Fokker-Planck theory of Spin-torque switching: Effective energy and transition-state rate theory

摘要

A number of devices have been proposed and synthesized that exploit the spin torque effect. These systems are most straightforward to analyze in the absence of thermal effects. However, thermal effects are very important for the understanding of many experimental results, and for the design of devices. In spin torque MRAM, for example, although the very high-current behavior can be modeled without including thermal fluctuations, switching with currents low enough to be practical has a strong thermal component. Another example is the spin torque oscillator, whose usefulness in devices depends on its linewidth, which is strongly affected by thermal fluctuations. The statistical theory of spin torque systems has previously been worked out using the Fokker-Planck equation, which describes the time evolution of the probability density ρ(M). In this paper we formulate the theory in terms of an effective energy, which has the advantage that the exact solution for the probability density has the familiar form exp(-V E_(eff)/k_BT) in terms of the effective energy. We also generalize Eyring's 1935 transition state theory of rates to the spin torque case; it appears that in many practical cases this is a very good approximation.