Dynamic exchange via spin currents in acoustic and optical modes of ferromagnetic resonance in spin-valve structures

Two ferromagnetic (FM) layers magnetically decoupled by a thick, normal metal spacer layer can be dynamically coupled via spin currents emitted by the spin pump and absorbed through the spin-torque effects at the neighboring interfaces. A decrease of damping in both layers due to a partial compensation of the angular momentum leakage in each layer was previously observed at the coincidence of the two FM resonances. In the case of nonzero magnetic coupling, such a dynamic exchange will depend on the mutual precession of the magnetic moments in the layers. A difference in the linewidth of the resonance peaks is expected for the acoustic and optical regimes of precession. However, the interlayer coupling (IC) hybridizes the resonance responses of the layers and therefore can also change their linewidths. The interplay between the two mechanisms has never been considered before. In the present work, the joint influence of the hybridization and nonlocal damping on the linewidth has been studied in weakly coupled NiFe/CoFe/Cu/CoFe/MnIr spin-valve multilayers. It has been found that the dynamic exchange by spin currents is different in the optical and acoustic modes, and this difference is dependent on the IC strength. In contrast to the acoustic precession mode, the dynamic exchange in the optical mode works as an additional damping source. A simulation in the framework of the Landau-Lifshitz-Gilbert formalism for two FM layers coupled magnetically and by spin currents has been done to separate the effects of the nonlocal damping from the resonance modes hybridization. In our samples, both mechanisms bring about linewidth changes of the same order of magnitude but lead to a distinctly different angular behavior. The obtained results are relevant for a broad class of coupled magnetic multilayers with ballistic regime of the spin transport.

Figure 1

Simulation of a FMR spectrum for a SV in the weak coupling regime (top curve). The middle and bottom curves show separated responses from the free and fixed layers in the SV. The layer parameters correspond to the first series of SVs: $d_1$ = 5.5 nm, ${\alpha }_1$ = 0.012, $M_{\mathrm{s}1}$ = 1155 emu/cm${}^3$, $K_1$ = 5.7·10${}^3$ erg/cm${}^3$; $d_2$ = 2.5 nm, ${\alpha }_2$ = 0.055, $M_{\mathrm{s}2}$ = 1175 emu/cm${}^3$, $K_2$ = 1.7·10${}^4$ erg/cm${}^3$, $E_{\mathrm{ex}}$ = 0.094 erg/cm${}^2$, and $E_{\mathrm{ic}}$ = 0.01 erg/cm${}^2$.

Figure 2

Experimentally measured angular dependences of the FMR peaks for the free and fixed layer: the first series where the interlayer coupling strength is varied by gradually changing the metal spacer thickness $d_{\mathrm{Cu}}$ (left panel); the second series where the mean FMR field of the free layer is varied by gradually changing the free layer effective magnetization, $M_{\mathrm{s}1}$ (right panel).

Figure 3

Linewidth of the free layer in the first sample series obtained from the experiment. Left panel: The angular dependence for different copper spacer thicknesses. The reference sample curve does not show steps. Right panel: The relative step height and mean linewidth versus the interlayer coupling strength.

Figure 4

Experimentally measured angular dependences of the linewidth for the free layer in the second sample series and in the respective reference samples.

Figure 5

Simulated angular dependences of the free layer's FMR linewidth. Four different regimes are shown: 00-00: $E_{\mathrm{ic}}$ = 0 and $A_{\mathrm{FNF}}^{\uparrow \downarrow }$ = 0; IC-00: $E_{\mathrm{ic}}$ = 0.01 erg/cm${}^2$ and $A_{\mathrm{FNF}}^{\uparrow \downarrow }$ = 0; 00-SP: $E_{\mathrm{ic}}$ = 0 and $A_{\mathrm{FNF}}^{\uparrow \downarrow }$ = 1.1 × 10${}^{15}$ cm${}^{-2}$; IC-SP: $E_{\mathrm{ic}}$ = 0.01 erg/cm${}^2$ and $A_{\mathrm{FNF}}^{\uparrow \downarrow }$ = 1.1 × 10${}^{15}$ cm${}^{-2}$. The layer parameters refer to the first series of SVs, as they are already listed in the caption to Fig. 1.

Figure 6

Simulation: linewidth of the free layer in the parallel and antiparallel orientation versus the interlayer coupling strength calculated with an account for spin conductivity and without it. The layer parameters are set for the first series of SVs, as they are already listed in the caption to Fig. 1.

Figure 7

Simulated angular behavior of the linewidth in the second series of SVs, with a gradual variation of the effective magnetization of the free layer, obtained considering the spin conductivity and without it. For the red and black curves, the fixed layer resonance does not cross that of the free layer anymore. The parameter set is the same as for the first series, with $M_{\mathrm{s}2}$ = 1525 emu/cm${}^3$ and $E_{\mathrm{ex}}$ = 0.12 erg/cm${}^2$.