single-wall carbon nanotubes phonon spectra: symmetry-based calculations

PHYSICAL REVIEW B 68, 045408 ~2003!

Single-wall carbon nanotubes phonon spectra: Symmetry-based calculations

E. Dobardzˇic,* I. Milosevic, B. Nikolic, T. Vukovic, and M. Damnjanovic´Faculty of Physics, University of Belgrade, P.O. Box 368, 11001 Belgrade, Serbia, Yugoslavia

~Received 21 December 2002; revised manuscript received 10 March 2003; published 10 July 2003!

The phonon dispersions and atomic displacements for single-wall carbon nanotubes of arbitrary chirality arecalculated. The full symmetry is implemented. The approach is based on the force constants of graphene, withthe symmetry imposed modifications providing the twisting acoustic mode exactly. The functional dependenceof frequencies of the Raman and infrared active modes on the wrapping angle and on the diameter arepresented. The armchair tubes are found to be infrared inactive under the light linearly polarized along the tubeaxis. Also the overbending absolute value and the wave vector dependence on the tube geometry are found andthe chirality selective method for the sample characterization is proposed. Finally, the specific heat calculationsare carried out.

DOI: 10.1103/PhysRevB.68.045408 PACS number~s!: 61.46.1w, 63.22.1m, 65.80.1n

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I. INTRODUCTION

Elastic and vibrational properties of the single-wall cabon nanotubes~SWCNTs!1 have been studied extensively bmany research groups2 and a large variety of the methodhave been applied: simple and modified zone folding3,4

nonorthogonal5 and orthogonal tight-binding Hamiltoniamodels fitted to the graphite,6 force constant7 and valenceforce field8 model, effective potential,9 andab initio10 calcu-lations. Recently,ab initio calculations~with comparison toother methods! for some SWCNTs have been reported.11 Inthis paper we present the results of the symmetry-bamethod of the SWCNT’s phonon dispersion calculations.using the full symmetry group of the SWCNT’s~Ref. 12! weeasily carry out numerical evaluations of the phonon spefor the tubes of any chiralities and diameters. Also,solved exactly the problem of adjusting the graphene foconstants to the SWCNT geometry, and get the rotatioacoustic mode.

Our preliminary results on the SWCNT’s phonon dispsions have already been highlighted within the Raman stering framework.13 In this paper we present detailed dscription of the method we are applying and prescomplete, thoroughly tested, results on the vibrational prerties of the SWCNT’s. This group theoretical concept eables also calculation of the dispersions of multiwall carbnanotubes: the spectra of the double-wall ones we expereport soon.

We adopt the model of ideally structured infinite and islated SWCNT~Ref.12! and apply the line group.14theoreticalapproach: the modified Wigner projectors procedure15,16

implemented into the computer program POLSym.17 Thisimproves both the precision and efficiency of the calcutions, enabling us to study the phonon dispersions andresponding eigenvectors for 1280 SWCNT’s~all the tubeswith diameters in the [email protected] Å, 50.0 Å#!. It is reliablestatistics for the investigations of the dependence of the vous phonon spectra properties on the parameters such atube helicity and diameter. Apart from the technical advatages, the full symmetry implementation gives many funmental results: the phonon bands assignment by the fulof the conserved quantum numbers, band degeneracy,

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the selection rules for various processes,18 extraction of theRaman and infrared~IR! active modes.

In what follows we firstly introduce the computationmethod and explain briefly its specific symmetry-based cowhich is not incorporated in the other force-constant meods~Sec. II!. Then we give the results that stem from the ligroup symmetry considerations only and the Raman andactive-modes frequencies as functions of the tube parame~Sec. III!. In Sec. IV we discuss the phonon dispersions, laexcitation energy dependence of the high-frequency Ramactive mode and related to that, the overbending wave veand chiral angle dependence, the heat capacity, sound veties, and some other details. The paper ends with the bsummary~Sec. V!.

II. METHOD

A. Computational scheme

The computational code POLSym, which we use in tstudy, was originally applied to the DNA molecule and prsentedin extensoelsewhere.17 It is based on the modifiedWigner projectors for induced representations15and linegroup symmetry.14

We consider an infinitely long, isolated, and perfecstructured SWCNT, impose no periodic boundary conditioon its geometry and take a benefit of the fact that a SWCis a single-orbit system~i.e., the cylindrical carbon web cabe obtained from any atom by action of the symmetry groof the tube considered!. This way we maximally reduce thedimension of the eigenvalue problem to be solved, to at m12. In principle, this dimension is the product of the following three factors:~i! number of the orbits of the system~1 forSWCNT’s!, ~ii ! dimension of the orbit representative interispace~3 for phonons!, and ~iii ! dimension of the relevanirreducible representation~1 or 2 for the chiral; 1, 2, or 4 forthe achiral SWCNT’s!. Each irreducible representation, i.eeach set of quantum numbers, defines the so-called tranoperators,15 which pull down the eigen problem to suchlow-dimensional space. The dispersions are automaticobtained as the eigenvalues of the reduced~pulled-down!operators. To get the displacements, the eigenvector ofpulled-down problem is mapped back to the entire space

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E. DOBARDZIC et al. PHYSICAL REVIEW B 68, 045408 ~2003!

the partial~over the irreducible space only! scalar producttaken. Full mathematical formalism of the tight-bindinmethod~being applicable to molecular dynamics calculatioas well! we use in this work is presented in Ref. 16 areferences therein.

