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

TL;DR

This paper introduces a group-theory method to relate the phonon spectra of carbon nanotubes to those of graphene, aiding in understanding their vibrational modes and improving Raman spectroscopy analysis.

Contribution

A novel group-theory approach using a virtual intermediate structure to connect nanotube and graphene phonon spectra, clarifying the origin of nanotube vibrational modes.

Findings

Identified the symmetry origin of nanotube phonon modes.

Determined the G-band structure for various nanotubes.

Facilitated experimental indexing via Raman spectroscopy.

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.