Probability maximization of magnetization switching by spin-orbit torques generated by a ferromagnet

Spin-orbit torques (SOTs) caused by spin currents generated in a ferromagnetic electrode enable a fast and deterministic magnetization switching. One SOT ($y$-SOT), polarized orthogonal to both electric current flowing in the electrode and easy axis of a ferromagnetic free layer, causes fast magnetization instability. The other SOT ($z$-SOT) points to the easy-axis direction and leads to the deterministic switching. Here, we evaluate the magnetization switching probability by these SOTs for various values of electric current density and the ratio of two SOTs from numerical simulation of the Landau-Lifshitz-Gilbert (LLG) equation. It is found that the switching probability is maximized when the electric current density is close to a critical value for the magnetization destabilization solely by the $y$-SOT. The origin of such a current dependence is investigated by analyzing temporal dynamics, spectra of the magnetization distribution, and a steady-state solution of the LLG equation. We reveal that the maximization of the switching probability originates due to two different switching behaviors. In the low current region, the magnetization in some trials remains near the initial state because of the weak $y$-SOT, and thus, the switching error occurs. The number of such trials, which represents the switching error, decreases as the electric current density increases because both $y$ and $z$-SOTs prompt the switching. In the large current region, the large $y$-SOT immediately tilts the magnetization toward the switched direction. However, since the $y$-SOT prefers an in-plane magnetized state, the switching error due to a probabilistic return to the initial state after turning off the electric current density occurs. The switching error in this region tends to increase with increasing current density. As a result, the switching probability is optimized when the electric current density is close to the critical value.