Charge and spin currents generated by dynamical spins

We demonstrate theoretically that a charge current and a spin current are generated by spin dynamics in the presence of spin-orbit interaction in the perturbative regime. We consider a general spin-orbit interaction including the spatially inhomogeneous case. Spin current due to spin damping is identified as one origin of generated charge current, but other contributions exist, such as the one due to an induced conservative field and the one arising from the inhomogeneity of spin-orbit interaction.