Ab initiotheory of moiré superlattice bands in layered two-dimensional materials

When atomically thin two-dimensional (2D) materials are layered, they often form incommensurate noncrystalline structures that exhibit long-period moir\'e patterns when examined by scanning probes. In this paper, we present an approach that uses information obtained from ab initio calculations performed on short-period crystalline structures to derive effective Hamiltonians that are able to efficiently describe the influence of the moir\'e pattern superlattices on electronic properties. We apply our approach to the cases of graphene on graphene (G/G) and graphene on hexagonal boron nitride (G/BN), deriving explicit effective Hamiltonians that have the periodicity of the moir\'e pattern and can be used to calculate electronic properties of interest for arbitrary twist angles and lattice constants.