Introduction

Dynamical magnetization in a ferromagnet emits a spin current1,2, enabling to explore the physics of spin transport in metals and semiconductors3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22. The dynamical spin-current emission has been achieved utilizing ferromagnetic metals, semiconductors and insulators23,24,25,26. In particular, the discovery of the spin-current emission from a magnetic insulator yttrium iron garnet, Y3Fe5O12, has drawn intense experimental and theoretical interests, opening new possibilities to spintronics based on metal/insulator hybrids, where angular momentum can be carried by both electrons and magnons.

A ferrimagnetic insulator yttrium iron garnet, Y3Fe5O12, is characterized by the exceptionally small magnetic damping, making it a key material for the development of the physics of nonlinear magnetization dynamics27,28,29. The nonlinear magnetization dynamics in Y3Fe5O12 has been extensively studied both experimentally and theoretically in the past half a century, benefited by the exceptional purity, high Curie temperature and simplicity of the low-energy magnon spectrum28,29,30,31. Recently, the nonlinear magnetization dynamics has been found to affect the spin-current emission from the magnetic insulator; the spin-current emission is enhanced by magnon scattering processes [see Fig. 1(a)], triggered by changing the excitation frequency or power of the magnetization dynamics14,15,32. These findings shed new light on the long-standing research on nonlinear magnetization dynamics, promising further development of spintronics and magnetics based on the magnetic insulator.

Figure 1
figure 1

Detection of spin-current emission.

(a) The magnon dispersion in Y3Fe5O12, where f and k are the frequency and wavenumber of magnons, respectively. The dispersion of the first 40 thickness modes 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 = fmin. (b) The experimental setup. The Pt/Y3Fe5O12 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/Y3Fe5O12 film at f0 = 7.6 GHz and Pin = 10 mW. μ0HR = 183 mT is the resonance field. Pabs is the definition of the magnitude of the microwave absorption intensity. The absorption peak structure comprises multiple signals due to spin-wave modes. (d) H dependence of the electric voltage V. VISHE is the magnitude of the electric voltage. The blue and red data were measured with the in-plane magnetic field H and −H, respectively. (e) H dependence of dV(H)/dH for the Pt/Y3Fe5O12 film. The damping constant of the Pt/Y3Fe5O12 film was roughly estimated to be 5 × 10−4 from f0 dependence of the linewidth at 5 mW.

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In this work, we demonstrate that the spin-current emission from Y3Fe5O12 is strongly affected by nonlinear magnetization dynamics at low temperatures. The spin-current emission is probed by the inverse spin Hall effect (ISHE) in a Pt film attached to the Y3Fe5O12 film11,33,34, which enables to measure temperature dependence of the spin-current emission from the magnetic insulator under various conditions. In spite of the simple structure of the metal/insulator bilayer, we found nontrivial variation of the spin-current emission; the temperature dependence of the spin-current emission strongly depends on the microwave frequency and excitation power. This result reveals that nonlinear spin-current emission due to three and four magnon scatterings emerges by decreasing temperature, even at constant magnon 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 Y3Fe5O12 (111) film (3 × 5 mm2) with a thickness of 5 μm was grown on a Gd3Ga5O12 (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 H2SO4 and H2O2 (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/Y3Fe5O12 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 electrodes11. Figure 1(c) shows the in-plane magnetic field H dependence of the microwave absorption P measured by applying a 10 mW microwave with the frequency of f0 = 7.6 GHz at T = 300 K. Under the ferromagnetic resonance condition H = HR, dynamical magnetization in the Y3Fe5O12 layer emit a spin current js 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 insulator35. 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/Y3Fe5O12 film, we have plotted dV/dH in Fig. 1(e), which allows rough estimation of the damping constant, .

Figure 2(a) shows temperature dependence of VISHE/Pabs, where VISHE and Pabs are the magnitude of the microwave absorption and electric voltage, respectively; VISHE/Pabs characterizes the angular-momentum conversion efficiency from the microwaves into spin currents. Notably, VISHE/Pabs increases drastically below T = 150 K by decreasing T at f0 = 4.0 GHz. This drastic change is irrelevant to the temperature dependence of the spin pumping and spin-charge conversion efficiency in the Pt/Y3Fe5O12 bilayer, such as the spin Hall angle θSHE, the spin pumping conductance geff, 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 σs. The spin Hall conductivity was obtained from the temperature dependence of VISHE/Pabs at 10 mW for f0 = 7.6 GHz shown in Fig. 2(a); the value of VISHE/Pabs is insensitive to the excitation power from 5 to 15 mW, indicating that the spin-current emission is reproduced with a liner spin-pumping model36:

Figure 2
figure 2

Temperature evolution of spin-current emission.

