General invariance and equilibrium conditions for lattice dynamics in 1D, 2D, and 3D materials

TL;DR

This paper establishes general invariance and equilibrium conditions for lattice dynamics in low-dimensional materials, explaining flexural phonon behavior and providing an effective approach for materials with long-range interactions, validated on 158 2D materials.

Contribution

It introduces a unified framework for understanding lattice vibrations in low-dimensional materials, including conditions for flexural phonons and handling long-range dipole interactions.

Findings

Quadratic dispersion of flexural phonons confirmed in low-dimensional materials.

Effective approach successfully applied to 158 2D materials.

Flexural modes can be purely out-of-plane with quadratic dispersion.

Abstract

The long-wavelength behavior of vibrational modes plays a central role in carrier transport, phonon-assisted optical properties, superconductivity, and thermomechanical and thermoelectric properties of materials. Here, we present general invariance and equilibrium conditions of the lattice potential; these allow to recover the quadratic dispersions of flexural phonons in low-dimensional materials, in agreement with the phenomenological model for long-wavelength bending modes. We also prove that for any low-dimensional material the bending modes can have a purely out-of-plane polarization in the vacuum direction and a quadratic dispersion in the long-wavelength limit. In addition, we propose an effective approach to treat invariance conditions in crystals with non-vanishing Born effective charges where the long-range dipole-dipole interactions induce a contribution to the lattice potential and stress tensor. Our approach is successfully applied to the phonon dispersions of 158 two-dimensional materials, highlighting its critical relevance in the study of phonon-mediated properties of low-dimensional materials.