B. Symmetry

Line groups14 ~known also as rod or monoperiodgroups! are the maximal subgroups of the Euclidean grothat leave the quasi-one-dimensional crystals invariant. Smetry groups of chiral (n1 ,n2), zig-zag (n,0), and armchair(n,n) tubes (C, Z, andA, for short! are nonsymmorphic linegroups:12 LC5Tq

r Dn5Lqp22, LZA5T2n1 Dnh5L2nn /mcm.

Here, n is the greatest common divisor ofn1 and n2 ,q52(n1

21n1n21n22)/nR is the order of the isogonal grou

principle axis, R53 if (n12n2)/3n is integer, otherwiseR51, while the parametersr and p describe the helicityby more complicated functions12 of n1 andn2. The generalline group element is, tsuv5(Cq

r una/q) tCnsUusv

v , whereCn

s (s50, . . . ,n21) is the rotation through 2ps/n aroundthe tube axis (z axis!, the Koster-Seitz symbol (Cq

r una/q) t

(t50,61, . . . ; a is the translational period of the tube! de-notes the rotation for 2rtp/q around the tube axis followedby the translation along it fornat/q, while U is the rotationthroughp ~thusu50,1) around the horizontal axis througthe center of a carbon hexagon perpendicularly to the tuWe fix a tube reference frame12 so thatx andz axes coincidewith theU and the tube axes, respectively. Achiral tubes han extra symmetry: vertical mirror reflectionsv in the xzplane~for C tubes one should takev50). Note that, havingboth sv and U symmetry,Z and A tubes have horizonta(xy) mirror plane symmetrysh5svU5Usv as well. Bymapping any carbon atom by the entire set of the transmations , tsu5, tsu0 @which are concretely defined by thwrapping indices (n1 ,n2)] one builds the correspondintube. In other words, SWCNT is a single-orbit system andcarbon atoms can be labeled as Ctsu .

The symmetry singles out the complete set of the qutum numbers of quasimomenta and parities. The pair of thzcomponents of linear and total angular quasimomentak andm is most frequently used, althoughm is not conserved.While k runs over the Brillouin zone~BZ! (2p/a,p/a#, mtakes on the integer values from (2q/2,q/2# ~note thatq iseven!. Besides the quasimomenta, there is a parity withspect to the twofold horizontal axis, and for the achiral tubonly, in addition, there are vertical and horizontal mirrparities.

Each normal mode is assigned by a setm of these quan-tum numbers~i.e., m is a particular choice ofk, m, and theparities values!. The same set singles out the irreducible reresentationDm ~i.e., umu-dimensional matrices16 explicatingthe law of the mode transformations under the symmetri!.Let emM5$ . . . ,emM(a), . . . % denotes such a mode, in acordance with the usual notation:19 the atoma is displacedby the vectoremM(a) with the coordinatesei

mM(a). Notethat M51, . . . ,umu distinguishes between the degenermodes forming the multiplet ofDm. Then the transformations , tsuv mix the modes only within the same multiplet:

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, tsuvemM5(M8

DM8Mm

~, tsuv!emM8.

Each modeemM is a collective vibration of the SWCNT atoms, i.e., it consists of the three-dimensional~3D! displace-mentsemM(tsu) of all the Ctsu atoms. Within the full sym-metry implemented tight-binding method,16the modifiedgroup projectors technique uses the symmetry to transferrelevant information to the three-dimensional configuratspace of initial atom C000 ~multiplied by the space of theconsidered irreducible representation!. The results of the ei-genvalue problem are the normal mode frequencies anddisplacementsemM(000) of this atom. Finally, the same technique employs symmetry to determine the displacementother atoms fromemM(000) in the generalized Blochform,15which for SWCNT ~being a single-orbit system! re-duces to

emM~ tsu!5(M8

DM8Mm

~, tsu!Dpv~, tsu!e

mM8~000!, ~1!

whereDpv denotes the vector representation of the symmegroup.

Working with the space of initial atom C000 only, themodified group projectors technique15,16 establishes the generalized Bloch form of the normal mode: conveniently, asany single-orbit system, all the atomic displacements aretermined by that of the C000 atom.

