress in the experimental aspects of this ® eld since 1997 has been


Figure 67. The temperature dependence of the thermopower of a bulk mat sample of as-grown SWNTs from 4.2 K to 300 K [54, 173].


rapid, encouraging theorists to develop more sophisticated models, especially inconnection with the various resonant Raman phenomena. Despite this rapidprogress, the ® eld is still at an early stage of development, from both an experimentaland theoretical standpoint. Thus, in writing this review article, the authors haveattempted to focus on those models and measurements that they feel will stand thetest of time and will help to guide future developments in this ® eld.

At this point it is not clear whether the applications of carbon nanotubes will besu� cient to insure long term interest in carbon nanotube-based materials. However,because of the unique properties of single-wall carbon nanotubes and their use as aprototype 1D materials system, it is likely that scienti® c interest in carbon nanotubeswill continue to be a focus of the condensed matter research community for a fewmore years to come.

AcknowledgementsThe authors gratefully acknowledge the helpful and informative discussions with

Drs G. Dresselhaus, J.-C. Charlier, A. Jorio, A. M. Rao, Professors M. Endo,R. Saito, H. Kataura, M. A. Pimenta, Mr K. A. Williams, and Ms Sandra Brown.They are also thankful to many other colleagues for their assistance with thepreparation of this article. The research of MD is supported by NSF grantDMR 98-04734, and the work of PCE by NSF grants OSR-9452895 and DMR 98-09686 and by the University of Kentucky Center for Applied Energy Research.

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