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We construct a general theoretical framework for describing curvature-inducedspin-orbit interactions on the basis of group theory. Our theory cansystematically determine the emergence of spin splitting in the band structureaccording to symmetry in the wavenumber space and the bending direction of thematerial. As illustrative examples, we derive the curvature-induced spin-orbitcoupling for carbon and silicon nanotubes. Our theory offers a strategy fordesigning valley-contrasting spin-orbit coupled materials by tuning theircurvatures.

Designing Valley-Dependent Spin-Orbit Interaction by Curvature