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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 $σ$ and $π$ bands are considered. In addition to the zero-energy $π$-edge states, $σ$-derived edge states are found for the three investigated ribbon geometries. The $σ$ states are also spin-filtered and localized at the boundaries of the ribbons. The calculated spin-orbit splittings are larger for the $σ$- than for the $π$-derived edge states. Due to this enhancement, spin-orbit splittings of the $σ$-states reach values of the order of a few Kelvin. These spin-filtered edge states are robust under $σ-π$ hybridization and curvature effects.