Vibrational modes and low-temperature thermal properties of graphene and   carbon nanotubes: A minimal force-constant model
  We present a phenomenological force-constant model developed for the description of lattice dynamics of sp2 hybridized carbon networks. Within this model approach, we introduce a new set of parameters to calculate the phonon dispersion of graphene by fitting the ab initio dispersion. Vibrational modes of carbon nanotubes are obtained by folding the 2D dispersion of graphene and applying special corrections for the low-frequency modes. Particular attention is paid to the exact dispersion law of the acoustic modes, which determine the low-frequency thermal properties and reveal quantum size effects in carbon nanotubes. On the basis of the resulting phonon spectra, we calculate the specific heat and the thermal conductance for several achiral nanotubes of different diameter. Through the temperature dependence of the specific heat we demonstrate that phonon spectra of carbon nanotubes show one-dimensional behavior and that the phonon subbands are quantized at low temperatures. Consequently, we prove the quantization of the phonon thermal conductance by means of an analysis based on the Landauer theory of heat transport. 
J. Zimmermann, P. Pavone, G. Cuniberti