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 $\mathcal{E}$, ${\ensuremath{\Delta}}_{\mathcal{E}}\ensuremath{\propto}\ensuremath{\Delta}$, where $\ensuremath{\Delta}$ 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 ${\ensuremath{\Delta}}_{\mathrm{curv}}\ensuremath{\propto}\ensuremath{\Delta}$. For the values of $\mathcal{E}$ and curvature profile reported in actual samples of graphene, we find that ${\ensuremath{\Delta}}_{\mathrm{int}}l{\ensuremath{\Delta}}_{\mathcal{E}}\ensuremath{\lesssim}{\ensuremath{\Delta}}_{\mathrm{curv}}$. The effect of spin-orbit coupling on derived materials of graphenelike fullerenes, nanotubes, and nanotube caps, is also studied. For fullerenes, only ${\ensuremath{\Delta}}_{\mathrm{int}}$ is important. Both for nanotubes and nanotube caps ${\ensuremath{\Delta}}_{\mathrm{curv}}$ is in the order of a few Kelvins. We reproduce the known appearance of a gap and spin-splitting in the energy spectrum of nanotubes due to the spin-orbit coupling. For nanotube caps, spin-orbit coupling causes spin-splitting of the localized states at the cap, which could allow spin-dependent field-effect emission.