Spin-orbit coupling in curved graphene, fullerenes, nanotubes, and nanotube caps

A continuum model for the effective spin-orbit interaction in graphene is derived from a tight-binding model which includes the $\ensuremath{\pi}$ and $\ensuremath{\sigma}$ bands. We analyze the combined effects of the intra-atomic spin-orbit coupling, curvature, and applied electric field, using perturbation theory. We recover the effective spin-orbit Hamiltonian derived recently from group theoretical arguments by Kane and Mele. We find, for flat graphene, that the intrinsic spin-orbit coupling ${\ensuremath{\Delta}}_{\mathrm{int}}\ensuremath{\propto}{\ensuremath{\Delta}}^{2}$ and the Rashba coupling due to a perpendicular electric field $\mathcal{E}$,…