Amplitude-phase coupling in a spin-torque nano-oscillator

The spin-torque nano-oscillator in the presence of thermal fluctuation is described by the normal form of the Hopf bifurcation with an additive white noise. By the application of the reduction method, the amplitude-phase coupling factor, which has a significant effect on the power spectrum of the spin-torque nano-oscillator, is calculated from the Landau–Lifshitz–Gilbert–Slonczewski equation with the nonlinear Gilbert damping. The amplitude-phase coupling factor exhibits a large variation depending on an in-plane anisotropy under the practical external fields.