Effective lattice model of graphene moire superlattices on hexagonal boron nitride

We propose an effective lattice model of graphene moire 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 (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 theta smaller than about 0.5 degrees. The gap at the primary Dirac points (Delta(P)) is maintained for theta < 0.5 degrees, while the gap at the secondary Dirac points (Delta(S)) decreases rapidly with small theta, and such distinct behavior of Delta(P) and Delta(S) is roughly consistent with recent experimental observations. In addition, decreasing interlayer distance can significantly enhance Delta(P). This lattice model is sufficiently transparent and flexible to be employed to investigate various configurations of MLG/BN moire superlattices.