C. Force-constant model

We start with the graphite force constants,3 and adjustthem~kinematically and dynamically! to the nanotube geometry. To clarify this two-step modification, in the dynamicmatrix19 we single out the 333 submatrixFb

a with elementsFb j

a i 5F i j (a,b), referring to the pair (a,b) of carbon at-oms. For the atomsa andb, the stretching direction is alongthe unit vectoruab,1& from a to b, the out-of-plane unitvector uab,2& is perpendicular both to the tube axis anduab,1&, while the in-plane unit vector isuab,3&5uab,1&3uab,2&. These three vectors are assumed to be the eigvectors of the matrixFb

a , with the force constantsciab as the

corresponding eigenvalues:Fbauab,i &5ci

abuab,i &. With thehelp of the eigenvectors’ coordinates in the tube’s refereframe uab,i &5(S1i

ab ,S2iab ,S3i

ab), the coordinate form of theextracted submatrix is found:Fb

a5( iciabuab,i &^ab,i u, i.e.,

Fb ja i 5(pcp

abSipabSjp

ab .As the first step, we make pure kinematic modificati

that provides the twisting mode exactly. Recall the rotatiosum rule:20

(b

~Rab1Fb2a j 2Rab2Fb1

a j !50, ;a, j 51,2,3 ~2!

where the sum runs over the relevant neighborsb of theatom a (bÞa), Fb i

a j is an element of the matrixFba , and

Rab i is the Cartesian component of the vectorRab from a tob in the equilibrium positions ~note that uab,1&

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SINGLE-WALL CARBON NANOTUBES PHONON . . . PHYSICAL REVIEW B68, 045408 ~2003!

5Rab /Rab). As uab,2& is orthogonal to the tube axis, itszcomponentS32

ab vanishes. Thus, coordinate form of Eq.~2!reduces to

(b

Rab~c2abSj 2

abS33ab2c3

abSj 3abS32

ab!50, j 51,2,3,;a,

i.e., to the pair of equations for eacha atom,

(b

Rabc2abS12

abS33ab50, (

bRabc2

abS22abS33

ab50. ~3!

Note that these conditions involve the out-of-plane foconstants only. As has already been emphasized a SWCNa single-orbit system, which enables to perform the calcutions~within the modified group projector technique! by con-sidering one carbon atom only, e.g., the orbit representaa5C000. Thus, Eq.~3! imposes two conditions altogetheThese are included exactly as follows. Up to the fourth lethere are 18 neighbors; collecting all the out-of-plane foconstants in the vectorc2

a5(c2a1 , . . . ,c2

a18), Eq. ~3! givesthe orthogonality conditionsc2

a•Sj

a50, j 51,2, where thevectors Sk

a5(Ra1Sk2a1S33

a1 , . . . ,Ra18Sk2a18S33

a18) comprise thecoordinate factors. Thus, any choice of the force constaproducing the vectorc2

a simultaneously orthogonal to thvectorsS1

a andS2a will exactly provide the twisting acoustic

mode. It is, therefore, natural to correct the physically suable set of the graphene force constants minimally, projing them out onto the orthocomplement of the plane defiby S1

a andS2a .

In order to include the dynamical response to the confiration changes~due to the rolling up of a graphene plane! wemake Ansatz that is analogous to that presented in the wof Saito et al.4 In the frame in whichz axis coincides withthe tube axis andx axis runs through the atoma, the eigen-basis ofFb

a reads

uab,1&5S 2cosc sinw

2,cosc cos

w

2,sinc D ,

uab,2&5S cosw

2,sin

w

2,0D , ~4!

uab,3&5S 2sinc sinw

2,sinc cos

w

2,2cosc D .

Here,w is a cylindrical coordinate of theb atom andc is theangle between the horizontalxy plane and the straight linethat connectsa and b. On the other hand, the graphenstretching, out-of-plane and in-plane unit vectorsuagbg ,i &are in the SWCNT’s geometry obtained by rolling upgraphene sheet:uagbg,1& becomes the tangent to the projetion of the line connectinga and b to the tube’s surfaceuagbg,2& is perpendicular to the tube ata, while againuagbg,3&5uagbg,1&3uagbg,2&. Their coordinates, in thesame reference frame as Eq.~4!, are the following:

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uagbg,1&5S 0,

cosw

2cosc

A12sin2w

2cos2c

,sinc

A12sin2w

2cos2c

D ,

uagbg,2&5~1,0,0!, ~5!

uagbg,3&5S 0,sinc

A12sin2w

2cos2c

,2

cosw

2cosc

A12sin2w

2cos2c

D .

