Symmetry of the carbon nanotube modes and their origin from the phonon branches of graphene

Abstract

A new group-theory method to relate a spectrum of carbon nanotubes with that of a graphene monolayer is elaborated. The spectrum reconstruction is performed using a virtual intermediate planar periodic structure. Selected irreducible representations of its space symmetry group and all those of the nanotube one are isomorphic and therefore span the correlated excitations. The method is applied to study the origin of the zone-center phonon modes of chiral and achiral carbon nanotubes. The structure of the G-band for particular types of nanotubes is determined. The results obtained could be useful for experimental indexing of the carbon nanotubes by means of Raman spectroscopy.