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

Abstract

The spin-orbit coupling in graphene induces spectral gaps at the high-symmetry points. The relevant gap at the $\ensuremath{\Gamma}$ point is similar to the splitting of the $p$ orbitals in the carbon atom, being roughly 8.5 meV. The splitting at the $\mathrm{K}$ point is orders of magnitude smaller. Earlier tight-binding theories indicated the value of this intrinsic gap of $1\text{ }\ensuremath{\mu}\text{eV}$, based on the $\ensuremath{\sigma}\text{\ensuremath{-}}\ensuremath{\pi}$ coupling. All-electron first-principles calculations give much higher values, between 25 and $50\text{ }\ensuremath{\mu}\text{eV}$, due to the presence of the orbitals of the $d$ symmetry in the Bloch states at $\mathrm{K}$. A realistic multiband tight-binding model is presented to explain the effects the $d$ orbitals play in the spin-orbit coupling at $\mathrm{K}$. The $\ensuremath{\pi}\text{\ensuremath{-}}\ensuremath{\sigma}$ 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 $\ensuremath{\pi}\text{\ensuremath{-}}\ensuremath{\sigma}$ 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.