Generalized spin-orbit torques in two-dimensional ferromagnets with spin-orbit coupling

Based on the spinor Boltzmann equation, we obtain the continuity equations of charge and spin accumulation in two-dimensional ferromagnets with Rashba or Dresselhaus spin-orbit coupling, in which the spin diffusion equation (the continuity equation of spin accumulation) gives the spin-orbit torques that have an extra term of charge density gradient that is paid little attention in previous studies. Numerical results indicate this term plays an important role in inhomogeneous systems. In order to study the magnetization dynamics of ferromagnets, we solve the Landau-Lifshitz-Gilbert equation together with the spin diffusion equation. Our results indicate that the magnetization in systems with Rashba SOI is easier to be switched by an external electric field than the system with Dresselhaus SOI, and for anisotropic ferromagnets, we study the critical external electric field that manipulate the reversal of magnetization.