Cyclic- and helical-symmetry-adapted phonon formalism within density functional perturbation theory

TL;DR

This paper introduces a novel first-principles method for calculating phonons in nanostructures with cyclic and helical symmetry, improving accuracy and efficiency for systems like carbon nanotubes.

Contribution

It develops a symmetry-adapted density functional perturbation theory framework for phonons in nanostructures with cyclic and helical symmetry, including new sum rules and implementation details.

Findings

Accurate phonon calculations for carbon nanotubes matching plane-wave results

Determined elastic moduli consistent with previous DFT and experiments

Derived scaling laws for phonon frequencies based on nanotube diameter

Abstract

We present a first-principles framework for the calculation of phonons in nanostructures with cyclic and/or helical symmetry. In particular, we derive a cyclic- and helical-symmetry-adapted representation of the dynamical matrix at arbitrary phonon wavevectors within a variationally formulated, symmetry-adapted density functional perturbation theory framework. In so doing, we also derive the acoustic sum rules for cylindrical geometries, which include a rigid-body rotational mode in addition to the three translational modes. We implement the cyclic- and helical-symmetry-adapted formalism within a high-order finite-difference discretization. Using carbon nanotubes as representative systems, we demonstrate the accuracy of the framework through excellent agreement with periodic plane-wave results. We further apply the framework to compute the Young's and shear moduli of carbon nanotubes, as well as the scaling laws governing the dependence of ring and radial breathing mode phonon frequencies on nanotube diameter. The elastic moduli are found to be in agreement with previous density functional theory and experimental results, while the phonon scaling laws show qualitative agreement with previous atomistic simulations.