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

TL;DR

This paper simplifies the derivation of the very low damping escape rate for classical spins, incorporating spin-transfer torque, by modifying Kramers' method to handle non-separable Hamiltonians directly from the Fokker-Planck equation.

Contribution

It extends Kramers' perturbative approach to classical magnetic spins with two degrees of freedom, simplifying the derivation of the VLD escape rate including spin-transfer torque.

Findings

Derived a simpler method for calculating the VLD escape rate for classical spins.

Included spin-transfer torque effects in the escape rate calculation.

Provided a direct derivation from the magnetic Fokker-Planck equation.

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 $(\vartheta, \varphi)$ is much simpler than those previously used.