Simple theory of low-temperature thermal conductivity in single- and double-walled carbon nanotubes

Low-temperature phonon thermal conductance (PTC) of any 1D system increases proportionally to the temperature. However, here we show that in single- and double-walled carbon nanotubes (CNTs), starting from 3--6 K, the PTC increases faster than the linear function, since the low-frequency modes of dispersion curves, which do not tend to zero together with the wave vector, are excited. To develop the PTC theory, we combine the Landauer's ballistic approach with the simple continuous model proposed for the calculation of the low-frequency phonon spectra of both free nanotubes and those interacting with an environment. The approach obtained is valid not only for commensurate double-walled CNTs, but also for incommensurate ones. The temperature-dependent relation between the PTC of double-walled CNT and those of its constituent SWNTs is obtained and discussed. The low-temperature heat transfer in bulk materials originated from CNTs is also considered and the upper limit of thermal conductivity of such materials is determined. We argue that the ideal material consisting of CNTs can challenge diamond only when the mean length of its defect-free nanotubes reaches at least one hundred of micrometers.