Tight-binding theory of the spin-orbit coupling in graphene structures

Spin-orbit coupling changes qualitatively the electronic band structure of graphene. Most important, the coupling induces spectral gaps at the K(K') points. Earlier theories estimated the \emph{intrinsic} gap of $1$ $\mu$eV for the single layer and several meVs for bi- and tri-layer graphene, based on $\sigma$-$\pi$ coupling. Our first-principles calculations give the value of 24 $\mu$eV for all these systems, due to the presence of the orbitals of the $d$ symmetry in the Bloch states of the $\pi$ bands. A realistic multiband tight-binding model is presented to explain the effects the $d$ orbitals play in the spin-orbit coupling of graphene and derive an effective single-orbital next-nearest-neighbor hopping model that accounts for the spin-orbit effects. We also study the \emph{extrinsic} spin-orbit coupling, due to an applied transverse electric field. In a single layer the \emph{extrinsic} effect is dominated by the $\pi$-$\sigma$ hybridization. In contrast, in the multi-layer structures the \emph{extrinsic} spin-orbit band splittings come from an interplay of the $d$-orbitals, the inter-layer hopping, and the electrostatic potential from the applied field.