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 $24\text{ }\ensuremath{\mu}\text{eV}$ (0.28 K) at the $K({K}^{\ensuremath{'}})$ point. It is shown that the previously accepted value of $1\text{ }\ensuremath{\mu}\text{eV}$, coming from the $\ensuremath{\sigma}\text{\ensuremath{-}}\ensuremath{\pi}$ 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 $10\text{ }\ensuremath{\mu}\text{eV}$ (0.11 K) per V/nm, coming from the $\ensuremath{\sigma}\text{\ensuremath{-}}\ensuremath{\pi}$ mixing. A ``miniripple'' configuration with every other atom shifted out of the sheet by less than 1% differs little from the intrinsic case.