Spin Hall Current and Spin-transfer Torque in Ferromagnetic Metal

We theoretically examine the spin-transfer torque in the presence of spin-orbit interaction (SOI) at impurities in a ferromagnetic metal on the basis of linear response theory. We obtained, in addition to the usual spin-transfer torque, a new contributioin $sim {bm j}_{rm SH}^{phantom{dagger}} cdot abla {bm n}$ in the first order in SOI, where ${bm j}_{rm SH}^{phantom{dagger}}$ is the spin Hall current driven by an external electric field. This is a reaction to inverse spin Hall effect driven by spin motive force in a ferromagnet.