Continuum approach for long-wavelength acoustic phonons in quasi-2D structures

TL;DR

This paper introduces a continuum elasticity approach to predict long-wavelength acoustic phonons in quasi-2D materials, providing accurate results for graphene, phosphorene, and carbon nanotubes, avoiding slow atomistic calculations.

Contribution

The authors develop a continuum elasticity model that accurately predicts long-wavelength phonon spectra in quasi-2D structures using ab initio elastic constants.

Findings

Accurately reproduces phonon spectra of graphene and phosphorene.

Correctly describes radial breathing mode dependence on diameter.

Provides simple formulas for acoustic mode frequencies.

Abstract

As an alternative to atomistic calculations of long-wavelength acoustic modes of atomically thin layers, which are known to converge very slowly, we propose a quantitatively predictive and physically intuitive approach based on continuum elasticity theory. We describe a layer, independent of its thickness, by a membrane, characterize its elastic behavior by a $(3{\times}3)$ elastic matrix as well as the flexural rigidity. We present simple quantitative expressions for frequencies of long-wavelength acoustic modes, which we determine using 2D elastic constants calculated by {\em ab initio} Density Functional Theory. The calculated spectra accurately reproduce observed and calculated long-wavelength phonon spectra of graphene and phosphorene, the monolayer of black phosphorus. Our approach also correctly describes the observed dependence of the radial breathing mode frequency on the diameter of carbon fullerenes and nanotubes.