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PRAMANA

— journal of

physics

c ? Indian Academy of SciencesVol. 67, No. 2

August 2006

pp. 305–317

Phonon and thermal properties of achiral single

wall carbon nanotubes

PRAPTI SAXENA1,∗and SANKAR P SANYAL1,2

1Condensed Matter Laboratory, Department of Physics, Barkatullah University,

Bhopal 462 026, India

2Department of Electronics, Barkatullah University, Bhopal 462 026, India

∗Corresponding author. E-mail: prapti194@rediffmail.com; spsanyal@sancharnet.in

MS received 11 August 2005; revised 17 April 2006; accepted 23 June 2006

Abstract.

single wall carbon nanotubes has been carried out using force constant model considering

up to third nearest-neighbor interactions. We have calculated the phonon dispersions,

density of states, radial breathing modes (RBM) and the specific heats for various zigzag

and armchair nanotubes, with radii ranging from 2.8˚ A to 11.0˚ A. A comparative study of

phonon spectrum with measured Raman data reveals that the number of Raman active

modes for a tube does not depend on the number of atoms present in the unit cell but

on its chirality. Calculated phonon modes at the zone center more or less accurately

predicted the Raman active modes. The radial breathing mode is of particular interest

as for a specific radius of a nanotube it is found to be independent of its chirality. We

have also calculated the variation of RBM and G-band modes for tubes of different radii.

RBM shows an inverse dependence on the radius of the tube. Finally, the values of specific

heat are calculated for various nanotubes at room temperature and it was found that the

specific heat shows an exponential dependence on the diameter of the tube.

A detailed theoretical study of the phonon and thermal properties of achiral

Keywords. Nanotubes; thermal properties; zone folding; phonons.

PACS Nos63.32.+m; 65.80.+n; 61.48.+c

1. Introduction

Carbon nanotubes are quasi-one-dimensional structures and are projected as one

of the most promising candidates of future nanoelectronics owing to their extra-

ordinary electronic, thermal, mechanical, elastic and electrical properties [1–10].

Although a satisfactory amount of theoretical work has already been published on

the lattice dynamical properties of carbon nanotubes, there is a great need for sim-

pler models that can explain accurately a satisfactory amount of experimental data

available on single wall carbon nanotubes (SWCNTs). So far, many theoretical

models have been proposed for the study of the vibrational spectra and phonon

density of state of SWCNTs. In prior efforts by Jishi et al [11], the force constant

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Prapti Saxena and Sankar P Sanyal

model using zone-folding approach was employed for the calculation of phonon dis-

persion relations of the nanotubes. In this method the graphite force constants are

modified to include the effect of curvature. However, only a few calculations of

phonon dispersions in nanotubes have been accomplished by zone-folding method

with the correction in the dynamical matrix to obtain two acoustic modes. Later,

Saito et al [12] and others have reported the phonon properties calculated using the

force constant model. In these approaches the authors have modified the force con-

stants in order to fulfill the rotational sum rule and to obtain the torsional acoustic

branch. Lately, Sanchez-Portal et al [13] have reported the ab initio and Yu et al

[14] have reported the tight binding calculations of vibrational properties of nan-

otubes. The latter approaches are free from the deficiencies of the first two models.

However, their work is limited to tubes with radius less than 7˚ A. Maultzsch et al

[15] have used the force constant approach but emphasized mainly on the behavior

of high-energy Raman active modes. In another approach, Popov et al [16] have

used the valence force field model for the calculations of phonon dispersion rela-

tions. This model takes into account the screw symmetry of the nanotubes, which

results in a reduction of the size of the dynamical matrix to six for all tube types

and allows one to study the vibrations of all observable nanotubes. Also they have

considered up to second nearest-neighbor interactions and focused mainly on the

breathing like modes in carbon nanotubes. All these models discussed above are

successful in some way or other but have some inherent deficiencies in explaining

phonon modes at lower frequencies. The torsional mode remained unexplained in

these studies.

