Thermal Properties of Spin Torque Oscillators: Singularity in Effective Energy

It is important to be able to calculate steady-state probability distributions in spin torque systems, for such applications as spin-torque oscillators and thermal switching of MRAM. It has been shown using a Fokker-Planck formulation of this problem that the steady state probability can be expressed in terms of an effective energy, which can be calculated analytically for uniaxial systems (including perpendicular-anisotropy thin film elements). For elements with in-plane anisotropy or in-plane fields, the effective energy has been calculated numerically and fit to a polynomial in the energy. The fit was not very good, and in the present paper we show that this is because the effective energy has a logarithmic singularity at the saddle point. Fitting to a form with the singularity built in gives a much better fit, using the same number of parameters. This makes computation of oscillator linewidths and switching rates in spin-torque systems much easier.