On a simple derivation of the very low damping escape rate for classical spins by modifying the method of Kramers

Abstract

The original perturbative Kramers’ method (starting from the phase space coordinates) (Kramers, 1940) of determining the energy-controlled-diffusion equation for Newtonian particles with separable and additive Hamiltonians is generalized to yield the energy-controlled diffusion equation and thus the very low damping (VLD) escape rate including spin-transfer torque for classical giant magnetic spins with two degrees of freedom. These have dynamics governed by the magnetic Langevin and Fokker–Planck equations and thus are generally based on non-separable and non-additive Hamiltonians. The derivation of the VLD escape rate directly from the (magnetic) Fokker–Planck equation for the surface distribution of magnetization orientations in the configuration space of the polar and azimuthal angles (ϑ,φ) is much simpler than those previously used.