Vibrations of single-wall carbon nanotubes: lattice models and low-frequency dispersion

We address a somewhat controversial question of the low-frequency dispersion of transverse acoustic (TA) modes of single-wall carbon nanotubes by studying an empirical lattice model with a set of elastic interactions no less comprehensive than the frequently used force constant model. Differently from the latter, however, our approach utilizes only geometrically invariant quantities such as variations of lengths and various angles, which makes the issue of force constant adjustment irrelevant. All the salient qualitative features of vibrational spectra of arbitrary nanotubes naturally follow from the vibrational Hamiltonian of graphene upon its isometric mapping onto a cylindrical surface and are calculated on the same footing with graphene. We find that the low-frequency dispersion of the TA modes is manifestly parabolic in agreement with the continuum analysis and with the earlier lattice study by Popov et al. but contrary to the conclusions of a linear dispersion reached in many other lattice calculations. (C) 2004 Elsevier B.V. All rights reserved.