Spin-current emission governed by nonlinear spin dynamics

Coupling between conduction electrons and localized magnetization is responsible for a variety of phenomena in spintronic devices. This coupling enables to generate spin currents from dynamical magnetization. Due to the nonlinearity of magnetization dynamics, the spin-current emission through the dynamical spin-exchange coupling offers a route for nonlinear generation of spin currents. Here, we demonstrate spin-current emission governed by nonlinear magnetization dynamics in a metal/magnetic insulator bilayer. The spin-current emission from the magnetic insulator is probed by the inverse spin Hall effect, which demonstrates nontrivial temperature and excitation power dependences of the voltage generation. The experimental results reveal that nonlinear magnetization dynamics and enhanced spin-current emission due to magnon scatterings are triggered by decreasing temperature. This result illustrates the crucial role of the nonlinear magnon interactions in the spin-current emission driven by dynamical magnetization, or nonequilibrium magnons, from magnetic insulators.

excitation frequency and power. This finding provides a crucial piece of information for understanding the spin-current emission from ferromagnetic materials and investigating the magnon interactions in the metal/insulator hybrid.

Results
Temperature evolution of spin-current emission. A single-crystal Y 3 Fe 5 O 12 (111) film (3 × 5 mm 2 ) with a thickness of 5 μm was grown on a Gd 3 Ga 5 O 12 (111) substrate by liquid phase epitaxy (purchased from Innovent e.V., Jena). After the substrates were cleaned by sonication in deionized water, acetone and isopropanol, a piranha etching process, a mixture of H 2 SO 4 and H 2 O 2 (with the ratio of 7:3), was applied, then to be able to remove any residuals an oxygen plasma cleaning was performed outside a sputtering chamber. On the top of the film, a 10-nm-thick Pt layer was sputtered in an Ar atmosphere. Prior to sputtering 10-nm-thick Pt layer, an argon plasma cleaning was also performed in-situ. The Pt/ Y 3 Fe 5 O 12 bilayer film was placed on a coplanar waveguide, where a microwave was applied to the input of the signal line as show in Fig. 1(b). Two electrodes were attached to the edges of the Pt layer. The signal line is 500 μm wide and the gaps between the signal line and the ground lines are designed to match to the characteristic impedance of 50 Ω . An in-plane external magnetic field H was applied parallel to the signal line, or perpendicular to the direction across the electrodes 11 . Figure 1  propagating along and opposite to the magnetic field is shown. The blue and red arrows represent the four and three magnon scatterings. The magnon dispersion shows that both the three and four magnon scatterings create secondary magnons with small group velocity. The lowest frequency is f = f min . (b) The experimental setup. The Pt/Y 3 Fe 5 O 12 film placed on the coplanar waveguide was cooled using a Gifford-McMahon cooler. (c) Magnetic field (H) dependence of the microwave absorption P for the Pt/Y 3 Fe 5 O 12 film at f 0 = 7.6 GHz and P in = 10 mW. μ 0 H R = 183 mT is the resonance field. P abs is the definition of the magnitude of the microwave absorption intensity. The absorption peak structure comprises multiple signals due to spin-wave modes. ( in-plane magnetic field H dependence of the microwave absorption P measured by applying a 10 mW microwave with the frequency of f 0 = 7.6 GHz at T = 300 K. Under the ferromagnetic resonance condition H = H R , dynamical magnetization in the Y 3 Fe 5 O 12 layer emit a spin current j s into the Pt layer, resulting in the voltage generation through the ISHE as shown in Fig. 1(d) 1,2 . The sign of the voltage is changed by reversing H, consistent with the prediction of the spin-current emission from the magnetic insulator 35 . Here, the absorption spectrum comprises multiple resonance signals due to spin-wave modes, including magnetostatic surface waves and backward-volume magnetostatic waves in addition to the ferromagnetic resonance. To extract the damping constant for the Pt/Y 3 Fe 5 O 12 film, we have plotted dV/dH in Fig. 1(e), which allows rough estimation of the damping constant, α ∼ × − 5 10 4 . Figure 2(a) shows temperature dependence of V ISHE /P abs , where V ISHE and P abs are the magnitude of the microwave absorption and electric voltage, respectively; V