(a) Temperature (T) dependence of VISHE/Pabs for the Pt/Y3Fe5O12 film at f0 = 7.6 (the black circles) and 4.0 GHz (the red circles). The data were measured with Pin = 10 mW microwave excitation. (b) T dependence of the electrical conductivity σ and the spin Hall conductivity σs for the Pt/Y3Fe5O12 film. (c) T dependence of Pabs/Pin, where Pabs is the microwave absorption intensity, for Pin = 10 mW and f0 = 4.0 GHz.

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where ωF = 3.0 mm and vF = 7.5 × 10−11 m3 are the width and volume of the Y3Fe5O12 film. d = 10 nm is the thickness of the Pt layer. μ0ΔH is the half-maximum full-width of the ferromagnetic resonance linewidth. For the calculation of σs, we used the measured parameters of the electrical conductivity σ and saturation magnetization Ms. The spin-diffusion length37 λ = 7.7 nm and spin pumping conductance38 geff = 4.0 × 1018 m−2 were assumed to be independent of temperature, as demonstrated previously39. The 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 report39. 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 VISHE/Pabs for f0 = 4 GHz shown in Fig. 2(a). Thus, the drastic change in VISHE/Pabs across 150 K at f0 = 4.0 GHz can be attributed to the change in the magnetization dynamics in the Y3Fe5O12 layer. In fact, by decreasing T, the microwave absorption intensity Pabs decreased clearly across T = 150 K as shown in Fig. 2(c), suggesting the change of the magnetization dynamics in the Y3Fe5O12 layer across T = 150 K.

Discussion

The origin of the temperature-induced drastic change of the spin-conversion efficiency VISHE/Pabs 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 f0/2 from the uniform magnon with f0 [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 emission14,32. The splitting is allowed only when f0/2 > fmin, where fmin is the minimum frequency of the magnon dispersion, because of the energy and momentum conservation laws. This condition can readily be found by finding fmin for the thin Y3Fe5O12 film from the lowest branch of the dipole-exchange magnon dispersion for the unpinned surface spin condition40:

where Ω = ωH + ωM(D/μ0Ms)k2, ωH = γμ0H, ωM = γμ0Ms and Q = 1 − [1 − exp(−kL)]/(kL). D = 5.2 × 10−13 Tcm2 is the exchange interaction constant, L = 5 μm is the thickness of the Y3Fe5O12 layer and k is the wavenumber of the magnons (see also the Supplementary Information). γ = 1.84 × 1011 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/Y3Fe5O12 film, calculated using Eq. (2). For the calculation, we used the saturation magnetization Ms at each temperature [see Fig. 3(c)], estimated from the ferromagnetic resonance field HR 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 literature32,41,42; although D varies with temperature, the variation is less than 4% in Y3Fe5O12 for T < 350 K and the shape of the magnon dispersion is not sensitive to the small variation of D41,42,43. Figure 3(a,b) demonstrate that the minimum frequency fmin decreases with decreasing temperature and the splitting condition f0/2 > fmin is satisfied below T = 150 K; the magnon redistribution is responsible for the enhancement of VISHE/Pabs. 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 3
figure 3

Magnon dispersion.

(a) The lowest-energy branch of the magnon spectra for the Pt/Y3Fe5O12 film calculated for the resonance condition at f0 = 4.0 GHz. The dispersions were calculated using γ = 1.84 × 1011 Ts−1. The dotted red line denotes f = f0/2 = 2.0 GHz. (b) The magnified view of the lowest-energy branch of the magnon spectra. (c) Temperature dependence of the saturation magnetization Ms estimated from the resonance field data.

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Figure 4(a,b) show temperature dependence of the spin-conversion efficiency VISHE/Pabs at different microwave excitation powers Pin for f0 = 7.6 and 4.0 GHz, respectively. At f0 = 4.0 GHz, the enhancement of VISHE/Pabs 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 VISHE/Pabs at T = 50 K for f0 = 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 VISHE/Pabs 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 VISHE/Pabs depends on the excitation power, especially below 150 K, at f0 = 7.6 GHz as shown in Fig. 4(a). These features for f0 = 7.6 and 4.0 GHz were confirmed in VISHE/Pabs 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 VISHE/Pabs at low temperatures.

Figure 4
figure 4

Temperature evolution of spin-current emission for different microwave powers.

