Thermal conductivity in bulk samples of single-walled carbon nanotube (SWNT) ropes in the temperature range 8 - 350 K, have been reported by Hone et al. [1]. The observed thermal conductivity is predominantly due to phonons: the contribution of electrons is less than two orders of magnitude in the entire temperature range [2]. The temperature dependent values of the thermal conductivity are much larger than that of crystalline fullerene, another allotrope of carbon [3]: at low temperature ~8 K it is more than an order which keeps increasing with increasing temperature and becomes more than two orders of magnitude at 300 K. It is smaller than the thermal conductivity of graphite along the z-direction [3] in the temperature range 8 - 150 K, beyond which, it exceeds and becomes almost twice that of graphite at 300 K. However, it is much lower than the thermal conductivity of graphite in the basal plane, in the temperature range 8 - 300 K. It is therefore, apparent that the folding of graphene sheets into a bucky ball or right circular cylindrical tubules, results in a remarkable change in its thermal conductivity, as is evident from the experimental measurements.

SWNT ropes consist of bundles of aligned SWNT of diameter around 1.25 nm. Each SWNT is in proximity with other SWNT and therefore, are bound to one another and give rise to three-dimensional dynamical modes, unlike the case of isolated SWNT, where, the dynamical modes are essentially two-dimensional because of thefolding of a graphene sheet into a capillary. The dynamics of a SWNT has been studied theoretically using a simplified model of the tube, and the appropriate dynamical matrix diagonalized, to obtain its vibrational modes [4]. However, there is hardly any theoretical work which attempts to explain the observed temperature variation of thermal conductivity in SWNT ropes, because of its complexity. One must nevertheless, try to understand the phenomenon because of its commercial utility and also, for academic purposes, which will lead to the understanding of its different physical properties. An attempt here, is therefore, made to understand both the thermal conductivity and its intimately related quantity, the specific heat [5,6] using different physical dynamical models, and then try to fix the dynamical model, using the basic physics of the sample, to explain consistently both the temperarure dependent thermal conductivity and the specific heat variation of the sample.

In the next section is briefly described the mathematical formalism, followed by the results and discussions of the computations. Important conclusions are recorded in the final section.

Thermal conductivity is expressed as:

where is the thermal conductivity in the direction (represent the directions in the Cartesian coordinates and also the polarization of the phonons).

where, and are the specific heat, velocity and the relaxation time of the phonons respectively in the direction.

Expressing specific heat in terms of phonon frequency distribution function (FDF), , and the average number of phonons of energy,

the expression for thermal conductivity turns out to be:

Different dynamical models of phonons are used to compute the specific heat and thermal conductivity.

We consider three dynamical models, which increasingly take into account the realistic nature of the SWNT ropes sample. The first model is celebrated isotropic threedimensional Debye model, which ignores the presence of anisotropic nature of the SWNT ropes. The model is given by expression (4) which when used in expression of the specific heat contained in expression (3), to compute the temperature variation of specific heat, one finds that the Debye temperature of 1500 K yields the specific heat in the temperature range 2 - 300 K, as shown by small dash curve (- - - - - -) in Figure 1. It is the model which yields Dulong Petit law i.e. the value of 3R at high temperatures in agreement with the experimental results shown by solid squares (■ ■ ■ ■); but at low temperatures yields quite different temperature variation of spe-

cific heat: the computed values are increasingly much lower than the corresponding experimentally observed values of specific heat in the entire temperature range down to 2 K. In order to take into account the anisotropy present in the system, one can use expression (5), which is referred to as “Extended Debye model” (EDM) where there are two values of, one along the basal plane and other perpendicular to it i.e.

and

respectively. The computed values of specific heat have been shown by large dash curve (___ ___ ___).

The EDM yields closer values to experimental ones in the temperature range 60 - 300 K, but gives much lower values at lower temperatures, even though these values are much higher than those given by isotropic Debye model. Finally, the anisotropic model given by expression (6) which not only takes into account the anisotropic planar modes reminiscent of graphene modes, but also the intertube coupling, which gives rise to three-dimensional dynamical modes at small values of frequencies, has been used to compute the specific heat. One finds that; and yields the best possible values of the specific heat as shown in Figure 1 by solid line (__________). As is evident from the Figure, this model explains quite well the temperature variation of specific heat in the entire temperature range 2 - 300 K (and is far superior to Debye and Extended Debye model). We therefore, use this model to study the temperature variation of thermal conductivity of SWNT ropes.

Thermal conductivity, unlike the study of specific heat in the present problem where number of phonons is fixed, involves phonon-phonon scattering and phonon-phonon interaction, which gives rise to phonon mean free path or phonon relaxation time. One has to use an appropriate form for the relaxation time. Taking a cue from thermal conductivity when phonon distribution function is three-dimensional Debye type, one can suggest the form of to be that given by expression (8) where, d is introduced as a variable having the physical connotation of the dimensionality of the system. For d = 2, the expression corresponds to three-dimensional case [5]. Using this form of in expression (3), one gets the final expression of thermal conductivity given by expression (7). Using the various values of the physical parameters obtained from the specific heat study and, the thermal conductivity is computed for different values of d, and its temperature dependent variation is plotted in Figure 2. As is evident from Figure 2, when the computed values given by solid curve (__________) explains reasonably well the temperature variation of thermal conductivity from 8 K to ~150 K. Above 150 K, experimental values are somewhat higher and maximum percentage deviation is ~ 40% at 300 K.

Since represents a two-dimensional nature of the system, it therefore appears that at high temperatures the behavior of thermal conductivity in SWNT ropes approaches that of two-dimensional material while at low temperatures, it corresponds to quasi two-dimensional because of the curvature of graphene sheets.

One may conclude that, it is possible to explain consistently both, the observed temperature variation of specific heat and thermal conductivity of SWNT ropes, us-

ing an appropriate anisotropic dynamical model and a suitable form of relaxation time that represents phononphonon scattering and interaction. The anisotropic dynamical model takes into account the presence of surface modes on the curved graphene sheets and intertube coupling. Further, the relaxation time, which is related to phonon mean free path, due to the phonon phonon interaction is found to be dependent both on phonon energy and the dimensionality. Its form also indicates the quasi two dimensional nature of SWNT ropes.