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

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 spin-polarized 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.

Figure 1

Geometry of the problem. A STT device consists of two ferromagnetic strata labeled the free and fixed layers, respectively, and a normal conducting spacer all sandwiched on a pillar between two ohmic contacts (Ref. 13). The fixed layer has a fixed magnetization along the direction ${\mathbf{e}}_P$. ${\mathbf{J}}_e$ is the spin-polarized current density, $\mathbf{M}$ is the magnetization of the free layer, and ${\mathbf{H}}_0$ is the applied magnetic field.

Figure 2

(a) Biaxial anisotropy potential $V(\vartheta ,\phi )$, Eq. (3). (b) Current-induced trajectory of the magnetization escape. Solid line: numerical solution of the deterministic Eq. (1), i.e., omitting the random field $\mathbf{h}$, for a strong spin-polarized current, damping $\alpha =0.01$, and typical values of other model parameters (see Sec. 5). Reversal of the magnetization from one metastable state to another typically occurs after many precessions about the $X$ axis, over saddle points. Thus having traversed the potential barrier, the magnetization decays to a new, stable direction of precession so that in this reverse direction the current accelerates the decay by increasing the effective damping, as is obvious from Eq. (1). (c) The same as in Fig. 2b for a weak spin-polarized current and the same values of other model parameters. Here, only the damped precessions of the magnetization about the stable direction exist.

Figure 3

3D plot of the nonconservative potentials $\Phi \left(\vartheta ,\phi \right)$, Eq. (18), (a) and ${\Phi }_{ap}\left(\vartheta ,\phi \right)$, Eq. (19), (b) for typical values of the model parameters ($P=0.3$, $J=6$, ${\phi }_P=0$, and ${\vartheta }_P=\pi /2$).

Figure 4

(a) 3D plot of the effective potential $V_{ef}\left(\vartheta ,\phi \right)$, Eq. (27), in the vicinity of the minima for $J=6$, $\alpha =0.02$, $h=0.15$, $\sigma =20$, $\delta =20$, ${\phi }_P=0$, and ${\phi }_{\xi }=0$.

Figure 5

$V_{ef}\left(\pi /2,\phi \right)$ vs the azimuthal angle $\phi$ for various spin-polarized current parameters $J$ $=$ 6, 0, −6, −12 and $\alpha =0.02$ (a); for various damping $\alpha$ $=$ 0.2, 0.02, 0.005 and $J$ $=$ −6 (b); for various external field parameters $h$ $=$ 0.0, 0.1, 0.2 and $J$ $=$ 6 (c); and $J$ $=$ −6 (d) ($\alpha =0.02$, $h=0.15$, $\sigma =20$, $\delta =20$, ${\phi }_P=0$, and ${\phi }_{\xi }=0$).

Figure 6

${\langle u_X\rangle }_0$ and ${\langle u_X^2\rangle }_0-{\langle u_X\rangle }_0^2$ vs the dimensionless current parameter $J$ for various values of damping $\alpha$ (a), external field parameter $h$ (b), external field orientation in the free layer ${\phi }_{\xi }$ (c), and spin-polarization orientation ${\phi }_P$ (d).

Figure 7

${\langle u_X\rangle }_0$ and ${\langle u_X^2\rangle }_0-{\langle u_X\rangle }_0^2$ vs the external field parameter $h$ for various values of the current parameter $J$ (a), the external field orientation ${\phi }_{\xi }$ (b), the spin-polarization orientation ${\phi }_P$ (c), and the inverse temperature parameter $\sigma$ (d).

Figure 8

Reversal time $\tau /{\tau }_0$ vs the damping parameter $\alpha$ for various values of the current $J$ (solid lines). Asterisks: escape rate formula, Eq. (30).

Figure 9

$\tau /{\tau }_0$ vs the inverse temperature parameter $\sigma \sim T^{-1}$ for various values of the current $J$ (a) and the external field $h$ (b). Asterisks: escape rate formula, Eq. (30).

Figure 10

$\tau /{\tau }_0$ vs the spin-polarized current parameter $J$ for various values of the external field $h=0,0.05,0.1,0.2$.

Figure 11

$\tau /{\tau }_0$ vs the azimuthal angles ${\phi }_{\xi }$ (a) and ${\phi }_P$ (b) for $J=6,0,-6,-12$.