On the other hand, very few theoretical approaches are reported so far on the

thermal properties of carbon nanotubes [5,6]. Berber et al [6] have used the mole-

cular dynamic simulation to predict the thermal conductivity of nanotubes. They

showed that the thermal conductivity of an isolated (10,10) carbon nanotube could

be as high as 6600 W/mK at room temperature. This exceptionally high value of

thermal conductivity reflects the one-dimensional conduction behavior of nanotubes

with least phonon scattering. Recent experimental results have, however, proved

the authenticity of these results [7–10]. Few authors [17,18] have also reported the

negative thermal expansion coefficient for nanotubes at room temperature. In view

of these facts there is a great need for an appropriate theoretical model, which can

be useful in the characterization of the nanotubes on the basis of these thermal

properties, as well.

Since carbon nanotubes are huge structures, it is unphysical to depend on mod-

els, which have large number of parameters. On the other hand, ab initio methods

are much time-consuming. Therefore, in order to simplify the problem and to get

reasonable quantitative results and understanding of the phonon dynamics and

thermal properties, we have employed the force constant approach for the calcu-

lation of phonon modes in carbon nanotubes. This model is based on Born–von

Karman theory [19] of lattice dynamics. In this model interatomic interactions are

represented in terms of sum of radial and angular bending forces extended up to

third nearest neighbors. The Brillouin zone of nanotube is defined by applying zone-

folding technique to the graphene layer. We have calculated the density of states

and compared them with the experimental Raman modes. Also, we have correctly

predicted the diameter dependence of the radial breathing mode (RBM) for all the

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Phonon and thermal properties of achiral SWCNTs

Figure 1. Various possible roll-up directions of graphene layer to generate

different types of nanotubes. a1 and a2 represent the primitive vectors of

graphene monolayer.

nanotubes. The variation of RBM with the radius of the tube further allows us to

predict the RBM for tubes of arbitrary chirality. We presented the results of radius

dependence of G-mode and discussed the overbending of high-energy longitudinal

optic modes in single wall nanotubes. Lastly, by using the phonon spectrum, we

have calculated the specific heat of both zigzag and armchair nanotubes and found

that their values show an exponential diameter dependence. We organize the paper

as follows: In §2 we give a brief description of the structure of achiral nanotubes

and the phenomena of zone folding. Section 3 deals with the theoretical model. In

§4 we present our results, followed by discussion.

2. Structure of nanotube

Nanotubes are generally classified on the basis of their angle of chirality (θ), which

is defined as the angle between the circumference and the nearest zigzag carbon

atoms. Accordingly they are classified into two groups, namely, achiral (armchair

(θ = 30◦) and zigzag (θ = 0◦)) and chiral (0◦< θ < 30◦) tubes. Conventionally,

they are characterized by the indices n and m of graphene lattice vectors a1and

a2, used in the definition of chiral vector (Ch) of nanotube. The chiral vector is

one of the unit vectors of one-dimensional carbon nanotubes and defines the roll-up

direction of graphene monolayer, defined as [20] (figure 1)

Ch= na1+ ma2≈ (n,m).

Armchair configuration is represented by the indices (n,n), while (n,0) and (n,m)

represent zigzag and chiral tubes respectively. Another unit vector, which is called

translational vector T, is parallel to the nanotube axis and is directed normal to the

chiral vector Chin the unrolled graphene lattice. The reciprocal lattice of nano-

tubes is defined by two vectors, q1and q2, such that vector q1is directed along the

circumference and is quantized to give N discrete wave-vectors while q2is directed

parallel to the translational vector, along the axis of the tube and is continuous.

Quantization of the reciprocal lattice vector q1around the circumference is known

as zone folding and is the result of boundary condition, which states that the

(1)

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Prapti Saxena and Sankar P Sanyal

wavelength of the mode multiplied by an integer should not be greater than the

circumference of the tube. In our calculations of phonon dispersion curves for the

carbon nanotubes are plotted along q2direction. Number of atomic pairs in the

unit cell of a nanotube is given by N = 2(n2+ m2+ nm)/dR where dR is the

greatest common divisor (gcd) of (2m + n) and (2n + m).