ISHE /P abs characterizes the angular-momentum conversion efficiency from the microwaves into spin currents. Notably, V ISHE /P abs increases drastically below T = 150 K by decreasing T at f 0 = 4.0 GHz. This drastic change is irrelevant to the temperature dependence of the spin pumping and spin-charge conversion efficiency in the Pt/Y 3 Fe 5 O 12 bilayer, such as the spin Hall angle θ SHE , the spin pumping conductance g eff , the spin diffusion length λ , and the electrical conductivity σ. Figure 2(b) shows the temperature dependence of the electrical conductivity σ and the spin Hall conductivity σ s . The spin Hall conductivity was obtained from the temperature dependence of V ISHE /P abs at 10 mW for f 0 = 7.6 GHz shown in Fig. 2(a); the value of V ISHE /P abs is insensitive to the excitation power from 5 to 15 mW, indicating that the spin-current emission is reproduced with a liner spin-pumping model 36 : spin Hall conductivity of the Pt layer shown in Fig. 2(b) increases with decreasing temperature above 100 K. Below 100 K, the spin Hall conductivity decreases with decreasing temperature. This feature is qualitatively consistent with the previous report 39 . Although the spin Hall conductivity varies with temperature, the variation of the spin Hall conductivity alone is not sufficient to explain the drastic increase of V ISHE /P abs for f 0 = 4 GHz shown in Fig. 2(a). Thus, the drastic change in V ISHE /P abs across 150 K at f 0 = 4.0 GHz can be attributed to the change in the magnetization dynamics in the Y 3 Fe 5 O 12 layer. In fact, by decreasing T, the microwave absorption intensity P abs decreased clearly across T = 150 K as shown in Fig. 2(c), suggesting the change of the magnetization dynamics in the Y 3 Fe 5 O 12 layer across T = 150 K.

Discussion
The origin of the temperature-induced drastic change of the spin-conversion efficiency V ISHE /P abs shown in Fig. 2(a) is enhanced spin-current emission triggered by the three magnon splitting. The three-magnon splitting creates a pair of magnons with the opposite wavevectors and the frequency f 0 /2 from the uniform magnon with f 0 [see also Fig. 1(a)]. The splitting process redistributes the magnons and changes the relaxation rate of the spin system, increasing the steady-state angular momentum stored in the spin system, or resulting in the stabilized enhancement of the spin-current emission 14,32 . The splitting is allowed only when f 0 /2 > f min , where f min is the minimum frequency of the magnon dispersion, because of the energy and momentum conservation laws. This condition can readily be found by finding f min for the thin Y 3 Fe 5 O 12 film from the lowest branch of the dipole-exchange magnon dispersion for the unpinned surface spin condition 40 : Tcm 2 is the exchange interaction constant, L = 5 μm is the thickness of the Y 3 Fe 5 O 12 layer, and k is the wavenumber of the magnons (see also the Supplementary Information). γ = 1.84 × 10 11 Ts −1 is the gyromagnetic ratio. In Fig. 3(a,b), we show the lowest branch of the magnon dispersion at different temperatures for the Pt/Y 3 Fe 5 O 12 film, calculated using Eq. (2). For the calculation, we used the saturation magnetization M s at each temperature [see Fig. 3(c)], estimated from the ferromagnetic resonance field H R data with Kittel's formula: , where the resonance condition is independent of the value of D. We assumed that D is independent of temperature, as demonstrated in literature 32,41,42 ; although D varies with temperature, the variation is less than 4% in Y 3 Fe 5 O 12 for T < 350 K and the shape of the magnon dispersion is not sensitive to the small variation of D [41][42][43] . Figure 3(a,b) demonstrate that the minimum frequency f min decreases with decreasing temperature and the splitting condition f 0 /2 > f min is satisfied below T = 150 K; the magnon redistribution is responsible for the enhancement of V ISHE /P abs . Thus, this result demonstrates that the enhanced spin-current emission can be induced not only by changing the excitation frequency or power of the magnetization dynamics, but also by changing temperature. Figure 4(a,b) show temperature dependence of the spin-conversion efficiency V ISHE /P abs at different microwave excitation powers P in for f 0 = 7.6 and 4.0 GHz, respectively. At f 0 = 4.0 GHz, the enhancement of V ISHE /P abs due to the three-magnon splitting below 150 K is observed for all the excitation