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

interaction in graphene is derived from a tight-binding model, which includes the $\pi$ and $\sigma$ bands. We analyze the combined effects of the intraatomic spin orbit coupling, curvature, and an 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 the intrinsic spin-orbit coupling $\Hi \propto \Delta^ 2$ and for the Rashba coupling due to an electric field and for flat graphene $\Delta_{\cal E} \propto \Delta$, where $\Delta$ is the intraatomic spin-orbit coupling constant for Carbon. Moreover we show that local curvature of the graphene sheet induces an extra spin-orbit coupling term $\Delta_{\rm curv} \propto \Delta$. For the values of $\cal E$ and curvature profile reported in actual samples of graphene, we find that $\Hi \lesssim \Delta_{\cal E} < \Delta_{\rm curv}$. The effect of spin orbit coupling on derived materials of graphene like fullerenes, nanotubes, and nanotube caps, where curvature effects are important, is also studied. For fullerenes only $\Hi$ is important. Both for nanotubes and nanotube caps $\Delta_R$ is of 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 for spin-dependent field-effect emission.