Generalized Spin Drift-Diffusion Formalism in the Presence of Spin-Orbit Interaction of Ferromagnets

We generalize the spin drift-diffusion formalism by considering spin-orbit interaction of a ferromagnet, which generates transverse spin currents in the ferromagnet. We consider quantum-mechanical transport of transverse spins in a spin-orbit coupled ferromagnet and develop a generalized drift-diffusion equation and boundary condition. By combining them, we identify previously unrecognized spin transport phenomena in heterostructures including ferromagnets. As representative examples, we show self-generated spin torque and self-generated charge pumping in ferromagnet-normal metal bilayers. The former is a torque exerting on a ferromagnet, originating from a transverse spin current leaving from the ferromagnet itself, whereas the latter is the Onsager reciprocity of the former. Our work not only provides a concise formalism for the effects of nondephased transverse spins in ferromagnets but also enables to design spintronic devices without an external spin source.