Chiral symmetry analysis and rigid rotational invariance for the lattice dynamics of single-wall carbon nanotubes

TL;DR

This paper develops a detailed vibrational potential model for single-wall carbon nanotubes that respects their symmetries, enabling precise calculation of phonon dispersion, mode frequencies, and polarization vectors based on chiral symmetry analysis.

Contribution

It introduces a symmetry-compliant vibrational potential model for SWCNTs, extending the valence force model to include rotational invariance and chiral symmetry considerations.

Findings

Accurate dispersion relations for acoustic and flexure modes

Systematic fitting of mode frequencies and velocities based on chiral parameters

Validation of the model with Raman and infrared active mode data

Abstract

In this paper, we provide a detailed expression of the vibrational potential for the lattice dynamics of the single-wall carbon nanotubes (SWCNT) satisfying the requirements of the exact rigid translational as well as rotational symmetries, which is a nontrivial generalization of the valence force model for the planar graphene sheet. With the model, the low frequency behavior of the dispersion of the acoustic modes as well as the flexure mode can be precisely calculated. Based upon a comprehensive chiral symmetry analysis, the calculated mode frequencies (including all the Raman and infrared active modes), velocities of acoustic modes and the polarization vectors are systematically fitted in terms of the chiral angle and radius, where the restrictions of various symmetry operations of the SWCNT are fulfilled.