The Ansatz is to modify the original graphene force costantsCi

agbg so to provide that the component of thei th force( i 51,2,3) alonguagbg ,i & is of the same intensity as in thgraphene plane. Simple geometry shows the required mfied constantsci

ab to be related to the original onesvia ciab

5Ciagbg/u^agbg ,i uab,i &u. The expansion over sine and co

sine ofw/2 yields

c1ab5C1

agbgS 22cosw

2 D ,

c2ab5C2

agbgF11sin2cS 12cosw

2 D G , ~6!

c3ab5C3

agbgF11cos2cS 12cosw

2 D G .In our calculations we first find the dynamically correct

graphene force constants by Eqs.~6!, and afterward projectthem. It should be noted that the kinematical correctiondominant, in the sense that if it is applied directly to tgraphene constants, avoiding the dynamical correction,results differ only slightly, except for the very thin tubes.general, due to the similar dynamical correction introducour results are close to that of Ref. 4, with exactly foacoustic mode.

III. ACOUSTIC, RAMAN, AND IR ACTIVE MODES

The quantum numbers of the momentum componentpz

~along the tube axis; tensor0A02) arek50 andm50. As for

the parities, it changes the sign upon thez reversal transfor-mations and is invariant under the vertical mirror reflectioConcerning the transversal momenta components, theythe 2D irreducible subspace of the0E1

(1) tensor; thus,k50, m561, and for the achiral tubes only~therefore in thebrackets! 1 for the horizontal mirror invariance. Hence, thpolar vectors, such as the linear momentum, transformcording to the representationDpv decomposing as0A0

2

10E1(1) . This further means that besidesk50 for all the

acoustic modes, for the longitudinal translationm vanishes aswell, while the two lateral translations are degenerate acharacterized bym561. Naturally, longitudinal translation

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changes sign upon thez reversal and is not affected by thvertical mirror, while the lateral ones, due to their degeeracy, have no defined parity besides the horizontal miinvariance. The twisting mode is, withm50, odd with re-spect to thesv and U transformations and, consequentinvariant under the horizontal mirror reflections~tensor0A0

2

for chiral and 0B01 for achiral SWCNT’s!. Bearing this in

mind, it is easy to extract the IR and Raman active mofrom the SWCNT’s dynamical representations.12

The linear momentum quantum numbers determine theactive modes as well: in the dipole approximation the indent electric field polarized parallel to the tube axis involvpz ; perpendicular linear polarization relatespx and py ,while the standard componentsp65(px7 i py)/2 describethe circular polarization. For the chiral tubes we find sixactive modes, out of which only one~of the 0A0

2 type! isactive under the parallel and all the others (0E1) under theperpendicular~either linear or circular! polarization of thelight. It should be stressed out that in the armchair confiration, there are no active modes for the parallel polarizatunlike the zig-zag case where there is one such mode. Uperpendicularly ~or xy circularly! polarized light, thereare three ~two! active modes in the armchair~zig-zag!configuration.

The momenta quantum numbers further determine thelection rules for first-order Raman scattering. The total Rman tensor transforms as the squareDpv

^ Dpv of the vectorrepresentation. Its symmetric and antisymmetric parts cospond to the symmetrized and antisymmetrized squa@Dpv#2520A0

110E110E2 and $Dpv%250A0210E1 for the

chiral SWCNT’s, while in the achiral cases@Dpv#2520A01

10E1210E2

1 and $Dpv%250B0110E1

2 . The antisymmetricpart of the Raman tensor should be relevant generally forchiral systems,21 as well as in the resonant scattering prcesses, which proved to be important in SWCNT’s.

By the use of the above decomposition, we extract theand Raman active modes from the SWCNT’s dynamirepresentations.12 As for the symmetric part of Raman tensoin accordance with Ref. 22, for the chiral tubes we findactive modes: 30A0

1 , 50E1, and 60E2. The achiral tubeshave eight Raman active modes out of which two are totsymmetric. The other six phonon modes are of the types0E1

2

and 0E21 ~three of each for the zig-zag and two and fo

respectively, for the armchair tubes!.When both the incoming and outgoing light are polariz

along the tube axis, the electron transitions occur betwthe subbands with the same quantum numbers, and onm50 phonons accompany such processes. Atk50 these aretotally symmetric modes0A0

1 : besides the low-frequencradial-breathing mode~RBM!, for the chiral tubes there artwo such high-frequency modes~HFM!. Nevertheless, beingodd in the vertical mirror parity, one of the HFM’s~second/third in Table I! becomes Raman inactive in the case ofzig-zag/armchair tubes; the remaining one is with out-phase displacements orthogonal/tangential to the tubecumference. In the case of the crossed polarization, thenon mode required is of the0E1

(2) type ~i.e. with m561),while when both the incoming and outgoing light are pol

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ized perpendicularly to the tube axis, the phonons inRaman process should satisfym50 or m562. They areeither totally symmetric~as in the parallel-parallel case! or ofthe 0E2

1 type, since the electron can be scattered fromm21 to m11 subband. As well as the symmetric one, tantisymmetric scattering with the crossed polarization isall the tubes enabled by the mode0E1

(2) . The antisymmetricscattering with both incoming and the scattered light polized perpendicularly is allowed in chiral and armchair tubonly, due to the extra~besides the twisting acoustic! 0A0

2 ,i.e., 0B0

1 mode; this is related to theDm50 electronic tran-sitions. The scattering with parallel polarization of the icoming and scattered light is totally symmetric for all thtube types; in the zig-zag tubes such is also the scattewith both polarizations being orthogonal.

