Damping Dependence of Spin-Torque Effects in Thermally Assisted Magnetization Reversal

Abstract

Index Terms-Escape rate, Nanomagnets, Reversal time of the magnetization, Spin-transfer torque.

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

“…and  is the nonconservative STT potential given by [20,21]   [20]. For a qualitative description of STT effects,  may be written (by expanding the logarithm) in the following simplified form [15,16,22]:…”

“…is the dimensionless STT parameter, I is the current which is taken as positive if electrons flow from the free into the fixed layer, e is the electronic charge, is Planck's reduced constant, and the coefficients P b and P c are model dependent and are determined by the spin-polarization factor (0 1) PP  [20]. For a qualitative description of STT effects,  may be written (by expanding the logarithm) in the following simplified form [15,16,22]:…”

“…As two degrees of freedom (namely, the polar and azimuthal angles ,  describing the magnetization orientation in configuration space) are involved, the spin Hamiltonian, unlike that of particles, is no longer separable and additive since the governing magnetic Langevin equation is essentially a form of the Larmor equation so that inertial effects play no role here [11,12]. Nevertheless, the role of inertia in the mechanical system is essentially mimicked in the magnetic system for noncircularly symmetric free energy potentials by the gyromagnetic term, which causes coupling or entanglement of We recall that the VLD spin escape rate both with and without STT has already been determined [12,13] either via involved vector manipulation [14] or else via long and complicated calculations with the energy and phase as variables in the magnetic Langevin equation involving multiplicative noise [13,15,16,17]. Both methods ultimately lead to the same energy-controlled-diffusion equation for giant classical spins with quasi-stationary solutions yielding the VLD rate.…”

“…We recall that the VLD spin escape rate both with and without STT has already been determined [12,13] either via involved vector manipulation [14] or else via long and complicated calculations with the energy and phase as variables in the magnetic Langevin equation involving multiplicative noise [13,15,16,17]. Both methods ultimately lead to the same energy-controlled-diffusion equation for giant classical spins with quasi-stationary solutions yielding the VLD rate.…”

“…and so automatically involving the Taniguchi et al[22] used these equations to derive the mean first passage time. This was then extended by Byrne et al[16] to include an external magnetic field of arbitrary strength.Furthermore, the VLD calculations were extended to include the Kramers turnover region in Kalmykov et al[15].Thus, the original VLD approach of Kramers (1940) based on calculating directly the averages over slightly perturbed orbits, when applied to magnetic nanoparticles, yields both a simpler and more easily visualized solution of the VLD problem than those hitherto existing.Eq. (A1) becomes in the new variables  The change of variables in the term St( ) W  can be accomplished in like manner.…”