Lattice match and lattice mismatch models of graphene on hexagonal boron nitride from first principles

The interface between graphene and hexagonal boron nitride (h-BN) substrate plays an important role in device applications. Previously, theoretical studies have suggested that a small gap is opened at Dirac cones of graphene, but there is no detectable band gap in experiments. To explain the experimental result, we used two models from the views of lattice match and lattice mismatch between graphene and h-BN by first-principles calculations. We first studied the landscapes of the sliding energy surface (SES) and band gap of graphene on h-BN substrate within a lattice match approximation, which mimics continuously variable stacking sequences in a long-period graphene/BN Moiré superstructure arising from minor lattice mismatch. The plausibility of the long-period Moiré superstructure was evidenced by the smooth SES. The main features of the SES landscape can be captured by means of a simple registry index method. For most stacking patterns, the interactions between graphene and h-BN substrate open a band gap at the Dirac cones of graphene. However, there are special stacking modes in the landscape that preserve the Dirac cones of graphene. To further simulate the long-period graphene/BN Moiré superstructure observed in experiments, we also employed a rotation model within the lattice mismatch approximation. At the equilibrium interlayer spacing, the Dirac cones of graphene are preserved in all the rotational graphene/BN superstructures. The zero-band-gap feature is independent of the translation and rotation of graphene with respect to the h-BN substrate, which clearly agrees with the results of zero band gap in experiments.