The diameter dependence of the Raman active-modequencies we give in Table I. The fit is made over 12SWCNT’s in the diameter [email protected] Å , 50.0 Å #, whichincorporates all the tubes between~2,2! and ~53,18!, in par-ticular, the zig-zag tubes from (4,0) to (63,0) and the archair from (2,2) to (36,36). In Fig. 1~right panel! we give,for the tube~15,4!, the dispersions of the branches that,k50, end up by the Raman and IR active modes.

The first entry in Table I is the RBM. However, it haszcomponent except in the armchair case. Otherwise, it sha nonvanishing component along the tube axis that depeon the chiral angleu and decreases with the diameterD. Forthe unit outward displacement, the longitudinal componof the RBM is well fitted by

zRBM~D,u!5S 0.197

D2

0.167

D3 D cos 3u ~7!

(D should be inserted in angstroms!. Indeed, in the armchaitubes, as all the atoms lay in one~of infinitely many! hori-zontal mirror plane, the symmetric modes~denoted by super-script 1) must have vanishing longitudinal component.contrast, for thesh-odd modes (2) the horizontal compo-nent necessarily disappears. As all thek50 modes are eitheeven or odd insh , they are either perpendicular or parallelthe tube axis. Analogously, the atoms in the zig-zag tubesin the vertical mirror planes and the symmetric~A! atomicdisplacements are necessarily within these planes, whilesv-antisymmetric ones~B! are tangential to the tube circumference. This refers to the entire subbands withm50,n,since unlike thesh which characterizes onlyk50 states forall m, the vertical mirror parity is defined for allk but form50,n only.

IV. PHONON DISPERSION CURVES

A. General characteristics

The number of the carbon atoms per unit cell of the natube isN52q, giving for the chiral tubes 6q double degen-erate phonon branches overkP@0,p/a#. Four of them areacoustic. In the achiral casesq52n and there are 12 doubland 6(n21) four-fold degenerate branches. The angular mmentum quantum numbers of the former arem50,n, while

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TABLE I. Frequencies of the Raman and IR active modes as functions of the tube diameterD and the chiral angleu. The classificationis based on the irreducible representations of the chiral tubes~first column!. In the second and third column the corresponding representaof the zig-zag and armchair tubes are given. For each type of SWCNT’s, all the processes in which the mode is active are indicaI andR stand for IR and Raman activity, respectively, with the polarization of incident and scattered light indicated in the superscript;@R# and$R%emphasize that the mode contributes to the symmetric and antisymmetric part only of the Raman tensor. In the last column we givefunctional dependencev(D,u) (v in cm21 for D in Å!. Obviously, the first term in the expansion is the limiting graphene value. Apart fthis, for the tubes~12,8!, ~17,0!, and~10,10! the concrete values of the IR and Raman active-mode frequencies are listed in the firscolumns, respectively.

Chiral ~12,8! Zig-zag ~17,0! Armchair ~10,10! v(D,u)

0A01 @R# ii,@R#'' 164 0A0

1 @R# ii,@R#'' 168 0A01 @R# ii,@R#'' 165

012243

D2

665

D3

1584 0B02 — 1584 0A0

1 @R# ii,@R#'' 158415882

757.6 cosu

D22

1069.9 cosu

D4

1587 0A01 @R# ii,@R#'' 1588 0B0

2 — 158815881

59.8 cos 6u

D2

0A02 $R%' ',I i 873 0A0

2 I i 873 0B01 $R%' ' 873

86811450

D22

3899.4

D4

0E1 Ri',R'i,I' 116 0E12 Ri',R'i 119 0E1

2 Ri',R'i 11701

1588.6

D2

5180.8 cos 2u

D4

232 0E11 I' 238 0E1

1 I' 233

013157.7

D2

139426sinSp3 12uDD6

871 0E12 Ri',R'i 871 0E1

1 I' 871

8651

1510.7 cosu

2

D22

5073.1 cosS p

121u D

D3

1583 0E12 Ri',R'i 1585 0E1

1 I' 1586

15882

1037.6 cosSp3 2uDD2

2

3311.6 cos2Sp19u

6 DD3

1586 0E11 I' 1584 0E1

2 Ri',R'i 1583

15881

48.8 cos2Sp6 2uDD

2

967.3 cos2Sp26u

12 DD2

0E2 @R#' ' 21 0E21 @R#'' 21 0E2

1 @R#'' 2101

4254.7

D22

5556.4

D3

231 0E22 — 236 0E2

2 — 23301

3190.9

D2

12121.5 cosSp6 1uDD3

365 0E21 @R#' ' 376 0E2

1 @R#' ' 367

015012.8

D2


865 0E22 — 865 0E2

1 @R#' ' 8668651

860.3

D22

10196.6 cosu

D3

1570 0E21 @R#' ' 1570 0E2

2 — 1569

158823891.2

D21

8395.2 cosSp6 1uDD3

1589 0E22 — 1587 0E2

1 @R#' ' 1589

15881

753.3 cos2Sp16u

12 DD2

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the latter are characterized bym51, . . . ,n21. In k50,only the chiral tubes branches assigned bymÞ0,q/2 remaindouble degenerate, while degeneracy is halved in ocases.16,18

