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

A group-theory method to relate the spectra of tubular and parent planar structures is elaborated. The spectra correlation is obtained using a virtual intermediate planar periodic structure, which has a local symmetry coinciding with that of the tube surface. The method is applied to deduce the zone-center phonon modes of carbon nanotubes from the dispersion branches of the graphene monolayer. The results obtained open an additional way for experimental indexing of the carbon nanotubes by means of Raman spectroscopy, and the origin of the Raman-active modes in the achiral nanotubes is determined. We demonstrate that the number of modes in the G band and the relation between their frequencies are governed by the nanotube type. The proposed theory can be easily explored to understand the phonon and electron spectra of many other tubular structures (boron nitride, transition metal dichalcogenides) which have emerged in the past decade. DOI: 10.1103/PhysRevB.87.035407