Nonlinear analysis of magnetization dynamics excited by spin Hall effect

We investigate the possibility of exciting self-oscillation in a perpendicular ferromagnet by the spin Hall effect on the basis of a nonlinear analysis of the Landau-Lifshitz-Gilbert (LLG) equation. In the self-oscillation state, the energy supplied by the spin torque during a precession on a constant energy curve should equal the dissipation due to damping. Also, the current to balance the spin torque and the damping torque in the self-oscillation state should be larger than the critical current to destabilize the initial state. We find that these conditions in the spin Hall system are not satisfied by deriving analytical solutions of the energy supplied by the spin transfer effect and the dissipation due to the damping from the nonlinear LLG equation. This indicates that the self-oscillation of a perpendicular ferromagnet cannot be excited solely by the spin Hall torque.