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 pi and 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 Delta(int)proportional to Delta(2) and the Rashba coupling due to a perpendicular electric field E, Delta E proportional to Delta, where Delta is the intra-atomic spin-orbit coupling constant for carbon. Moreover we show that local curvature of the graphene sheet induces an extra spin-orbit coupling term Delta(curv)proportional to Delta. For the values of E and curvature profile reported in actual samples of graphene, we find that Delta(int)