Low-frequency phonons in carbon nanotubes:: A continuum approach

Low-frequency phonons in carbon nanotubes are studied using a continuum model which allows consideration of an arbitrary wall thickness for the nanotube. Phonon dispersion relations are calculated for two archetypal examples of carbon nanotubes, the (5,5) and (10,10) tubes. The dependence of the radial breathing mode frequency at Gamma on the inverse nanotube diameter is verified within this model; furthermore, we prove it to hold for all pure modes within the thin-shell approximation. The effect of the nanotube wall thickness on the eigenfrequencies of carbon nanotubes is also analyzed, and a criterion to fix this parameter within a continuum model is presented. We compare our results to other continuum approaches, and show that by choosing the appropriate parameters, excellent agreement with recent first-principles calculations can be achieved.