Three-phonon Umklapp process in zigzag single-walled carbon nanotubes

The rate of relaxation of zigzag single-walled carbon nanotubes is calculated by consideration of three-phonon Umklapp process. The results show that the relaxation rate increases exponentially with phonon frequency at low frequency. The linear dependence of the relaxation rate on temperature is obtained. It is shown that the value of the phonon mean free path reaches a few micrometres, which is consistent with the estimated experimental result.