Low-energy band structure very sensitive to the interlayer distance in Bernal-stacked tetralayer graphene

We have investigated Bernal-stacked tetralayer graphene as a function of interlayer distance and perpendicular electric field by using density functional theory calculations. The low-energy band structure was found to be very sensitive to the interlayer distance, undergoing a metal-insulator transition. It can be attributed to the nearest-layer coupling that is more sensitive to the interlayer distance than are the next-nearest-layer couplings. Under a perpendicular electric field above a critical field, six electric-field-induced Dirac cones with mass gaps predicted in tight-binding models were confirmed, however, our density functional theory calculations demonstrate a phase transition to a quantum valley Hall insulator, contrasting to the tight-binding model prediction of an ordinary insulator.