Abstract
When a magnon, the quanta of a spin excitation, is created in a magnet, this quasiparticle can split into two magnons, which triggers an angular momentum flow from the lattice to the spin subsystem. Although this process is known to enhance spincurrent emission at metal/magnetic insulator interfaces, the role of interacting magnons in spintronic devices is still not wellunderstood. Here, we show that the enhanced spincurrent emission is enabled by spindamping tuning triggered by the redistribution of magnons. This is evidenced by timeresolved measurements of magnon lifetimes using the inverse spin Hall effect. Furthermore, we demonstrate nonlinear enhancement of the spin conversion triggered by scattering processes that conserve the number of magnons, illustrating the crucial role of spindamping tuning in the nonlinear spincurrent emission. These findings provide a crucial piece of information for the development of nonlinear spinbased devices, promising important advances in insulator spintronics.
Introduction
For more than half a century, the physics of spin dynamics has provided insight into the fundamental aspects of magnetism^{1,2,3,4}. The collective excitations of localized spins coupled by the magnetic dipole and quantum exchange interactions are called spin waves or, in quantized form, magnons^{5}. These quasiparticles, associated with the elementary magnetic excitation, are responsible for a flow of spin angular momentum, a spin current, in magnetic insulators, which are electrically inactive due to the frozen charge degrees of freedom, but magnetically active as a result of the interacting localized spins. In contrast to spin currents carried by conduction electrons in metals and semiconductors, which disappear within very short distances, spin currents in magnetic insulators persist over macroscopic distances owing to the substantially long lifetimes of the magnons relative to that of the conduction electron spins.
The recent discovery of spincarrier conversion from magnons to conduction electrons through dynamical spin exchange coupling at metal/magnetic insulator interfaces sheds new light on the physics of nonlinear spin dynamics, leading to the combination of spintronics with interacting magnons. In a metal/magnetic insulator junction, the excitation of nonequilibrium magnons in the magnetic layer emits a spin current into the attached metal through spin pumping^{6,7,8,9,10,11,12,13,14,15,16}. The spincurrent emission has been found to be enhanced through nonlinear magnon interactions^{13,17}. Although the physics behind the spincurrent enhancement is still not clear, this illustrates the crucial role of nonlinear spin dynamics and interacting magnons in spintronic devices.
In this work, we demonstrate that the stabilized enhancement of the spincurrent emission is enabled by spindamping tuning, resulting from the redistribution of magnons due to the nonlinear magnon interactions. Of particular interest is the nonlinear threemagnon splitting, the splitting of one magnon into a pair of magnons (see Fig. 1a). Since each magnon carries spin angular momentum, ℏ in the exchange limit, it is natural to expect that the splitting directly enhances the spincurrent emission. However, this picture is valid only for the time scale of the spinlattice relaxation; the steadystate enhancement of the spincurrent emission is governed by the spin damping. Our direct measurements of the spin damping demonstrate that the stabilized enhancement of the spincurrent emission is enabled by the long lifetime of the dipoleexchange secondary magnons created at the inflection point of the dispersion curve, where the negative dipolar dispersion is compensated by the positive exchange dispersion. We demonstrate, furthermore, that the spincurrent emission can be enhanced even in the absence of the magnon splitting; we found enhanced spincurrent emission triggered by scattering processes that conserve the number of magnons. These findings demonstrate the crucial role of magnon lifetimes in spintronic devices, opening a way for controlling nonlinear spincurrent emission through spindamping tuning.
Results
Enhancement of spincurrent emission
The nonlinear spincurrent emission from a magnetic insulator is investigated by measuring the voltage generated through the inverse spin Hall effect (ISHE) in a Pt/Y_{3}Fe_{5}O_{12} (Pt/YIG) film, shown in Fig. 1b (details are outlined in Methods). Figure 1c shows the inplane magnetic field, H, dependence of the microwave absorption intensity, P, measured by applying a 5 mW microwave with a fixed frequency of f_{0}=5 GHz at T=250 K. Figure 1c shows that at H=H_{R}=81.3 kA m^{−1}, H fulfills the ferromagnetic resonance condition: , where γ is the gyromagnetic ratio and M_{s} is the saturation magnetization of the YIG layer. Around H=H_{R}, we found voltage difference, V, between the ends of the Pt layer as shown in Fig. 1c. This provides evidence of spincurrent emission from the YIG layer into the Pt layer; nonequilibrium magnons, excited by the magnetic resonance, emit a spin current into the Pt layer^{18,19}, which is converted into an electric voltage by the ISHE in the Pt layer (see Fig. 1b)^{20,21,22}.
