Critical current density of a spin-torque oscillator with an in-plane magnetized free layer and an out-of-plane magnetized polarizer

Spin-torque induced magnetization dynamics in a spin-torque oscillator with an in-plane (IP) magnetized free layer and an out-of-plane (OP) magnetized polarizer under IP shape-anisotropy field ($H_{\rm k}$) and applied IP magnetic field ($H_{\rm a}$) was theoretically studied based on the macrospin model. The rigorous analytical expression of the critical current density ($J_{\rm c1}$) for the OP precession was obtained. The obtained expression successfully reproduces the experimentally obtained $H_{\rm a}$-dependence of $J_{\rm c1}$ reported in [D. Houssameddine $et$ $al$., Nat. Mater. 6, 447 (2007)].