Spin-orbit interaction in curved graphene ribbons

We study the electronic properties of electrons in flat and curved zigzag graphene nanoribbons using a tight-binding model within the Slater Koster approximation, including spin-orbit interaction. We find that a constant curvature across the ribbon dramatically enhances the action of the spin-orbit term, strongly influencing the spin orientation of the edge states: Whereas spins are normal to the surface in the case of flat ribbons, this is no longer the case for curved ribbons. This effect is very pronounced, the spins deviating from the normal to the ribbon, even for very small curvature and a realistic spin orbit coupling of carbon. We find that curvature results also in an effective second neighbor hopping that modifies the electronic properties of zigzag graphene ribbons. We discuss the implications of our findings in the spin Hall phase of curved graphene ribbons.