Thermal conductivities of single-walled carbon nanotubes calculated from the complete phonon dispersion relations

The thermal conductivity of an individual single-walled carbon nanotube is calculated with the analysis of all possible combining and splitting umklapp scattering processes based on complete phonon dispersion relations. The relaxation rate for the transverse acoustic phonons that undergo the combination umpklapp process is found to increase with many singularities as temperature rises. The calculated mean free path suggests that the combining (splitting) umklapp process is predominant in the low (high) frequency regime. The phonons with a very high frequency contribute little to thermal conduction due to the splitting umklapp process. The calculated thermal conductivity for (10,10) carbon nanotube is $474\mathrm{W}∕\mathrm{m}\mathrm{K}$ at $300\mathrm{K}$.

Figure 1

A schematic illustration of the possible (a) combining U process and (b) splitting U process for a phonon mode $\mathbf{q}$. The vector ${\mathbf{G}}_{\Vert }$ is the reciprocal vector in the axial direction. The points $O_1$ and $O_2$ are the centers of the first and second BZs consisting of $N$ equally spaced lines parallel to the nanotube axis, respectively.

Figure 2

Phonon dispersion relations of the armchair (10,10) carbon nanotube. There are 120 phonon branches represented by the cutting lines belonging to $N=20$ segments labeled by $j$, each has six lines.

Figure 3

Construction for the interaction of three phonons. (a) The combining umklapp process and (b) the splitting umklapp process. Point $P$ in (a) denotes the solution belonging to the channel $\mathrm{TA}+\mathrm{TA}\to \mathrm{TW}$ (TA the transverse acoustic mode and TW the twisting acoustic mode).

Figure 4

The temperature dependence of the relaxation rate of TA phonons undergoing a combining umklapp process at different frequencies.

Figure 5

The relaxation rate of TA phonons undergoing a combining umklapp process (a) at 40, 60, and $300\mathrm{K}$ and a splitting umklapp process (b) at $300\mathrm{K}$.

Figure 6

(a) The MFP of TA phonons at different temperature. (b) The weighted average MFP at $300\mathrm{K}$.

Figure 7

The thermal conductivity as a function of temperature.