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

The spin-orbit coupling in graphene induces spectral gaps at the high-symmetry points. The relevant gap at the Γ point is similar to the splitting of the p orbitals in the carbon atom, being roughly 8.5 meV. The splitting at the K point is orders of magnitude smaller. Earlier tight-binding theories indicated the value of this intrinsic gap of 1 μeV, based on the σ-π coupling. All-electron first-principles calculations give much higher values, between 25 and 50 μeV, due to the presence of the orbitals of the d symmetry in the Bloch states at K. A realistic multiband tight-binding model is presented to explain the effects the d orbitals play in the spin-orbit coupling at K. The π-σ coupling is found irrelevant to the value of the intrinsic spin-orbit-induced gap. On the other hand, the extrinsic spin-orbit coupling (of the Bychkov-Rashba type), appearing in the presence of a transverse electric field, is dominated by the π-σ hybridization, in agreement with previous theories. Tight-binding parameters are obtained by fitting to first-principles calculations, which also provide qualitative support for the model when considering the trends in the spin-orbit-induced gap in graphene under strain. Finally, an effective single-orbital next-nearest-neighbor hopping model accounting for the spin-orbit effects is derived.