Dynamical coupling between ferromagnets due to spin transfer torque: Analytical calculations and numerical simulations

We use a combination of analytical calculations and numerical simulations to demonstrate that electrical current flowing through a magnetic bilayer induces dynamical coupling between the layers. The coupling originates from the dependence of the spin transfer torque on the relative orientations of their magnetic moments. When the appropriate resonance conditions are satisfied, such coupling significantly modifies the behaviors of both layers, affecting the stability of the current-induced dynamical regimes and the efficiency of current-induced magnetic reversal.

Figure 1

Stability diagram for the analytical model of two-layer system with varied $A=S_1/S_2$. (a) $g_1=g_2$, bistable regions are hatched. (b) $g_1=-g_2$, the region of stable precession is filled. For $A<1$, $I_{c0}$ is defined by Eq. (1). For $A>1$, $I_{c0}$ is defined by Eq. (1) with $S_2$ replaced by $S_1$ to emphasize the dominance of this layer’s dynamics.

Figure 2

(Color online) (a)–(c) Normalized spectral intensity for the dynamics of the in-plane hard-axis component of ${\mathbf{s}}_2$. (a) $A=0.2$, (b) $A=0.67$, (c) $A=1$, and $g_1=-g_2$. Lighter colors correspond to higher intensity. The same scale is used in all three plots. (d) Resonant behaviors of dynamical coupling for fixed $t_2=4\text{nm}$ and $M_2=800\text{emu}/{\text{cm}}^3$. Symbols connected by line: $I_c$ vs $M_1$ at a fixed $A=0.67$. $I_c$ was determined by the condition that in-plane precession angle exceeds $0.2°$ while ramping the current at a rate of 0.01 mA/ns. Solid curve: Small-angle precession frequency of $F_1$. Dashed horizontal line shows small-angle precession frequency of $F_2$. The calculations were performed at $H=1\text{kOe}$ along the nanopillar easy axis.