3. Theoretical model

The present model has been worked out in the framework of Born–von Karman

theory of lattice dynamics [19,21]. Two types of forces are assumed to exist between

the atoms, the central force and the angular force. Central forces act along the line

joining the two atoms while angular forces are proportional to the angle, which

the line joining the displaced atoms makes with the line joining their equilibrium

position. Components of total radial and angular forces, which are also called

non-central forces, are given by

Fk(??) = −α?

p[uk(?) − uk(??)]

+(αp− α?

p)λk(??)

?

j

λj(??)[uj(??) − uj(?)],

(2)

where, uk(?) and uk(??) are the components of instantaneous displacement of ?th

and ??th atoms respectively for k = 1,2,3. αpand α?

force constants respectively where, p = 1,2,3 for the first, second and third nearest

neighbors. λj(??) are the direction cosines of ??th atom with respect to ?th atom,

with j = 1,2,3.

Thus F1(??) denotes the x-component (k = 1) of the force acting on the ?th par-

ticle arising from the displacement of the ??th atom. After Fourier transformation

of the atomic displacements and the force constant matrix, the dynamical problem

is reduced to the solution of the system of 6N equations for the tube having 2N

atoms in its unit cell.

pare the central and angular

|Dkk?(q) − mω2δkk?| = 0.

Equation (3) is the eigenvalue equation and describes adequately the phonon dis-

persion relation. One can define the frequency distribution function, g(ω), as the

phonon density of states obtained from the solution of eq. (3), such that g(ω)dω

represents the number of vibrational modes of the lattice in the frequency range ω

and ω + dω.

?

where Ω is the volume of the unit cell of the nanotube, dSqis the surface element

of constant frequency surface in q-space, ∇ωqis the group velocity.

We have used Lorentzian distribution function and performed the multiple-peak

fitting to obtain the exact frequency distribution function, which is given by

(3)

g(ω)dω =

Ω

(2π)3

dSq

|∇ωq|,

(4)

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Phonon and thermal properties of achiral SWCNTs

g(ω)dω =

Ω

(2π)3

?

q

δ

(ω − ωq)2+ δ2,

(5)

where ωq is the central frequency of the Lorentzian for a given value of phonon

wave vector q while δ is the full-width at half-maximum for a single fitted peak.

Ω is the volume of the unit cell of the nanotube. Once the frequency distribution

function is framed one can calculate the specific heat from the relation given by

?ωmax

where

??ω

where Cq(ω) is the heat capacity of lattice wave with wave vector q and angu-

lar frequency ω. kB and ? are the Boltzmann’s constant and Planck’s constant

respectively.

CV(T) =

0

Cq(ω)g(ω)dω,

(6)

Cq(ω) = kB

kBT

?2

exp(?ω/kBT)

[exp(?ω/kBT) − 1]2,

(7)

4. Result and discussions

In the present paper we have reported the phonon dispersion curves for a number of

zigzag and armchair single wall carbon nanotubes with their radius ranging from 2.8

˚ A to 11.0˚ A. The values of model parameters are derived by associating them to the

known experimental results on Raman scattering [22] and those reported by Bond

polarizability model of SWCNT [20]. The best-fit model parameters for (10,10)

armchair single wall carbon nanotube are summarized in table 1. It can be seen

Table 1.

wall carbon nanotube.

Structural and model parameters for single

Structural parameters

C–C bond length (aC–C)

Length of primitive vectors of

graphene (|a1| = |a2| = a)

1.44˚ A

√3 × aC–C

Model parameters

(central and angular

force constants in

dynes/cm)

α1

α2

α3

α?

α?

α?

9705.0

4070.0

−2775.0

3975.0

−247.5

−635.0

1

2

3

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Prapti Saxena and Sankar P Sanyal

Table 2. Experimental and theoretical Raman mode vibrational frequencies

for (10,10) single wall carbon nanotube.

