Phase-Field Crystal Method for Bilayer Graphene

Bilayer graphene has been a subject of intense study in recent years. Its most common structure is AB (Bernal) stacking, where one layer is shifted relative to the other, leading to distinct electronic behaviors compared to the less stable AA stacking or the fascinating twisted configurations. We extend a structural phase field crystal method to include an external potential based on the generalized stacking-fault energy that accounts for the effect from a bottom layer of graphene. Both of the favored stacking variants AB and BA are found with randomly generated initial phase fields. Using the width of the boundaries between different stacking variants as a function of the interactions between the two layers, we quantify the exact strength of the external potential by comparing the phase field crystal simulations with the results from atomistic simulation. Finally, we simulate a circular grain of one stacking phase enclosed by the other and find that, depending on the initial phase field, the center domain may shrink to form a uniform stacking phase, or may evolve to a relaxed state of a hexagon region or a triangular region that at each vertex the graphene structure is defected.