Eigenstate Fokker-Planck equation for magnetic nanostructures

A Fokker-Planck equation (FPE) approach is presented to calculate the magnetization probability densities in nanomagnetic systems characterized by complex magnetization dynamics. The formulation is based on an eigenstate expansion of the solutions of the linearized Landau-Lifshitz-Gilbert equation (LLGE), which allows considering statistical properties of nonuniform magnetization dynamics, such as finding write error rates in spin transfer torque magnetic random access memories induced by thermal noise. The formulation includes finding eigenstates and eigenfrequencies of the structure of interest, formulating an FPE describing the probability density dynamics of the excitation coefficients of the eigenstate solutions, and using this equation to find derived quantities. Compared to using the stochastic LLGE, the presented approach directly obtains statistical properties. Compared to generally used FPE approaches, the presented formulation allows handling nonuniform dynamics. The formulation of using the eigenstate-based FPE is applicable to many types of physics.