Reduced model for precessional switching of thin-film nanomagnets under the influence of spin torque

We study the magnetization dynamics of thin-film magnetic elements with in-plane magnetization subject to a spin current flowing perpendicular to the film plane. We derive a reduced partial differential equation for the in-plane magnetization angle in a weakly damped regime. We then apply this model to study the experimentally relevant problem of switching of an elliptical element when the spin polarization has a component perpendicular to the film plane, restricting the reduced model to a macrospin approximation. The macrospin ordinary differential equation is treated analytically as a weakly damped Hamiltonian system, and an orbit-averaging method is used to understand transitions in solution behaviors in terms of a discrete dynamical system. The predictions of our reduced model are compared to those of the full Landau-Lifshitz-Gilbert-Slonczewski equation for a macrospin.

Figure 1

Solutions of macrospin equation (30) for $\alpha =0.01,\mathrm{\Lambda }=0.1$. In (a), $p_{\perp }=0.2,\sigma =0.03$: decaying solution; in (b), $p_{\perp }=0.2,\sigma =0.06$: limit cycle solution [the initial conditions in (a) and (b) are $\theta \left(0\right)=3.5$, to better visualize the behavior]. In (c), $p_{\perp }=0.3,\sigma =0.08$: switching solution; in (d), $p_{\perp }=0.6,\sigma =0.1$: precessing solution.

Figure 2

Switching solution (blue line) and its discrete approximation (green circles). Parameters: $\alpha =0.01,\mathrm{\Lambda }=0.1,p_{\perp }=0.3,\sigma =0.08$. (a)The solution $\theta \left(t\right)$. (b) The trajectory for this solution in the $\mathcal{H}\text{−}\theta$ plane. The red line in (b) shows $V\left(\theta \right)$.

Figure 3

Precessing solution (blue line) and its discrete approximation (green circles). Parameters: $\alpha =0.01,\mathrm{\Lambda }=0.1,p_{\perp }=0.6,\sigma =0.1$. (a) The solution $\theta \left(t\right)$. (b) The trajectory for this solution in the $\mathcal{H}\text{−}\theta$ plane. The red line in (b) shows $V\left(\theta \right)$.

Figure 4

Precession vs switching prediction from the discrete map. Parameters: $\alpha =0.01,\mathrm{\Lambda }=0.1,p_{\perp }=0.35,\sigma =0.08$. Values of ${\mathcal{H}}_n-V({\theta }_C+\pi )$ to the left of the dashed line switch after the next period, the trajectory becoming trapped in the well around $\theta =0$. Values to the right begin to precess, and converge to a precessional fixed point of the discrete map.

Figure 5

Macrospin solution phase diagrams for $\alpha =0.01,\mathrm{\Lambda }=0.1$: (a) The result of iterating the discrete map (30) with initial energy ${\mathcal{H}}_n=V\left(\pi \right)$ corresponding to the initial condition (31); (b) the result of a direct numerical solution of the macrospin ODE (30) with initial condition (31). Dark regions to the left of the figures indicates solutions which do not escape their initial potential well, and the vertical dashed white line shows the computed value of the minimum current required to escape, ${\sigma }_s=\alpha /\left(2\mathrm{\Lambda }\right)$. The black bands represent solutions which decay, like in Fig. 1, while the dark gray bands represent limit cycle solutions like in Fig. 1. In the rest of the figures, the green points indicate switching in the negative direction like in Fig. 1, gray indicate switching in the positive direction, and white indicates precession like in Fig. 1. The solid black curves are the analytic predictions of boundaries of the regions (as indicated in the figure) by using the discrete map, and the dashed line is the prediction of the boundary below which switching in the positive direction is possible.

Figure 6

Switching solutions for full PDE model (20, 21, 22) in a strip geometry with assumed 1D variations, for three different strip widths. Parameters $d=2.5$ nm, $\ell =5$ nm, $\alpha =0.01,{\beta }_{*}=0.01,p_{\perp }=0.2$. (a) Width $L=30$ nm; (b) width $L=60$ nm; (c) width $L=90$ nm. The blue curves display the magnetization angle $\theta$, averaged across the strip width, as functions of time. The green curves show the values of $\theta$ at the boundary. The red curves show the values of $\theta$ at the center of the strip.

Figure 7

Spatial variations in $\theta$ in PDE solutions for strip of width 90 nm. The other parameters are as in Fig. 6. Color represents the values of $\theta (x,t)$.