powers as shown in Fig. 4(b). The drop in V ISHE /P abs at T = 50 K for f 0 = 4.0 GHz is induced by the decrease of the spin Hall conductivity shown in Fig. 2(b); below 100 K, the spin Hall conductivity, or the spin Hall angle, decreases with decreasing temperature, whereas the spin-current enhancement through the magnon splitting increases by decreasing temperature. The competition gives rise to the peak structure in V ISHE /P abs around 70 K for 4.0 GHz. This result also shows that the enhancement factor is almost independent of the excitation power. In contrast, notably, the variation of V ISHE /P abs depends on the excitation power, especially below 150 K, at f 0 = 7.6 GHz as shown in Fig. 4(a). These features for f 0 = 7.6 and 4.0 GHz were confirmed in V ISHE /P abs measured with the reversed external magnetic field [see the experimental data for −H in Fig. 4(a,b)], indicating that the change of the spin-current emission from the magnetic insulator is responsible for the nontrivial behavior of V ISHE /P abs at low temperatures.
To understand the temperature and power dependences of V ISHE /P abs at f 0 = 7.6 GHz in details, we plot [V ISHE /P abs ] 100 mW /[V ISHE /P abs ] 5 mW in Fig. 4(c). For the spin-current emission in the linear magnetization dynamics regime, V ISHE /P abs is constant with P in , or [V ISHE /P abs ] 100 mW /[V ISHE /P abs ] 5 mW = 1 because the emitted spin current is proportional to P in 35 . Since the three-magnon splitting is prohibited at f 0 = 7.6 GHz, [V ISHE /P abs ] 100 mW /[V ISHE /P abs ] 5 mW ≈ 1.2, at T = 300 K, demonstrates enhanced spin-current emission without the splitting of a pumped magnon.
The observed enhancement of the spin-current emission at T = 300 K is induced by the four magnon scattering, where two magnons are created with the annihilation of two other magnons [see also Fig. 1(a)] 44,45 . The four-magnon scattering emerges at high microwave excitation powers P in > P th , known as the second order Suhl instability 46 , where P th is the threshold power of the scattering. Although this process conserves the number of magnons, the magnon redistribution can decrease the relaxation rate of the spin system through the annihilation of the uniform magnons with large damping η 0 and creation of dipole-exchange magnons with small damping η q . This results in the steady-state enhancement of the angular momentum stored in the spin system, or the enhanced spin-current emission 32    Y 3 Fe 5 O 12 film, the damping η 0 of the uniform magnon at low excitation powers is mainly dominated by the two-magnon scattering; the temperature dependence of the ferromagnetic resonance linewidth is almost independent of temperature as shown in the inset to Fig. 5, indicating that the damping η 0 is not dominated by the temperature peak processes or the Kasuya-LeCraw mechanism 47 . In contrast, the damping η q of the secondary magnons created by the four-magnon scattering is dominated by the Kasuya-LeCraw mechanism, since the two-magnon scattering events are suppressed due to the small group velocity; the group velocity of the secondary dipole-exchange magnons created at the same frequency as the uniform magnon can be close to zero because of the exchange-dominated standing spin-wave branches [see Fig. 1(a)] 44,48-50 . The exchange-dominated branches, i.e. the thickness modes, show the energy minimum not only at the bottom of the dispersion but also at the excitation frequency. Therefore, in the present system, the damping η 0 of the uniform magnon is dominated by the temperature-independent two-magnon scattering, whereas the damping η q of the secondary magnon is dominated by temperature-dependent three-particle confluences, such as the Kasuya-LeCraw process 47 . In the presence of the four magnon scattering, the total number of the nonequilibrium magnons N t is expressed as 32 where η q is defined as the average decay rate to the thermodynamic equilibrium of the degenerate secondary magnons for simplicity. The imaginary part of the susceptibility is expressed as .  Here, η sp is the decay constant of the uniform precession to degenerate magnons at f 0 due to scattering on sample inhomogeneities. Under the assumption that the spin-pumping efficiency is insensitive to the wavenumber k of the nonequilibrium magnons, that is V ISHE ∝ j s ∝ N t , Eq. (3) is directly related to the spin-conversion efficiency: V ISHE /P abs ∝ N t /P abs .
