Phonon-phonon interactions in single-wall carbon nanotubes

The lifetime of phonon modes undergoing three-phonon interactions in single-wall carbon nanotubes has been calculated. The calculations are based upon an approach using Fermi’s golden rule and a quasielastic continuum model for the anharmonic potential. In deriving the relaxation-rate equation, the effects of both normal and umklapp processes have been included, and a clear distinction between class 1 and class 2 events is made. Dispersion relations for the phonon modes of a carbon nanotube were obtained from analytic expressions developed by Zhang et al. The lifetime of the lowest two optical modes is found to be comparable to the lifetime of the acoustic modes. The results show that the relaxation rate is dominated by normal processes at low temperatures and umklapp processes at room temperature and above. A linear relationship between the relaxation rate and temperature is obtained for temperatures greater than $200\mathrm{K}$. The relaxation-rate contribution decreases (increases) with an increase in the tube radius for normal (umklapp) processes. These results are suggested to have interesting implications for mean-free-path and thermal-conductivity calculations.

Figure 1

(Color online) The phonon-dispersion curves of the six lowest branches for a carbon nanotube of radius $R$. ${\omega }_B$ is the frequency of the breathing mode at the zone center. Here the letter corresponds to the phonon branch: $L=\text{longitudinal}$, $T=\left(\text{doubly}\text{degenerate}\right)$ transverse, $W=\text{twist}$, $\mathit{Br}=\text{breathing}$, and $\sigma =\sigma$ (the lowest nonzero mode).

Figure 2

(Color online) The inverse relaxation time of different phonon modes as a function of frequency $\omega$ for a (10,10) carbon nanotube at $300\mathrm{K}$ for (a) $\sigma$ and $\mathit{Br}$ modes with normal processes included; (b) $L$, $T$, and $W$ modes only undergoing umklapp processes. ${\omega }_B$ is the frequency of the breathing mode at the zone center. Here the letter corresponds to the mode ($L=\text{longitudinal}$, $T=\left(\text{doubly}\text{degenerate}\right)$ transverse, $W=\text{twist}$, $\mathit{Br}=\text{breathing}$, and $\sigma =\sigma$ (the lowest nonzero mode).

Figure 3

(Color online) The relaxation rate of different modes undergoing normal processes as a function of $K_{z}R$ for a (10,10) carbon nanotube at $300\mathrm{K}$. [$L=\text{longitudinal}$, $T=\left(\text{doubly}\text{degenerate}\right)$ transverse, $W=\text{twist}$, $\mathit{Br}=\text{breathing}$, and $\sigma =\sigma$ (the lowest nonzero mode).]

Figure 4

(Color online) The relaxation rate of different modes undergoing umklapp processes as a function of $K_{z}R$ for a (10,10) carbon nanotube at $300\mathrm{K}$. [$L=\text{longitudinal}$, $T=\left(\text{doubly}\text{degenerate}\right)$ transverse, $W=\text{twist}$, $\mathit{Br}=\text{breathing}$, and $\sigma =\sigma$ (the lowest nonzero mode).]

Figure 5

(Color online) The relaxation rate of various modes undergoing selected processes for a (10,10) carbon nanotube at $10\mathrm{K}$. [$L=\text{longitudinal}$, $T=\left(\text{doubly}\text{degenerate}\right)$ transverse, $W=\text{twist}$, $\mathit{Br}=\text{breathing}$, and $\sigma =\sigma$ (the lowest nonzero mode).]

Figure 6

(Color online) The inverse relaxation time of phonon modes with frequency $\omega ={\omega }_B∕2$ as a function of temperature $T$ for a (10,10) carbon nanotube. [$L=\text{longitudinal}$, $T=\left(\text{doubly}\text{degenerate}\right)$ transverse, $W=\text{twist}$.]