Real-space and plane-wave hybrid method for electronic structure calculations for two-dimensional materials

We propose a computational approach to combining the plane-wave method and the real-space treatment to describe the periodic variation in the material plane and the decay of wave functions from the material surfaces. The proposed approach is natural for two-dimensional material systems and thus may circumvent some intrinsic limitations involving the artificial replication of material layers in traditional supercell methods. In particular, we show that the proposed method is easy to implement and, especially, computationally effective since low-cost computational algorithms, such as iterative and recursive techniques, can be used to treat matrices with block tridiagonal structure. Using this approach we show first-principles features that supplement the current knowledge of some fundamental issues in bilayer graphene systems, including the coupling between the two graphene layers, the preservation of the $\ensuremath{\sigma}$ band of monolayer graphene in the electronic structure of the bilayer system, and the differences in low-energy band structure between the AA- and AB-stacked configurations.