When atomically thin two-dimensional (2D) materials are layered they often form incommensurate non-crystalline structures that exhibit long-period moir{\' e} patterns when examined by scanning probes. In this paper we present an approach which uses information obtained from {\it 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.
@article{arxiv.1312.7723,
title = {Ab-Initio Theory of Moir\'e Superlattice Bands in Layered Two-Dimensional Materials},
author = {Jeil Jung and Arnaud Raoux and Zhenhua Qiao and Allan H. MacDonald},
journal= {arXiv preprint arXiv:1312.7723},
year = {2015}
}