Automatic numerical evaluation of vacancy-mediated transport for arbitrary crystals: Onsager coefficients in the dilute limit using a Green function approach

TL;DR

This paper presents a comprehensive numerical method for calculating vacancy-mediated diffusion and Onsager coefficients in arbitrary crystal structures using a Green function approach, applicable in the dilute limit.

Contribution

It introduces a general, efficient Green function-based algorithm for evaluating transport coefficients in any crystal structure, leveraging symmetry and Dyson equations.

Findings

Validated convergence of the Green function integration method.

Computed tracer correlation factors for diverse crystal types.

Demonstrated the method's efficiency and accuracy across multiple crystal structures.

Abstract

A general solution for vacancy-mediated diffusion in the dilute-vacancy/dilute-solute limit for arbitrary crystal structures is derived from the master equation. A general numerical approach to the vacancy lattice Green function reduces to the sum of a few analytic functions and numerical integration of a smooth function over the Brillouin zone for arbitrary crystals. The Dyson equation solves for the Green function in the presence of a solute with arbitrary but finite interaction range to compute the transport coefficients accurately, efficiently and automatically, including cases with very large differences in solute-vacancy exchange rates. The methodology takes advantage of the space group symmetry of a crystal to reduce the complexity of the matrix inversion in the Dyson equation. An open-source implementation of the algorithm is available, and numerical results are presented for the convergence of the integration error of the bare vacancy Green function, and tracer correlation factors for a variety of crystals including wurtzite (hexagonal diamond) and garnet.