Sincem is related to the isogonal group~not a subgroup inthe nonsymmorphic SWCNT’s symmetry group!, it is notconserved quantum number.12,18 Particularly, it is not con-served in the Umklapp processes. Therefore, whenbranches~and not onlyk50 modes! are studied, it is advantageous to switch to the conserved, so-called helical quannumbersk and m of the helical and the angular momentThe range (2qp/na,qp/na# of k, i.e., the helical BZincludesq/n Brillouin zones; on the contrary, the rangemP~2n/2,n/2# is q/n times smaller than that ofm. Hence,a single band assigned by the helical (mk) quantum numberssplits into q/n bands in the linear (km) quantum numbersassignation. The orderq of the principle axis of the isogonagroup is usually very high, and the reduction of the numof branches by a factorq/n in the helical assignation athe cost of use of theq/n Brillouin zones may facilitatethe visualization and interpretation of the branch structur13

Nevertheless, the two sets of quantum numbers areuniquely related:

k5k1m2rp

na1k

2qp

na, m5m modn, ~8!

k5 k2m2rp

na1k

2p

a, m5~m2kp!modq. ~9!

FIG. 1. Phonon dispersion curves for the~15,4! tube. Left panel:Bands assigned by the helical quantum numbers. Sinceq5602, n51, and a574 Å, the positive half of the helical BZ is

@0,25.6 Å21#, while m50 for all bands; TA modes show up atk

513.6 Å21, and LA and TW ones show up atk50, in accordancewith Eq. ~8!. Folding left panel to one 602th, one gets bandssigned by linear quantum numbers:kP@0,0.04 Å21# and m52300, . . .,301. Right panel: In the linear BZ scheme thbranches that ink50 end up with the states listed in Table I together with the acoustic ones are presented. Note very small dision of all the bands. Also, the acoustic dispersion curves are gin the inset.

04540

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e

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Here the integersk and k are introduced to providekP(2qp/na,qp/na# and k P (2p/a,p/a#, respectively. InFig. 1 ~left panel! we give the phonon dispersions for th~15,4! tube in the helical BZ scheme. Asn51, all the bandsare labeled by zero helical angular momentum quantnumber~no pure rotations!!. Also, according to the relation~8! LA and TW modes carry zero helical momentum quatum number, while TA modes are characterized by nonvishing k.

Concerning the obtained results, we stress out thatcalculations are performed on the same 1280 tubes as beThe acoustic branches are obtained exactly: in the chtubes the transversal acoustic~TA! branches~beginning atk50 with the double degenerate TA mode! are assigned bykE1 and kE21 representations, the longitudinal acous~LA ! and twisting acoustic~TW! branches bykE0, both ofthem being double degenerate over the positive half of Bin the achiral tubes, mirror planes cause additional degeracy of the single fourfold TA branch assigned by the reresentationkG1, while the double degenerate LA and TWbranches are assigned bykE0

A and kE0B ~differing by the ver-

tical mirror parity!, respectively. All the acoustic branchenear theG point are linear ink, with the slopes, i.e., thesound velocities, being basically tube independent:vTA59.41 km/s,vLA520.37 km/s,vTW514.98 km/s. These results agree with the previously reported ones,3,4despite somearguments on the quadratick dependence of TA branch.23,24

Acoustic dispersions of the~15,4! tube we give in the inseof the right panel of Fig. 1.

B. Specific heat

Theoretical predictions of the specific heat of some pticular isolated SWCNT~Refs. 23,25! are in a reasonablygood agreement with the heat capacity measurementbundles of SWCNT’s.26–28 On the other side, up to ouknowledge, the fine dependence of this quantity on the natube parameters has not been systematically studied. Inwork we also calculate the specific heat for 225 isolaSWCNT’s of all the chiralities and a wide range odiameters.

Close tok50 the SWCNT acoustic branch we find to blinear in k. As this point is relevant in the specific heat cotext, we interrelate the graphene and SWCNT’s acoubranches. As it can easily be seen, the graphene TA mturns into the tube’s RBM upon rolling up the sheet into tcylinder. Similarly, the degenerated in-plane translations crespond to the TW and LA acoustic modes of a SWCNTthe translational directions are taken so to be parallelorthogonal relative to the chiral vector. Finally, note that t~degenerate! SWCNT’s TA modes are not related even to tgraphene BZG point, but to the certain edge point along thKM direction.