Raman frequencies (cm−1)

Bond polarizability model

Ref. [20]

Rao et al [22]

Exp. (Theoret.)Mode Present model

E2g

E1g

A1g

E2g

E1g

A1g

E2g

A1g

E2g

E1g

E2g

16.45*

92.19*

165.56*

373.50*

1569*

1587.28*

1592.77*

687.07

859.17

1352.85

1528.6

17

118

165

368

1585

1587

1591

–

–

–

–

(22)

116(117)

186(165)

377(368)

1567(1584)

1593(1585)

1609(1590)

673(670)

855(866)

1347(1374)

1550(1543)

*Represents the frequencies used for fitting force constants.

from table 1 that the values of central force constants are significantly larger than

their angular counterparts for the first and second nearest-neighbor interactions,

while for the third nearest-neighbor interactions angular forces dominate over the

central one. This shows that the effect of curvature is more prominent in the case

of third nearest neighbor. This is quite justified when one considers the cylindrical

structures, like nanotubes, as emphasized by Saito et al [20]. The Raman modes,

which are used for fitting, together with the calculated frequencies from the present

model are listed in table 2.

The phonon band structure, in general, reflects the symmetry as well as the num-

ber of atoms present in the unit cell and dynamical response of atomic species that

constitutes the nanotubes. However, the order and characteristics of the resulting

symmetries are ascertained from the way the two-dimensional hexagonal sheet’s

boundaries are connected to each other to form the cylindrical structure. In arm-

chair and zigzag type of single wall carbon nanotubes, the translational periodicity

is reflected, and hence gives rise to maximum symmetry. Due to the optimum sym-

metry properties they possess maximal degeneracy [23]. Therefore, for (10,10) and

(10,0) nanotubes, instead of 120 modes one gets only 66 phonon modes at the zone

center, out of which 54 are doubly degenerate while 12 are non-degenerate. Low

frequency E2gmode for (10,10) carbon nanotube obtained from our calculations is

16.45 cm−1, which is also reported as 17 cm−1theoretically [20]. The dispersion

curves for (10,10) and (10,0) nanotubes are shown in figures 2a and 3a, respec-

tively. Both of them contain 40 atoms in their unit cells, and hence 120 phonon

modes are obtained for each of them. At the zone center we have obtained four

acoustic modes, for both the tubes, out of which the lowest two acoustic modes are

doubly degenerate and transverse in nature. The highest one is the longitudinal

acoustic mode. The fourth mode is a special feature of nanotubes and comes into

existence due to the movement of atoms around the axis of the tube. Since we

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Phonon and thermal properties of achiral SWCNTs

Figure 2. (a) Dispersion curves of (10,0) nanotubes plotted in axial q-vector.

(b) Calculated phonon density of states.

reported in ref. [5].

(c) Phonon density of states as

Figure

q-vector. (b) Calculated phonon density of states. (c) Phonon density of

states as reported in ref. [20].

3. (a) Dispersion curves of (10,10) nanotubes plotted in axial

consider the nanotubes as one-dimensional continuous cylindrical structures, this

type of mode is possible and is referred to as ‘twisting or torsional mode’. One

important finding of our calculations is that we have obtained four acoustic modes

for all the tube structures having radius greater than 5˚ A while for nanotubes

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with lower radius only three acoustic modes are observed. This can be explained

from the fact that for lower radius the radial dimensions become comparable to

the interatomic dimensions. Therefore, this type of atomic displacements around

the circumference become less probable and the structure behaves truly nanoscopic

contrary to the bulk behavior. However, the possibility of cross circumference

interactions for lower radius cannot be ruled out completely. The total phonon

density of states for both the tubes are plotted and compared with the results

reported by Cao et al [5] and Saito et al [20] in figures 2b–2c and 3b–3c. It can

be noticed from these figures that various high- and low-frequency peak positions

of SWCNT agree well with the observed Raman peaks. However, there is some

discrepancy, which is obvious in the two plots at the medium frequency range. The

reason of this deviation is the values of Raman frequencies that we have taken for

fitting the model parameters, which cover only high and low frequency modes (table

2). Also in refs [5] and [20], they have used the force constant directly obtained

for graphene sheet while in our case we have used the force constants obtained for

nanotubes, by fitting the Raman modes of (10,10) nanotube only. This could be

another reason for inconsistency between the two plots in the intermediate energy

range.

