Spin torque driven dynamics of a coupled two-layer structure: Interplay between conservative and dissipative coupling

The general concepts of spin wave theory are adapted to the spin torque driven dynamics of a self-polarized system based on two layers coupled via interlayer exchange (conservative coupling) and mutual spin torque (dissipative coupling). An analytical description of the nonlinear dynamics is proposed and validated through numerical simulations. In contrast to the single layer model, the phase equation of the coupled system has a contribution coming from the dissipative part of the LLGS equation. It is shown that this is a major contribution to the frequency mandatory to describe well the most basic features of the dynamics of this coupled system. Using the proposed model a specific feature of coupled dynamics is addressed: the redshift to blueshift transition observed in the frequency current dependence of this kind of exchange coupled systems upon increasing the applied field. It is found that the blueshift regime can only occur in a region of field where the two linear eigenmodes contribute equally to the steady state mode (i.e., high mode hybridization). Finally, a general perturbed Hamiltonian equation for the coupled system is proposed.

Figure 1

Schematics of (a) the single layer picture and (b) the self-polarized SyF system. Frequency versus supercriticality $j_{\mathrm{app}}/j_{\mathrm{c}}(j_{\mathrm{app}}$ is applied current densitiy and $j_c$ is critical current density) at $T=0\mathrm{K}$ at different values of field for (c) the single layer case and (d) for the self-polarized SyF obtained by solving numerically the LLGS equation.

Figure 2

Conservative coefficients $N_1,N_2$, $F$, $S$, $T$ as a function of the applied field, for the material parameters given in Sec. 2a and for different ratios of the thicknesses of both layers (a) $t_1=3.9\mathrm{nm},t_2=3.95\mathrm{nm}$, (b) $t_1=2.9\mathrm{nm},t_2=3.95\mathrm{nm}$, (c) $t_1=4.2\mathrm{nm},t_2=3.95\mathrm{nm}$, (d) $t_1=3.9\mathrm{nm},t_2=2.7\mathrm{nm}$.

Figure 3

Analysis of the steady state dynamics in (a) the $m_{x,y,z}$ components and (b)–(d) the $b$ basis, obtained by numerical simulations for zero applied field and $j_{\mathrm{app}}=450\times {10}^{10}\mathrm{A}/{\mathrm{m}}^2$. (b) Imaginary vs real part of $b_1$ and $b_2$. Time evolution of the phases (c) and instantaneous frequencies (d). Horizontal lines in (d) correspond to the average frequency values.

Figure 4

$\langle \psi \rangle$ evolution with supercriticality at different fields.

Figure 5

Frequency of the two modes of the coupled system (the higher frequency steady state mode and the damped lower frequency mode) as a function of the supercriticality at different fields.

Figure 6

Frequency current dependence at (a) and (b) $H=0\mathrm{kA}/\mathrm{m}$ and (c) and (d) $H=3\mathrm{kA}/\mathrm{m}$ obtained from the (a)–(c) $\langle d{\varphi }_1/dt\rangle$ and (b)–(d) $\langle d{\varphi }_2/dt\rangle$ equations. Black symbols correspond to values obtained numerically using Eq. (9), and the rest of the symbols correspond to values obtained analytically including different terms in the simplified model of Eq. (14) (see text). Orange symbols correspond to the values obtained using the full equation (14). Red symbols represent the weight of the term which results from the $S$ term of the Hamiltonian, which has not been included in the truncated model.

Figure 7

Frequency current dependence at low (a) and (c) and large (b) and (d) field obtained using the simplified model of Eq. (14), from the $\langle d{\varphi }_1/dt\rangle$ and $\langle d{f}_2/dt\rangle$ equations (blue and red lines, respectively). Black line corresponds to values obtained numerically using Eq. (9). In (a) and (b) the strength of the RKKY interaction is set to $J_{\mathrm{RKKY}}=-2\times {10}^{-4}\mathrm{J}/{\mathrm{m}}^2$, while in (c) and (d) the layers thicknesses are set to $t_1=3.9\mathrm{nm},t_2=4.2\mathrm{nm}$. The rest of the material parameters used are the same as in the main text.

Figure 8

Mode asymmetry at $H=0\mathrm{kA}/\mathrm{m}$ as a function of $\psi$ along a trajectory period at different currents.

Figure 9

(a) $p_2$ vs $p_1$ at $H=0\mathrm{kA}/\mathrm{m}$ and (b) $H=3\mathrm{kA}/\mathrm{m}$. (c) Current dependence of the mode asymmetry (averaged over a period), (d) $\sqrt{\frac{p_1}{p_2}}$, (e) ${\tilde{\mathrm{\Delta }}}_{2,0}sin\left(\psi \right)$, and (f) $\sqrt{\frac{p_1}{p_2}}{\tilde{\mathrm{\Delta }}}_{2,0}sin\left(\psi \right)$, at $H=0\mathrm{kA}/\mathrm{m}$ (black symbols), $H=1\mathrm{kA}/\mathrm{m}$ (red symbols), and $H=3\mathrm{kA}/\mathrm{m}$ (blue symbols).