Atomic corrugation and electron localization due to Moiré patterns in twisted bilayer graphenes

We report on unprecedentedly large-scale density-functional calculations that clarify atomic and electronic structures of twisted bilayer graphene (BLG). We find the existence of the critical twist angle from either the AB or the AA stacking BLG, above which the two graphene layers are essentially decoupled and below which the atomic planes are corrugated and the Dirac electrons are localized. We also find a magic angle at which the Fermi velocity of the Dirac electron vanishes. We clarify that the Moir\'e pattern in tBLG with a tiny twist angle generates inhomogeneity for the electron systems and thus causes the drastic modification of the electronic properties, leading to flat bands at the Fermi level. Sensitivity to the Moir\'e of the valence-electron density and the electron state near the Fermi level is discussed.