Amplitude expansion of the phase-field crystal model for complex crystal structures

TL;DR

This paper introduces a generalized amplitude expansion of the phase-field crystal model that can describe virtually any lattice symmetry, enabling detailed simulations of complex crystal structures and defects at diffusive timescales.

Contribution

The authors develop a comprehensive amplitude-based framework for the PFC model that captures a wide range of lattice symmetries, including non-Bravais lattices, with applications to complex crystal structures.

Findings

Allows modeling of complex and non-standard lattice symmetries.

Enables coarse-grained simulations of dislocations in various lattices.

Supports adaptive real-space numerical methods for large systems.

Abstract

The phase-field crystal (PFC) model describes crystal lattices at diffusive timescales. Its amplitude expansion (APFC) can be applied to the investigation of relatively large systems under some approximations. However, crystal symmetries accessible within the APFC model are limited to basic ones, namely triangular and square in two dimensions, and body-centered cubic and face-centered cubic in three dimensions. In this work, we propose a general, amplitudes-based description of virtually any lattice symmetry. To fully exploit the advantages of this model, featuring slowly varying quantities in bulk and localized significant variations at dislocations and interfaces, we consider formulations suitable for real-space numerical methods supporting adaptive spatial discretization. We explore approaches originally proposed for the PFC model which allow for symmetries beyond basic ones through extended parametrizations. Moreover, we tackle the modeling of non-Bravais lattices by introducing an amplitude expansion for lattices with a basis and further generalizations. We study and discuss the stability of selected, prototypical lattice symmetries. As pivotal examples, we show that the proposed approach allows for a coarse-grained description of the kagome lattice, exotic square arrangements, and the diamond lattice, as bulk crystals and, importantly, hosting dislocations.