Vibrational heat capacity of carbon nanotubes in low and ultra-low temperature regions

We develop a continuous theory of low-frequency dynamics for single-walled carbon nanotubes (SWCNTs) weakly interacting with the environment. In the frame of the approach proposed we obtain temperature dependence of SWCNTs specific heat in the low (T<40 K) and ultra-low (T<2.5 K) temperature ranges. We take into account the main term in the coupling between SWCNT and the environment that slightly increases the frequencies of those SWCNT modes, which possess predominantly radial polarization. The coupling drastically decreases the density of phonon states in the lowest frequencies region. The theoretically predicted fall of the specific heat in the interval T<2.5 K properly explains available experimental data in contrast to the preceding approaches. The theory proposed can be the basis for studies of low-temperature heat capacity and phonon dynamics of many other single-walled and multi-walled tubular structures (boron nitride, transition metal dichalcogenides) which have emerged in the past decade.