Elastic Constants and Bending Rigidities from Long-Wavelength Perturbation Expansions

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.

Figures (7)

• Figure 1: (a) Phonon dispersion and (b) independent components of the elastic tensor for silicon as a function of supercell size. The inset of panel (a) shows the enlarged acoustic branches around the $\Gamma$ point along the K and L directions. The dotted lines in the panel (b) are the reference values of elastic constants from the finite-difference calculations using the $\textsc{thermo\_pw}$ code dal2016.
• Figure 2: (a) Phonon dispersion with enlarged inset around the $\Gamma$ point and (b) independent components of elastic tensor for NaCl as a function of supercell size where the standard (DD) long-range has been removed. In the panel (b), the elastic constants of NaCl obtained without the long-range removal are shown in the inset, and the dotted lines are the reference values from the $\textsc{thermo\_pw}$ finite-difference calculations.
• Figure 3: Phonon dispersions of BaTiO$_3$ as a function of supercell size using (a) the standard (DD) interpolation and the multipolar interpolation with (b) DD+DQ+QQ+DO+D$\epsilon$D where the inset shows an enlarged version of the acoustic branches.
• Figure 4: (a-f) Independent components of the clamped-ion elastic tensor of BaTiO$_3$ as a function of supercell size. The elastic constants are calculated based on the short-range IFCs in real space by increasing the level of treatment for the long-range multipolar interactions. The dotted horizontal line in each subplot is the corresponding reference value from the finite-difference calculations using the thermo_pw code.
• Figure 5: (a) Phonon dispersion and (b) independent components of the elastic and bending rigidity tensors in graphene as a function of supercell size. The inset of panel (a) shows the enlarged acoustic branches. The dotted lines in the panel (b) represent the theoretical elastic constants $C_{11}$/$C_{12}$ and bending rigidity $D_{11}$ taken from Ref. andrew2012 and Ref. kumar2020, respectively.