Spin orbit in curved graphene ribbons

We study the electronic properties of electrons in flat and curved zigzag graphene ribbons using a tight-binding model within the Slater Koster approximation. We find that curvature dramatically enhances the action of spin orbit effects in graphene ribbons and has a strong effect on 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 in the case of curved ribbons. We find that for the edge states, the spin density lies always in the plane perpendicular to the ribbon axis, and deviate strongly from the normal to the ribbon, even for very small curvature and the small 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 finding in the spin Hall phase of curved graphene Ribbons.