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 and 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 ₈₍ₓ^2 and the Rashba coupling due to a perpendicular electric field E, ₄, where 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 ₂ₔₑₕ. For the values of E and curvature profile reported in actual samples of graphene, we find that ₈₍ₓ<₄₂ₔₑₕ. The effect of spin-orbit coupling on derived materials of graphenelike fullerenes, nanotubes, and nanotube caps, is also studied. For fullerenes, only ₈₍ₓ is important. Both for nanotubes and nanotube caps ₂ₔₑₕ 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.