Close to theG point TA graphene acoustic branch is qudratical, while the other two~degenerate! subbands are lineain the wave vector. The van Hove singularity at the RBfrequency for all the SWCNT types and diameters methat the corresponding band is quadratic ink in k50 vicinity.Further, in the zone folding method, the graphene bands

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FIG. 2. The heat capacityCph dependence ontemperature~upper panel! for 225 tubes, wherediametersD range from 2.8 Å to 20.5 Å, i.e.,starting with ~2,2! and ending up at~24,4!. Thethickness of the curve we interpret by meansthe upper inset, which illustrates, at various fixetemperatures, rather slow variation of the speciheat Cph with the tube diameter. High-temperature limit can be seen from the lower iset.

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hep tod

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are sliced in the direction described by chiral vector dcretely, in the steps corresponding to the quantum numbem.Thus, for sufficiently dense set ofm values, the ‘‘branchesalong-m’’ with fixed k50 can be considered. Naturally, foTA branch~i.e., the one that matches the RBM! suchm bandsshould be quadratic. We verified this on the chiral tu~53,18! having 49 044 bands: the series of0Em bands that form50 gives RBM is quadratic inm nearm50. Among theacoustic modes, only the twisting modem branch is qua-dratic in m50.

Further it is relevant to determine precisely the tempeture region in which only the acoustic modes contributethe SWCNT specific heat. The phonon branch which cosponds to the minimal optical frequency is assigned askE2

for chiral, andkG2 for achiral tubes, starting atk50 with theRaman active0E2 and 0E2

1 modes~see Table I!. If vmino is

the frequency of the lowest-energy optical mode, then fortemperatures belowTo'\vmin

o /6kB the specific heat is determined by the acoustic phonon branches only.23,26We find itsdependence on the tube diameter:To5(7.210.05D11.045D2)21103 K (D in Å!. The minimumvmin

o is at ko ,which besides being chirality dependent~e.g., for achiraltubesko50) rapidly decreases with the tube diameter. Tgether with the above discussion on the RBM mode, tshows that the different regimes of the specific heatCph be-havior are diameter dependent and that the crossover tographeneT2 dependence is continuously reached by theear regime narrowing with the tube diameter.

In Fig. 2 the calculated specific heat temperature depdence up to 300 K is presented. It matches nicely the msured values.27 The increase of the curve width with the temperature indicates that slight differences ofCph between thetubes at higher temperatures are to be expected. The spheatCph of a SWCNT shows the diameter dependence. Yas it is enlightened in the upper inset, the dependence ondiameter is saturated for considerably large tubes. Inlower inset of Fig. 2 we give the high-temperature limit thagrees reasonably well with the expected value 3kB /m forthe carbon systems.

04540

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The chirality shows no signature in the tubes specific heThis is illustrated in Fig. 3 where the specific heat tempeture dependence coincide for the tubes~10,10!, ~12,8!, ~15,4!and ~11,9! with the same diameterD513.6 Å but differentwrapping angle. Besides, the log-log plot up to 100 K, tspecific heat low- and high- temperature dependence, uTo(13.6 Å)55 K and 300 K are depicted in the insertepanels.

C. Overbending

As it is well known, in contrast to almost all covalensolids, graphite phonon branch that corresponds to the lotudinal high-energy optic mode has a local minimum at theGpoint. The feature is usually referred as ‘‘overbending,’’ sinthe local maxima thus appear at some general points ofBZ around theG point. According to our calculations thoverbending pertains to the SWCNT’s as well. As the ovbending seems to be essential for exact theoretical intertation of the laser excitation energy dependence and doupeak structure of the high-energy mode ('1600 cm21) infirst-order Raman spectra,29 we analyze its characteristic

FIG. 3. The heat capacityCph for the four SWCNT’s of thesame diameterD513.6 Å. In all the temperature regions their cpacities coincide. NearT50 ~upper panel! capacity is linear inT.

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thoroughly, namely, these Raman spectra features havequite recently attributed to the double-resonant process cing from the phonon modes close to theG point. Since thedispersion aroundk50 is generally quite weak, the excitation energy dependence is expected to be very sensitivthe vibrational spectra details. In particular, the accurateoretical investigation when related to the experiment mhelp the SWCNT’s sample characterization. On the one sfrom the Raman scattering measurements the slope ofexcitation energy dependence of the high-energy modequency can be obtained. On the other side, the quantheory of the double resonant Raman processes~which is,however, beyond the scope of this paper! relates this dependence to the overbending positionk0, its absolute valueDE,its slope and the phonon eigenvectors~all of which can beeasily calculated for any required spectrum of the SWCNdiameters and chiralities!.