(a) Temperature T dependence of VISHE/Pabs at f0 = 7.6 GHz for the in-plane magnetic field H (the upper panel) and reversed in-plane magnetic field −H (the lower panel). (b) T dependence of VISHE/Pabs at f0 = 4.0 GHz for the in-plane magnetic field H (the upper panel) and −H (the lower panel). (c) T dependence of [VISHE/Pabs]100 mW/[VISHE/Pabs]5 mW at f0 = 7.6 GHz. [VISHE/Pabs]100 mW and [VISHE/Pabs]5 mW are VISHE/Pabs measured at Pin = 100 mW and 5 mW, respectively.

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To understand the temperature and power dependences of VISHE/Pabs at f0 = 7.6 GHz in details, we plot [VISHE/Pabs]100 mW/[VISHE/Pabs]5 mW in Fig. 4(c). For the spin-current emission in the linear magnetization dynamics regime, VISHE/Pabs is constant with Pin, or [VISHE/Pabs]100 mW/[VISHE/Pabs]5 mW = 1 because the emitted spin current is proportional to Pin35. Since the three-magnon splitting is prohibited at f0 = 7.6 GHz, [VISHE/Pabs]100 mW/[VISHE/Pabs]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 Pin > Pth, known as the second order Suhl instability46, where Pth 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 η0 and creation of dipole-exchange magnons with small damping ηq. This results in the steady-state enhancement of the angular momentum stored in the spin system, or the enhanced spin-current emission32. In the Pt/Y3Fe5O12 film, the damping η0 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 η0 is not dominated by the temperature peak processes or the Kasuya-LeCraw mechanism47. In contrast, the damping ηq 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)]44,48,49,50. 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 η0 of the uniform magnon is dominated by the temperature-independent two-magnon scattering, whereas the damping ηq of the secondary magnon is dominated by temperature-dependent three-particle confluences, such as the Kasuya-LeCraw process47. In the presence of the four magnon scattering, the total number of the nonequilibrium magnons Nt is expressed as32

Figure 5
figure 5

Microwave power dependence of spin-current emission at different temperatures.

Microwave excitation power Pin dependence of [VISHE/Pabs]/[VISHE/Pabs]5 mW at f0 = 7.6 GHz for different temperatures. The in-plane magnetic field is H for the upper panel and −H for the lower panel, respectively. The inset shows T dependence of the half-maximum full-width μ0ΔH of ferromagnetic resonance for the Pt/Y3Fe5O12 film.

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where ηq 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

where

Here, ηsp is the decay constant of the uniform precession to degenerate magnons at f0 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 VISHE ∝ js ∝ Nt, Eq. (3) is directly related to the spin-conversion efficiency: VISHE/Pabs ∝ Nt/Pabs.

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 [VISHE/Pabs]100 mW/[VISHE/Pabs]5 mW ≈ 1.2 at T = 300 K to [VISHE/Pabs]100 mW/[VISHE/Pabs]5 mW ≈ 2.4 at 50 K. Figure 5 shows microwave excitation power Pin dependence of VISHE/Pabs for f0 = 7.6 GHz at different temperatures. This result clearly shows that the threshold power Pth of the spin-current enhancement decreases with decreasing temperature, which is the origin of the nontrivial behavior of the temperature dependence of VISHE/Pabs 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, VISHE/Pabs 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/Y3Fe5O12 film at T = 75 K [see the orange circles in Fig. 6]. At T = 300 K, a clear threshold is observed around Pin = 40 mW. The threshold power of the four-magnon scattering is given by47 , where hth is the threshold microwave field and σq 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 f0 = γμ0H and the coupling strength can be approximated as σq = γμ0Ms. Thus, the threshold power for the four-magnon scattering is proportional to

Figure 6
figure 6

Threshold power of spin-current enhancement.

Microwave excitation power Pin dependence of [VISHE/Pabs]/[VISHE/Pabs]5 mW at f0 = 7.6 GHz for T = 300 K and T = 75 K.

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Equation (6) predicts that the threshold power of the spin-current enhancement decreases with decreasing temperature, since Ms increases by decreasing temperature as shown in Fig. 3(c). Although the damping η0 of the uniform magnon is almost independent of temperature as shown in the inset to Fig. 5, the damping ηq of the dipole-exchange magnon tends to decrease the threshold power, since ηq, dominated by the Kasuya-LeCraw process is approximately proportional to temperature47. 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 VISHE/Pabs around 100 K for f0 = 7.6 GHz [see Fig. 4(a)].

In summary, we have demonstrated that the spin-current emission from a Y3Fe5O12 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 splitting15. 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/Y3Fe5O12 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.

Additional Information

How to cite this article: Tashiro, T. et al. Spin-current emission governed by nonlinear spin dynamics. Sci. Rep. 5, 15158; doi: 10.1038/srep15158 (2015).