Effective lattice model of graphene moiré superlattices on hexagonal boron nitride

Abstract

We propose an effective lattice model of graphene moir\'e superlattices on hexagonal boron nitride (MLG/BN) based on ab initio calculations to study the evolution of electronic properties of relaxed MLG/BN superlattices with relative twist angle ($\ensuremath{\theta}$). The parameters for obtaining hopping terms and on-site energies in the lattice Hamiltonian are directly extracted from the ab initio electronic structure of shifted commensurate bilayers. The effect of the in-plane strain induced by the structural relaxation can be taken into account straightforwardly for the lattice model. We consider completely free MLG/BN heterostructures and those with constant interlayer distances. We find that the structural distortion due to the relaxation exhibits similar magnitude for $\ensuremath{\theta}$ smaller than about $0.{5}^{\ensuremath{\circ}}$. The gap at the primary Dirac points (${\mathrm{\ensuremath{\Delta}}}_{P}$) is maintained for $\ensuremath{\theta}l0.{5}^{\ensuremath{\circ}}$, while the gap at the secondary Dirac points (${\mathrm{\ensuremath{\Delta}}}_{S}$) decreases rapidly with small $\ensuremath{\theta}$, and such distinct behavior of ${\mathrm{\ensuremath{\Delta}}}_{P}$ and ${\mathrm{\ensuremath{\Delta}}}_{S}$ is roughly consistent with recent experimental observations. In addition, decreasing interlayer distance can significantly enhance ${\mathrm{\ensuremath{\Delta}}}_{P}$. This lattice model is sufficiently transparent and flexible to be employed to investigate various configurations of MLG/BN moir\'e superlattices.