Spin-orbit gap of graphene: First-principles calculations

Even though graphene is a low-energy system consisting of a two-dimensional honeycomb lattice of carbon atoms, its quasiparticle excitations are fully described by the $(2+1)$-dimensional relativistic Dirac equation. In this paper we show that, while the spin-orbit interaction in graphene is of the order of $4\phantom{\rule{0.3em}{0ex}}\mathrm{meV}$, it opens up a gap of the order of ${10}^{\ensuremath{-}3}\phantom{\rule{0.3em}{0ex}}\mathrm{meV}$ at the Dirac points. We present a first-principles calculation of the spin-orbit gap, and explain the behavior in terms of a simple tight-binding model. Our result also shows that the recently predicted quantum spin Hall effect in graphene can occur only at…