Lattice Green function for extended defect calculations: Computation and error estimation with long-range forces

TL;DR

This paper presents a method to accurately compute the lattice Green function for extended defect calculations, enabling better error estimation and improved modeling of long-range elastic fields in materials.

Contribution

It introduces a technique combining cutoff dynamical matrices and long-range behavior analysis to accurately calculate the lattice Green function and estimate associated errors.

Findings

Accurate calculation of lattice Green function up to a finite range.

Error estimation method for periodic boundary condition calculations.

Identification of a key length scale for defect modeling accuracy.

Abstract

Computing the atomic geometry of lattice defects--point defects, dislocations, crack tips, surfaces, or boundaries--requires an accurate coupling of the local strain field to the long-range elastic field. Periodic boundary conditions used by classical potentials or density-functional theory may not accurately reproduce the correct bulk response to an isolated defect; this is especially true for dislocations. Recently, flexible boundary conditions have been developed to produce the correct long-range strain field from a defect--effectively "embedding" a defect in a finite cell with infinite bulk response, isolating it from either periodic images or free surfaces. Flexible boundary conditions require the calculation of the bulk response with the lattice Green function (LGF). While the LGF can be computed from the dynamical matrix, for supercell methods (periodic boundary conditions) it can only be calculated up to a maximum range. We illustrate how to accurately calculate the lattice Green function and estimate the error using a cutoff dynamical matrix combined with knowledge of the long-range behavior of the lattice Green function. The effective range of deviation of the lattice Green function from the long-range elastic behavior provides an important length scale in multiscale quasi-continuum and flexible boundary-condition calculations, and measures the error introduced with periodic boundary conditions.