Spin-torque oscillators with thermal noise: A constant energy orbit approach

We study the magnetization dynamics of spin-torque oscillators in the presence of thermal noise and as a function of the spin-polarization angle in a macrospin model. The macrospin has biaxial magnetic anisotropy, typical of thin film magnetic elements, with an easy axis in the film plane and a hard axis out of the plane. Using a method that averages the energy over precessional orbits, we derive analytic expressions for the current that generates and sustains out-of-plane precessional states. We find that there is a critical angle of the spin polarization necessary for the occurrence of such states and predict a hysteretic response to applied current. This model can be tested in experiments on orthogonal spin-transfer devices, which consist of both an in-plane and out-of-plane magnetized spin polarizers, effectively leading to an angle between the easy and spin-polarization axes.