Spin-torque effects in thermally assisted magnetization reversal: Method of statistical moments

Abstract

Thermal fluctuations of nanomagnets driven by spin-polarized currents are treated via the Landau-Lifshitz-Gilbert equation generalized to include both the random thermal noise field and the Slonczewski spin-transfer torque term. By averaging this stochastic (Langevin) equation over its realizations, the explicit infinite hierarchy of differential-recurrence relations for statistical moments (averaged spherical harmonics) is derived for arbitrary demagnetizing factors and magnetocrystalline anisotropy for the generic nanopillar model of a spin-torque device comprising two ferromagnetic strata representing the free and fixed layers and a nonmagnetic conducting spacer all sandwiched between two ohmic contacts. The influence of thermal fluctuations and spin-transfer torques on relevant switching characteristics, such as the stationary magnetization, the magnetization reversal time, etc., is calculated by solving the hierarchy for wide ranges of temperature, damping, external magnetic field, and spinpolarized current indicating new spin-torque effects in the thermally assisted magnetization reversal comprising several orders of magnitude. In particular, a pronounced dependence of the switching characteristics on the directions of the external magnetic field and the spin polarization exists.

Spin-transfer torque effects in the dynamic forced response of the magnetization of nanoscale ferromagnets in superimposed ac and dc bias fields in the presence of thermal agitation

“…1b). This potential, in general, represents an energyscape with two minima and two saddle points compelling the magnetization to align in a given direction in either of the energy minima in the equatorial or XY plane [28]. As in Ref.…”

“…[28], Z is taken as the hard axis while the X-axis is the easy one. Furthermore, the non-conservative potential  due to the spin-polarized current may sensibly be approximated [28] for all polar angles  ,  and arbitrary orientation of the unit vector …”

“…These observables comprise the stationary distribution of the magnetization orientations, the effective potential, the in-plane component of the magnetization of the free layer, and the static susceptibility. In particular, these time-and frequency-independent observables have been studied [28] for wide ranges of the spin-polarized current, the dissipative coupling to the thermal bath, the anisotropy parameters and the magnitude and orientation of the applied external dc field, which was supposed constant in time. Besides the calculation of these stationary observables, the reversal time of the in-plane component of the magnetization of the free layer has also been evaluated [28] via the smallest nonvanishing eigenvalue of the corresponding FokkerPlanck operator [29] again as a function of the parameters mentioned.…”

“…His primary objective was to securely anchor Néel's conjectures [1] concerning the nature of the superparamagnetic relaxation of a single domain ferromagnetic particle within the framework of the theory of stochastic processes 4 (in essence, Brown's theory of the magnetization relaxation in magnetic nanoparticles [2] is an analog of the Debye theory [22,23] Now we have previously treated [28] STT effects on certain out-of-equilibrium time-and frequency-independent stationary observables in the presence of a dc bias field alone via the generic nanopillar model (Fig. 1) by solving the magnetic Langevin equation (1) using the statistical moment method [27].…”

“…Now in Ref. 28, the external applied (bias) field was supposed time-independent, i.e., it represents a dc field applied in the infinite past. Hence, the results of Ref.…”