Field-free self-oscillation of magnetization enabled by the fieldlike spin-orbit torque

Abstract

Fieldlike spin-orbit torque, which behaves like a torque from a magnetic field, but relies on the current together with the damplike one, was recognized to influence magnetization switching. However, its role on magnetic oscillations remains to be explored. By linear stability analysis and energy averaging technique, we obtain analytic formulas for the stable boundaries of various states. Then, a phase diagram is constructed, which is controlled by the current and a tunable ratio $\beta$ of the fieldlike torque to the dampinglike one. We find that some new stable, bistable and dynamic states come forth. Especially, a bias-field-free self-oscillation emerges for negative $\beta$, which enables the average energy balance between the dampinglike torque and the intrinsic damping. There occur two kinds of oscillations, analogous to the usual in-plane and out-of-plane precessions. If increasing the current, the frequency declines for the former, and subsequently rises for the latter. Varying $\beta$ slightly affects the frequency range, but dramatically alters the adjustable range of current. In addition, the phase diagram indicates that the switching direction reverses with $\beta$ stepping over $- 1/\alpha$ with $\alpha$ being the damping constant, and the switching current decreases with $\vert \beta \vert$ increasing.