Low-temperature specific heat of nanotube systems

The low-temperature specific heat of single-walled carbon nanotubes (isolated and bundled) and multiwalled carbon nanotubes is calculated within force-constant dynamical models. It is shown that, due to the quadratic dependence of the frequency $\omega$ of the transverse-acoustic phonons on the wave number q, the phonon density of states has a singularity of the type $1/\sqrt{\omega }$ and the very low-temperature specific heat varies as $\sqrt{T}$ with the temperature T. With the increase of the diameter of the system, the contributions of the longitudinal- and torsional-acoustic phonons to the specific heat begin to prevail and the latter becomes linear in T. These results are confirmed by the recent experimental data.