In this study, we derive analytical expressions for the critical switching current density induced by spin Hall effect in magnetic structures with the first-and second-order perpendicular magnetic anisotropy. We confirm the validity of the expressions by comparing the analytical results with those obtained from a macrospin simulation. Moreover, we find that for a particular thermal stability parameter, the switching current density can be minimized for a slightly positive second-order perpendicular magnetic anisotropy and the minimum switching current density can further be tuned using an external magnetic field. The analytical expressions are of considerable value in designing high-density magnetic random access memory and cryogenic memory.
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