Theoretical condition for switching the magnetization in a perpendicularly magnetized ferromagnet via the spin Hall effect

TL;DR

This paper derives a theoretical formula for the threshold current needed to switch perpendicular magnetization in ferromagnets via the spin Hall effect, supported by numerical simulations and perturbation analysis.

Contribution

It introduces a new analytical formula for the threshold current, linking magnetic field strength and current to magnetization switching.

Findings

Switching occurs when the steady-state magnetization is outside an unstable region.

The threshold current depends on the magnetic field and anisotropy.

Magnetic field should exceed 15% of the anisotropy field for switching.

Abstract

A theoretical formula is derived for the threshold current to switch a perpendicular magnetization in a ferromagnet by the spin Hall effect. The numerical simulation of the Landau-Lifshitz-Gilbert equation indicates that magnetization switching is achieved when the steady-state solution of the magnetization in the presence of the current is outside an energetically unstable region. Based on the numerical result, an analytical theory deriving the threshold current is developed by focusing on the first-order perturbation to the unstable state. The analytical formula clarifies that the magnitude of the magnetic field applied to the current direction should be larger than 15\% of the perpendicular magnetic anisotropy field, and the current is less than the derived threshold value.