Interlayer hybridization in graphene quasicrystal and other bilayer graphene systems

The incommensurate 30 degrees twisted bilayer graphene (BG) possesses both relativistic Dirac fermions and quasiperiodicity with 12-fold rotational symmetry arising from the interlayer interaction [Ahn et al., the interlayer states interact with each other is of vital importance for identifying and subsequently engineering the quasicrystalline order in the layered structure. Herein, via symmetry and group representation theory we unravel the interlayer hybridization selection rules governing the interlayer coupling in both untwisted and twisted BG systems. Compared with the only allowed equivalent hybridization in D6h untwisted BG, D6 twisted BG permits equivalent and mixed hybridizations, and D6d graphene quasicrystal allows both equivalent and nonequivalent hybridizations. The energy-dependent hybridization strengths in graphene quasicrystal and D6 twisted BG show two remarkable characteristics: (i) near the Fermi level the weak hybridization owing to the relatively large energy difference between Dirac bands from top and bottom layers, and (ii) in high-energy regions the electron-hole asymmetry of hybridization strength with stronger interlayer coupling for holes, which arises from the non-nearest-neighbor interlayer hoppings and the wave-function phase difference between pairing states. These hybridization-generated band structures and their hybridization strength characteristics are verified by the calculated optical conductivity spectra. Our theoretical study paves a way for revealing the interlayer hybridization in van der Waals layered systems.