Theory of Thermally Induced Phase Noise in Spin Torque Oscillators for a High-Symmetry Case

We derive equations for the phase noise spectrum of a spin torque oscillator in the macrospin approximation for the highly symmetric geometry where the equilibrium magnetization, applied field, anisotropy, and spin accumulation are all collinear. This particular problem is one that can be solved by analytical methods, but nevertheless illustrates several important general principles for phase noise in spin torque oscillators. In the limit, where the restoring torque is linearly proportional to the deviation of the precession amplitude from steady-state, the problem reduces to a sum of the Wiener-Lévy (W-L) and Ornstein-Uhlenbeck (O-U) processes familiar from the physics of random walks and Brownian motion. For typical device parameters, the O-U process dominates the phase noise and results in a phase noise spectrum that is nontrivial, with 1/ω² dependence at low Fourier frequencies, and 1/ω⁴ dependence at high Fourier frequencies. The contribution to oscillator linewidth due to the O-U process in the low temperature limit is independent of magnetic anisotropy field Hk and scales inversely with the damping parameter, whereas in the high temperature limit the oscillator linewidth is independent of the damping parameter and scales as √(|Hk|) . Numerical integration of the fully nonlinear stochastic differential equations is used to determine the temperature and precession amplitude ranges over which our equations for phase noise and linewidth are valid. We then expand the theory to include effects of spin torque asymmetry. Given the lack of experimental data for nanopillars in the geometry considered here, we make a rough extrapolation to the case of nanocontacts, with reasonable agreement with published data. The theory does not yield any obvious means to reduce phase noise to levels required for practical applications in the geometry considered here.