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

Spin-transfer torque (STT) effects on the stationary forced response of nanoscale ferromagnets subject to thermal fluctuations and driven by an ac magnetic field of arbitrary strength and direction are investigated via a 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 STT effects are treated via Brown's magnetic Langevin equation generalized to include the Slonczewski STT term thereby extending the statistical moment method [Y. P. Kalmykov et al., Phys. Rev. B 88, 144406 (2013)] to the forced response of the most general version of the nanopillar model. The dynamic susceptibility, nonlinear frequency-dependent dc magnetization, dynamic hysteresis loops, etc. are then evaluated highlighting STT effects on both the low-frequency thermal relaxation processes and the high-frequency ferromagnetic resonance, etc., demonstrating a pronounced dependence of these on the spin polarization current and facilitating interpretation of STT experiments.

Figure 1

(a) 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 [12]. 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, ${\mathbf{H}}_0$ is the dc bias magnetic field, and $\mathbf{H}cos\omega t$ is the applied ac field. (b) Free energy potential presented in the standard form of superimposed easy-plane and in-plane easy-axis anisotropies.

Figure 2

Real and imaginary parts of the normalized linear susceptibility $\chi \left(\omega \right)/\chi$ vs the normalized frequency $\omega {\tau }_N$ for various spin-polarized current parameters $J=6,0,-3,-6,-12$ and for various orientations of the applied fields ${\phi }_{\xi }=0$ (a) and ${\phi }_{\xi }=\pi /4$ (b) with the anisotropy parameter $\sigma =20$ and the dc field parameter ${\xi }_0=2$. Solid lines: Matrix continued fraction solution. Asterisks: Approximate Eq. (14) with the reversal time $\tau ={\lambda }_1^{-1}$ calculated using the independent method of Ref. [28].

Figure 3

(a) Time-independent (dc) component of the magnetization $M_0/M_S$ vs the spin-polarized current parameter $J$ for (a) various ac field amplitudes $\xi =0.01,1,2$, and 3 $(\xi =0.01$ represents linear response) and $\omega {\tau }_N=1$ and for (b) various dimensionless frequencies $\omega {\tau }_N={10}^{-1},10,$ and ${10}^3$ at $\xi =2$ at $\sigma =5,{\phi }_{\xi }=0$, and ${\xi }_0=0$.

Figure 4

(a) Real and imaginary parts of the nonlinear susceptibility $\mathrm{Re}\left[\chi \left(\omega \right)\right]$ and $-\mathrm{Im}\left[\chi \left(\omega \right)\right]$ vs $\omega {\tau }_N$ for various ac field amplitudes $\xi$ and $J=-3,{\phi }_{\xi }=\pi /4,\sigma =20$, and ${\xi }_0=2$ using the matrix continued fraction solution. Solid lines: Linear response. Dashed lines: Nonlinear response. (b) The high-frequency parts of the spectra alone.

Figure 5

Real and imaginary parts of the nonlinear susceptibility $\mathrm{Re}\left[\chi \left(\omega \right)\right]$ and $-\mathrm{Im}\left[\chi \left(\omega \right)\right]$ vs $\omega {\tau }_N$ for various spin-polarized current parameter $J=6,0,-3,-6,-12[\sigma =20,{\phi }_{\xi }=\pi /4,{\xi }_0=2$, and $\xi =4]$ using the matrix continued fraction solution.

Figure 6

DMH loops for various spin-polarized current parameter $J=-1,0,1$ for (a) $\xi =2$ and (b) $\xi =5$ with $\omega {\tau }_N=1,{\phi }_{\xi }=0,\alpha =0.01,\sigma =5,{\xi }_0=0$,

Figure 7

DMH loops for various spin-polarized current parameter $J=-1,0,1$ (a) and $J=-2,0,2$ (b) with $\omega {\tau }_N=1,{\phi }_{\xi }=0,\alpha =0.01,{\xi }_0=0,\xi =3$, and $\sigma =5$.

Figure 8

DMH loops for various spin-polarized current parameters $J=-1,0,1$ and frequencies $\omega {\tau }_N={10}^{-2}$ (a), ${10}^{-1}$ (b), 1 (c), 10 (d), ${10}^2$ (e), and 3981 (f) with ${\phi }_{\xi }=0,\alpha =0.01,{\xi }_0=0,\xi =5$, and $\sigma =5$.

Figure 9

DMH loops for various spin-polarized current parameters $J=-1,0,1$ and frequencies $\omega {\tau }_N={10}^{-2}$ (a), ${10}^{-1}$ (b), 1 (c), 10 (d), ${10}^2$ (e), and 3981 (f) with ${\phi }_{\xi }=\pi /4,\alpha =0.01,{\xi }_0=0,\xi =5$, and $\sigma =5$.

Figure 10

Normalized area of the DMH loop $A_n$ [Eq. (12)] vs spin-polarized current parameter $J$ (a) for various ac field amplitudes $\xi$ and $\omega {\tau }_N=1$ and (b) for various frequencies $\omega {\tau }_N$ and $\xi =2$ with ${\phi }_{\xi }=0,\alpha =0.01,{\xi }_0=0,$ and $\sigma =5$.