The above model reveals that the spin-current enhancement due to the four-magnon scattering is responsible for the nontrivial behavior of the voltage generation shown in Fig. 4(a). As shown in Fig. 4(c), the nonlinearity of the spin-current emission is enhanced by decreasing temperature, from [V ISHE /P abs ] 100 mW /[V ISHE /P abs ] 5 mW ≈ 1.2 at T = 300 K to [V ISHE /P abs ] 100 mW /[V ISHE /P abs ] 5 mW ≈ 2.4 at 50 K. Figure 5 shows microwave excitation power P in dependence of V ISHE /P abs for f 0 = 7.6 GHz at different temperatures. This result clearly shows that the threshold power P th of the spin-current enhancement decreases with decreasing temperature, which is the origin of the nontrivial behavior of the temperature dependence of V ISHE /P abs shown in Fig. 4(a,c). The threshold power of the spin-current enhancement through the four-magnon process is very low at low temperatures, making it difficult to observe the threshold behavior. In fact, V ISHE /P abs deviates from the prediction of the linear model even at the lowest microwave excitation power that is necessary to detect the ISHE voltage in the Pt/Y 3 Fe 5 O 12 film at T = 75 K [see the orange circles in Fig. 6]. At T = 300 K, a clear threshold is observed around P in = 40 mW. The threshold power of the four-magnon scattering is given by 47 η γ η σ ∝ = ( / ) ( / ) P h 2 q q th th 2 0 2 , where h th is the threshold microwave field and σ q is the coupling strength between the uniform and secondary magnons. For simplicity, we neglect the surface dipolar interactions, or L → ∞. Under this approximation, the ferromagnetic resonance condition is given by f 0 = γμ 0 H and the coupling strength can be approximated as σ q = γμ 0 M s . Thus, the threshold power for the four-magnon scattering is proportional to Equation (6) predicts that the threshold power of the spin-current enhancement decreases with decreasing temperature, since M s increases by decreasing temperature as shown in Fig. 3(c). Although the damping η 0 of the uniform magnon is almost independent of temperature as shown in the inset to Fig. 5, the damping η q of the dipole-exchange magnon tends to decrease the threshold power, since η q , dominated by the Kasuya-LeCraw process is approximately proportional to temperature 47 . At high power excitations, the competition between the increase of the spin-current enhancement due to the four-magnon scattering and the decrease of the spin Hall effect by decreasing temperature gives rise to the peak structure in V ISHE /P abs around 100 K for f 0 = 7.6 GHz [see Fig. 4(a)].
In summary, we have demonstrated that the spin-current emission from a Y 3 Fe 5 O 12 film is strongly affected by nonlinear magnetization dynamics at low temperatures. The spin-current emission has been demonstrated to be enhanced even in the absence of the three-magnon splitting 15 . The experimental results presented in this paper are consistent with this result and further extend the physics of the nonlinear spin-current emission from the magnetic insulator. Our study reveals that the spin-current enhancement arises from both the three and four magnon scatterings depending on the excitation frequency and temperature. We show that the enhanced spin-current emission can be triggered by decreasing temperature, which is evidenced by our systematic measurements for the Pt/Y 3 Fe 5 O 12 film; the spin-current emission can be enhanced not only by changing the magnon excitation frequency or power, but also by changing temperature. This result demonstrates the generality of the crucial role of magnon interactions in the spin-current emission, combining the long-standing research on nonlinear spin physics with spintronics.