Structural phase field crystal approach for modeling graphene and other two-dimensional structures

This paper introduces a new structural phase field crystal (PFC) type model that expands the PFC methodology to a wider class of structurally complex crystal structures than previously possible. Specifically, our approach allows for stabilization of graphene, as well as its coexistence with a disordered phase. It also preserves the ability to model the usual triangular and square lattices previously reported in two-dimensional (2D) PFC studies. Our approach is guided by the formalism of classical field theory, wherein the free-energy functional is expanded to third order in PFC density correlations. It differs from previous PFC approaches in two main features. First, it utilizes a hard-sphere repulsion to describe two-point correlations. Second, and more important, is that it uses a rotationally invariant three-point correlation function that provides a unified way to control the formation of crystalline structures that can be described by a specific bond angle, such as graphene, triangular, or square symmetries. Our approach retains much of the computational simplicity of previous PFC models and allows for efficient simulation of nucleation and growth of polycrystalline 2D materials. In preparation for future applications, this paper details the mathematical derivation of the model and its equilibrium properties and uses dynamical simulations to demonstrate defect structures produced by the model.