Amongst the 6N phonon modes obtained from our calculations for the nanotube

having 2N atoms in its unit cell, only a few are Raman and infra-red active [22,24].

The Raman active vibrational modes are mainly symmetry dependent.

calculations we have obtained a higher frequency band around 1600 cm−1(G-band),

which arises from six tangential bond-stretching vibrational modes of symmetry

2A1, 2E1 and 2E2 for all the tubes [25]. It can be argued that these modes are

only approximately longitudinal or transverse [26]. However, the order of the three

pairs of frequencies depends on the particular model [27]. The variation of G-band

modes with radius for various armchair and zigzag tubes are presented in figures

4a and 4b, respectively.

Overbending [28] is one of the special features of graphene that persists even

when the graphene layer is rolled-up to form a tube. When the longitudinal high-

energy optic mode has its local minima at Γ point, we refer to it as overbending.

The magnitude of overbending is the difference of energy of phonon branch local

minima at q = 0 and the maxima at any non-zero value of q. Due to this softening

of LO phonon, a peak appears at the low-energy side of the high-energy band

of Raman spectra of nanotubes. In this way one can interpret the double peak

structure of the high-energy sub-band of carbon nanotubes. In the case of achiral

tubes it was found that the magnitude of overbending is higher for zigzag tubes

(∼12.7 cm−1) than for the armchair tubes (∼10.9 cm−1).

The seventh and the most important low-frequency (A1g) Raman active mode

of nanotubes is called the radial breathing mode (RBM). It’s value is definite and

different for each tube and also varies with tube radius. When all the atoms move

in a phase perpendicular to the tube axis changing the radius of the tube, one gets

the RBM [29]. It involves the collective movement of atoms towards and away from

the central axis of SWCNTs (see figure 5). The radial breathing mode emerges from

the periodicity of the graphene lattice, which is imposed on nanotubes on account

of wrapping. Therefore, the frequency of this mode is directly influenced by the

perimeter of the tube. We have calculated the radial breathing modes for a family

In our

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Phonon and thermal properties of achiral SWCNTs

of zigzag and armchair nanotubes. For (10,10) nanotube, which has a larger radius,

the value of RBM is obtained as 165 cm−1, while for (10,0) nanotube, which has

smaller radius, it’s value is 282 cm−1. Both the values are in good agreement with

the values reported by various authors [22,29–31].

The variation of RBM with the radius of the tubes is given in figure 6. By fitting

power trend line to these results we have obtained an inverse relationship between

the radius and the RBM for both the tubes. A general representation of the relation

between RBM and radius of the tube can be given as

ω(RBM)= AR−1.

(8)

We have calculated different values of proportionality constants A for the two types

of nanotubes such that, A = 1134 cm−1˚ A for armchair nanotubes and A = 1092

cm−1˚ A for zigzag nanotubes. On averaging the values of A for all achiral tubes

one gets A = 1107.8 cm−1˚ A, which is in the range reported by various authors

[29,31].

Finally we have calculated the specific heats for these sets of nanotubes. For this

purpose, we have formulated the expression of vibrational density of state (g(ω)) by

using Lorentzian function (eq. (5)). Since lots of Lorentzian peaks are possible for

a given value of wave vector q we have used the method of averaging. The variation

of specific heat at 300 K with the radius of the nanotube is shown in figure 7. The

values of specific heats of armchair nanotubes are comparatively larger than those

for the zigzag nanotubes having the same value of n. This variation comes from a

basic structural detail, which is the difference in volumes of the unit cells of these

tubules in spite of them possessing the same number of carbon atoms in their unit

cells. We have calculated the dependence of specific heat on the radius of tubes for

both nanotubes.

