Elastic Constants and Bending Rigidities from Long-Wavelength Perturbation Expansions

TL;DR

This paper introduces an efficient first-principles method to calculate elastic and bending rigidity tensors of materials, accounting for long-range electrostatic effects and multipolar interactions, validated on various crystalline and 2D materials.

Contribution

It develops a novel formulation combining long-wavelength perturbation theory with cluster expansion to accurately compute elastic and bending properties, including electrostatic effects.

Findings

Accurately predicts elastic tensors for multiple materials.

Shows multipolar interactions up to octupoles are needed for 3D elastic tensors.

Higher multipoles are essential for converging 2D bending rigidities.

Abstract

Mechanical and elastic properties of materials are among the most fundamental quantities for many engineering and industrial applications. Here, we present a formulation that is efficient and accurate for calculating the elastic and bending rigidity tensors of crystalline solids, leveraging interatomic force constants and long-wavelength perturbation theory. Crucially, in the long-wavelength limit, lattice vibrations induce macroscopic electric fields which further couple with the propagation of elastic waves, and a separate treatment on the long-range electrostatic interactions is thereby required to obtain elastic properties under the appropriate electrical boundary conditions. A cluster expansion of the charge density response and dielectric screening function in the long-wavelength limit has been developed to efficiently extract multipole and dielectric tensors of arbitrarily high order. We implement the proposed method in a first-principles framework and perform extensive validations on silicon, NaCl, GaAs and rhombohedral BaTiO$_3$ as well as monolayer graphene, hexagonal BN, MoS$_2$ and InSe, obtaining good to excellent agreement with other theoretical approaches and experimental measurements. Notably, we establish that multipolar interactions up to at least octupoles are necessary to obtain the accurate short-circuit elastic tensor of bulk materials, while higher orders beyond octupole interactions are required to converge the bending rigidity tensor of 2D crystals. The present approach greatly simplifies the calculations of bending rigidities and will enable the automated characterization of the mechanical properties of novel functional materials.