Structural relaxation effects on the low-energy electronic structure of twisted bilayer graphene

The structural relaxation and electronic band structure of the moire pattern formed by twisted bilayer graphene (TBG) with a wide range of twist angles are calculated by plane-wave-based and linear-scaling localized-orbit-based first-principles methods. We provide the relaxation patterns for the interlayer separation and the in-plane displacements (with respect to the ideal TBG structure) covering the whole moiré pattern. Our results show that as the twist angle becomes small, one single graphene sheet undergoes a gradual transition from a nearly flat to a corrugated layer due to different layer interactions in different regions of the moiré pattern. The in-plane displacement exhibits a vortex-like pattern, and the displacement direction is reversed from the AA to AB stacking region. Overall, the atomic relaxation leads to the shrinkage of the AA stacking area in favor of the more energetically stable AB/BA stacking domains. For the electronic band structure, we find that the energy states at the Γ point above and below the four flat bands near the Fermi level split further away after structural relaxation. At larger twist angles, the splitting is mainly contributed by the out-of-plane relaxation. Approaching the magic angle, the in-plane and out-of-plane relaxation have nearly equal contributions to the splitting. However, at the magic angle, the splitting actually becomes smaller for vertical relaxations. Our calculations emphasize the necessity of building a large-scale moiré supercell to explore the full relaxation in each degree of freedom.