Figure 4. Variation of G-band modes with the chiral index n of (a) armchair

nanotubes and (b) zigzag nanotubes.

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Prapti Saxena and Sankar P Sanyal

Figure 5. RBM mode (unique to SWNTs).

Figure 6. Variation of radial breathing mode with the radius of (n,n) nano-

tubes (line with triangles) and (n,0) nanotubes (line with circles).

CV = CV0+ AVe(−R/R0),

(9)

where CV0, AV and R0are the constants such that for armchair nanotubes CV0=

749.93 J/Kg-K, AV = −3600 J/Kg-K and R0= 2˚ A while for zigzag nanotubes

CV0= 778.47 J/Kg-K, AV = −3200 J/Kg-K and R0= 1.5˚ A.

The temperature dependence of specific heat for (10,10) (armchair) and (10,0)

(zigzag) SWCNT is shown in figure 8. It is observed that specific heat exhibits

almost the same behavior with temperature for both tube types. However, the

magnitude of CV is greater for (10,10) carbon nanotube as compared to (10,0)

nanotube. It is also useful to note that at very low temperatures, up to 8 K,

the specific heat increases almost linearly with temperature. At low temperatures,

CV probes only the lowest energy phonons, i.e. only the low-energy modes get

activated. These are the acoustic modes, whose dispersion can often be expressed

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Phonon and thermal properties of achiral SWCNTs

Figure 7. The variation of specific heat with the chiral index n of nanotubes

at 300 K.

Figure 8. Temperature dependence of specific heat of (10,10) and (10,0)

carbon nanotubes. Inset shows the variation of specific heats at low temper-

atures.

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Prapti Saxena and Sankar P Sanyal

as a power law,

g(ω) ∝ kα.

(10)

For a single such mode, eq. (7) simplifies to

CV ∝ Td/α,

(11)

where d is the dimensionality of the system. For a linearly dispersing mode (α = 1)

(in eq. (10)) the specific heat is linear in T for a 1-D system, and shows the

familiar Debye T3behavior for a 3-D system. Therefore, the linear behavior at low

temperature is a direct consequence of the 1-D quantized nature of the nanotube

phonon band structure. Above 8 K, the slope of CV increases gradually, as the

optical sub-bands start to contribute. Around 30 K there is a sharp rise in CV,

which indicates that at this temperature the first high-energy optical sub-band gets

activated and the specific heat increases drastically with temperature. Hone et al [9]

and Dobardzic et al [28] have also depicted this type of low-temperature behavior

(below 30 K) for thermal conductivity. As the temperature increases further, the

value of specific heat increases to finally reach its classical limit.

In conclusion, we have developed a lattice vibrational theory to explain the

phonon dispersion in achiral single wall carbon nanotubes using a phenomeno-

logical force constant model. Its extension for chiral tubes is in progress and will

be reported subsequently. Also, we have successfully predicted the behavior of ra-

dial breathing mode and its dependence on the radius of the nanotube. We have

obtained all the Raman modes in close proximity to those reported earlier from

experiments. We have also developed the relation between specific heat and the

radius of the tube that could be further employed for thermal characterization of

nanotubes. Finally, we have provided an approach, which could be further used to

predict the phonon behavior of non-carbon [21,31] as well as filled nanotubes.

Acknowledgement

The authors are grateful to the Department of Science and Technology (through

Project No. SR/S5/NM-12/2005 Nanoscience and Technology Initiative) for finan-

cial support to this work. PS is also thankful to DST for SRF and to Prof. Stefano

Bellucci (INFN, Roma), Dr Sandro Scandolo and Dr Ralph Gabeur (Condensed

Matter Division, ICTP, Italy) for valuable discussions. One of us (SPS) is thankful

to Prof. G P Shrivastava and Steven Hepplstone of the University of Exeter, UK

and Prof. Peter Entel, University of Duisburg, Germany for useful discussions and

Royal Society, London and DAAD, Bonn, for financial support.

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