Intrinsic spin-orbit interactions in flat and curved graphene nanoribbons

Recent theoretical and experimental works on carbon nanotubes and graphene samples have revealed that spin-orbit interactions, though customarily ignored in carbon-based materials, are more important and complex than it was thought. We study the intrinsic spin-orbit coupling effects on graphene nanoribbons, both flat and bent. Calculations are performed within the tight-binding model with the inclusion of a four-orbital basis set. Thereby the full symmetry of the honeycomb lattice and the hybridization of $\sigma$ and $\pi$ bands are considered. In addition to the zero-energy $\pi$-edge states, $\sigma$-derived edge states are found for the three investigated ribbon geometries. The $\sigma$ states are also spin-filtered and localized at the boundaries of the ribbons. The calculated spin-orbit splittings are larger for the $\sigma$- than for the $\pi$-derived edge states. Due to this enhancement, spin-orbit splittings of the $\sigma$-states reach values of the order of a few Kelvin. These spin-filtered edge states are robust under $\sigma-\pi$ hybridization and curvature effects.