Lattice Deformation, Low Energy Models and Flat Bands in Twisted Graphene Bilayers

Twisted graphene bilayers show a complex electronic structure, further modified by interaction effects. The main features can be obtained from effective models, which make use a few phenomenological parameters. We analyze the influence of effects at the atomic scale, such as interlayer hopping and lattice relaxation, on the electronic bands. We assume that the twist angle and the size of the Moir\'e pattern is fixed, as it is usually the case in experiments. We obtain a strong dependence of the electronic structure on details of the models at the atomic scale. We discuss how to incorporate this dependence on effective models