The phonon branch exhibiting the overbending issigned bym50 quantum number for all the tubes. The coresponding irreducible representations16are kE0 for C, kE0

A

for Z, and kE0B for A tubes. The displacement of the ato

C000 as the function ofk is presented in Fig. 5. Again, thBloch-like mode displacements of the other sites are giby Eq. ~1!.

FIG. 4. The overbendingDE as a function of the chiral angleufor the 754 SWCNT’s~among those with 2.8 Å<D<50.0 Å, thetubes with maximalD for given u are chosen!.

04540

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We find the overbendingDE5E(k0)2E(0) ~the energydifference of the phonon branch local minimum atk50 andmaximum at a nonvanishingk0) to be strongly helicity de-pendent. On the other side, for a fixed chiral angle, it dcreases with the diameter but rapidly saturates yet forvery thin tubes, hence showing no further diameter depdence whatsoever. In Fig. 4 the overbendingDE as a func-tion of the chiral angleu is presented for the tubes that fainto the diameter dependence saturation region. The funcis well fitted by

DE~u!521.15819.57

120.245 cos 6u@cm21#.

Hence, the maximal overbending (11.5 cm21) is exhibitedby the zig-zag and the minimal (6.50 cm21) by the armchairtubes. Further, we find that the local maximum appearsbandm50 at k0520.01810.518/(120.143 cos 6u) @Å21#;the D dependence rapidly saturates, and only for very smtubes’ deviations from the above result appear~less than1.5% of the helical BZ!. By applying Eq.~8! one easily findsthe correspondingk0 and m0 values. Only for the achiratubes the overbending occurs in them50 band, while for theother tubesm increases with the diameter in steps ofn alongthe rayn(n1 ,n2). In the zig-zag (n,0) tubes these bands staat k50 by the mode assigned as0A0

1 without tangentialcomponent. In the armchair tubes, thek50 mode of theband is assigned by0B0

2 , and must be purely longitudinaThen, the horizontal component of the displacements aralong the BZ, as presented in Fig. 5; the minimum ozdisplacement corresponds to the maximum of the band.cise data on these quantities are needed for thorough qtum theoretical modeling of the double resonant high-eneRaman mode,29 which might be useful for SWCNT’scharacterization.

V. SUMMARY

We have calculated the phonon dispersions and atodisplacements for achiral and chiral SWCNT’s within thsymmetry-based force-constant approach. The calculat

r-

-

FIG. 5. The overbending displacement coodinates of the atom C000 for the allowed values@0,p/a# of k. Displacements for all the other atoms may easily be found by use of Eq.~1!. Leftand right: two degenerate modes.

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are of high precision, but still extremely efficient due to tcomplete incorporation of the symmetry. Besides the aumatic assignment by the symmetry-based quantum numbthe symmetry enables to express the normal mode displments of the atoms along the tube, in the form of a Blolike wave ~1!, determined by the normal mode transformtion law ~irreducible representation! and the displacement oa single atom. In particular, this approach gives the exsolution to the twisting mode.

Extensive calculations have included sample of 12tubes, i.e., all SWCNT’s with the diameters between 2.8and 50.0 Å. The exhaustive data enabled to find the futional dependence of the frequencies of the Raman andactive modes on the chiral angle and diameter, as weldetermining the corresponding eigenvectors for all of theAs the IR and Raman active phonons are characterizedk50, their classification is performed according to the isgonal symmetry group, i.e.,Dq in the chiral andD2nh in theachiral cases. As only the later contains the space inversfor the chiral tubes there are simultaneously the IR andman active modes. Also, the antisymmetric Raman tencomponents are found, for their expected significance in

*Electronic address: [email protected]; http://www.ff.co.ynanoscience

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04540

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ct

0

c-IRas.

by-

n,a-ore

resonant scattering. Finally, it is stressed out that underlight polarized linearly along the tube axis the armchair tubare IR inactive.

The calculated phonon dispersions data we have alsoto address the laser excitation energy dependence ofhigh-energy Raman active mode and have found the futional dependence of the overbending and of the wave veat which the slope of the frequency shift changes the signthe proposed double resonant model for the high-energyman mode proves to be true, our results can be useful forSWCNT’s sample characterization. Finally, we have fouthe SWCNT’s specific heat temperature~up to 4000 K! anddiameter~at fixed temperatures! dependence and have detemined the temperatureTo ~as a function of the tube diamete!below which the population of the optical branches canneglected when estimating the SWCNT specific heat.

ACKNOWLEDGMENTS

We appreciate discussions with C. Thomsen, S. Reand J. Maultzsch.

,

l-

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t.

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2 , 0E1 and 0E2 for chiral tubes, while for the achiraonesA1g, A2u, E1u, E1g and E2g correspond to0A0

1 , 0A02 ,

0E11 , 0E1

2 and 0E21 .

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single-wall carbon nanotubes phonon spectra: symmetry-based calculations

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