Abstract
Moir\'e patterns in van der Waals (vdW) heterostructures bring novel physical effects to the materials. We report theoretical investigations of the Moir\'e pattern formed by graphene (Gr) on hexagonal boron nitride $(h\text{BN})$. For both the two-dimensional (2D) flat-sheet and the freestanding three-dimensional (3D) wavelike film geometries, the behaviors of Dirac electrons are strongly modulated by the local high-symmetry stacking configurations of the Moir\'e pattern. In the 2D flat sheet, the secondary Dirac cone (SDC) dispersion emerges due to the stacking-selected localization of SDC wave functions, while the original Dirac cone (ODC) gap is suppressed due to an overall effect of ODC wave functions. In the freestanding 3D wavelike Moir\'e structure, we predict that a specific local stacking in the Moir\'e superlattice is promoted at the expense of other local stackings, leading to an electronic structure more similar to that of the perfectly matching flat Gr/$h\text{BN}$ than that of the flat-sheet 2D Moir\'e pattern. To capture the overall picture of the Moir\'e superlattice, supercells containing 12 322 atoms are simulated by first principles.
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