A Mean-field model for spin dynamics in multilayered ferromagnetic media
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In this paper, we develop a mean-field model for describing the dynamics of spintransfer torque in multilayered ferromagnetic media. Specifically, we use the techniques of Wigner transform and moment closure to connect the underlying physics at different scales and reach a macroscopic model for the dynamics of spin coupled with the magnetization within the material. This provides a further understanding of the linear response model proposed by Zhang, Levy, and Fert [Phys. Rev. Lett., 88 (2002), 236601], and in particular we get an extra relaxation term which helps to stabilize the system. We develop efficient numerical methods to overcome the stiffness appearing in this new mean-field model and present several examples to analyze and show its validity.