Band-structure topologies of graphene: Spin-orbit coupling effects from first principles

The electronic band structure of graphene in the presence of spin-orbit coupling and transverse electric field is investigated from first principles using the linearized augmented plane-wave method. The spin-orbit coupling opens a gap of 24 μeV (0.28 K) at the K(K^') point. It is shown that the previously accepted value of 1 μeV, coming from the σ-π mixing, is incorrect due to the neglect of d and higher orbitals whose contribution is dominant due to symmetry reasons. The transverse electric field induces an additional (extrinsic) Bychkov-Rashba-type splitting of 10 μeV (0.11 K) per V/nm, coming from the σ-π mixing. A ``miniripple'' configuration with every other atom shifted out of the sheet by less than 1% differs little from the intrinsic case.