Geometric spin–orbit coupling and chirality-induced spin selectivity

We report a new type of spin–orbit coupling (SOC) called geometric SOC. Starting from the relativistic theory in curved space, we derive an effective nonrelativistic Hamiltonian in a generic curve embedded into flat three dimensions. The geometric SOC is O ( m ^−1 ), in which m is the electron mass, and hence much larger than the conventional SOC of O ( m ^−2 ). The energy scale is estimated to be a hundred meV for a nanoscale helix. We calculate the current-induced spin polarization in a coupled-helix model as a representative of the chirality-induced spin selectivity. We find that it depends on the chirality of the helix and is of the order of 0.01 ℏ per nm when a charge current of 1 μ A is applied.