Microscopic theory of spin-orbit torques in two dimensions

We formulate a general microscopic approach to spin-orbit torques in thin ferromagnet/heavy-metal bilayers in linear response to electric current or electric field. The microscopic theory we develop avoids the notion of spin currents and spin-Hall effect. Instead, the torques are directly related to a local spin polarization of conduction electrons, which is computed from generalized Kubo-Středa formulas. A symmetry analysis provides a one-to-one correspondence between polarization susceptibility tensor components and different torque terms in the Landau-Lifshitz-Gilbert equation for magnetization dynamics. The spin-orbit torques arising from Rashba or Dresselhaus type of spin-orbit interaction are shown to have different symmetries. We analyze these spin-orbit torques microscopically for a generic electron model in the presence of an arbitrary smooth magnetic texture. For a model with spin-independent disorder we find a major cancellation of the torques. In this case the only remaining torque corresponds to the magnetization-independent Edelstein effect. Furthermore, our results are applied to analyze the dynamics of a skyrmion under the action of electric current.