The amplitude of the spincurrent emission critically depends on the microwave excitation frequency, f_{0}, because of the nonlinear dynamics of the interacting spins in the YIG layer. Figure 1d shows the field dependence of V(H)/P_{abs}, measured at T=250 K for various f_{0} values, where P_{abs} is the microwave absorption intensity (see Fig. 1c). Here V(H)/P_{abs} characterizes the conversion efficiency of the angular momentum created by the microwave field into the spin current carried by the conduction electrons in the Pt layer. Notably, V(H)/P_{abs} abruptly increases below f_{0}=f_{c}=3.4 GHz, while it is nearly constant for f_{0}>3.5 GHz. To study the f_{0} dependence of V(H)/P_{abs} in detail, we plot κ(f_{0}, T)≡V_{ISHE}(f_{0}, T)/P_{abs}(f_{0}, T) at different f_{0} values in Fig. 1e, where V_{ISHE} is the magnitude of the electric voltage. Figure 1e shows that κ(f_{0}, T=250 K) for f_{0}>3.5 GHz is wellreproduced with a calculation based on the linear spin pumping model^{23} (see the red line in Fig. 1e), demonstrating that the enhancement of κ(f_{0}, T=250 K) at low f_{0} originates from the nonlinearity of the spin dynamics in the YIG layer: the threemagnon splitting^{17}. Here the microwave excitation power P_{in}=5 mW is about 3 orders of magnitude larger than the threshold power of the threemagnon splitting^{5,17}.
Spincurrent emission at various temperatures
Importantly, the conversion efficiency κ is enhanced at low temperatures. Figure 2a shows the f_{0} dependence of the conversion efficiency κ(f_{0}, T)=V_{ISHE}(f_{0}, T)/P_{abs}(f_{0}, T) at different temperatures T, where κ(f_{0}, T) is divided by κ(f_{0}=5 GHz, T) to omit the temperature dependent parameters relevant to the conversion efficiency from the spin current to the electric voltage such as the spin Hall angle, electrical conductivity and spin diffusion length in the Pt layer (see the inset to Fig. 2a). Notably, κ(f_{0}, T) clearly increases with decreasing temperature under the threemagnon splitting (see also Fig. 2b). Figure 2a also shows that the cutoff frequency, f_{c}, below which κ(f_{0}, T) increases abruptly, increases with decreasing temperature (see Fig. 2c).
We first discuss the T dependence of f_{c}, which further supports that the threemagnon splitting is responsible for the enhancement of the spincurrent emission increased at low temperatures. The threemagnon splitting creates a pair of magnons with opposite wavevectors and frequency f_{0}/2 from the uniform magnon with a frequency of f_{0} (see Fig. 1a), following the energy and momentum conservation laws. Thus, the splitting process is only allowed for f_{0}/2>f_{min}, where f_{min} is the minimum frequency of the magnon dispersion as shown in Fig. 1a. f_{min} for the thin YIG film used in the present study can be obtained from the dipoleexchange magnon dispersion. The lowest branch of the magnon dispersion is expressed as^{24}
where Ω=ω_{H}+ω_{M}(D/μ_{0}M_{s})K^{2}, ω_{H}=γμ_{0}H, ω_{M}=γμ_{0}M_{s}, K^{2}=λ^{2}+k^{2}=(π/L)^{2}+k^{2}, and Q=(k/K)^{2}[1+(λ/K)^{2}(2/kL) [1+exp(−kL)]]. D is the exchange interaction constant, L is the thickness of the YIG layer and k is the wavenumber of the magnons. To obtain the exact condition f_{0}/2>f_{min} for the Pt/YIG film, a detailed calculation using equation (1) is necessary as discussed below. Before discussing in detail the condition for the threemagnon splitting based on the exact magnon dispersion, we first neglect the surface dipolar interactions in equation (1) for simplicity. f_{min} under this approximation can be obtained in the limit of L→∞ as f_{min}≈γμ_{0}H. This relation with the resonance frequency shows that the condition f_{0}/2>f_{min} for the threemagnon splitting is satisfied when f_{0}<(2/3) γμ_{0}M_{s}. This predicts that f_{c} increases with M_{s} due to the change in the magnon dispersion. We estimated M_{s} of the Pt/YIG film from the resonance field H_{R} using the Kittel formula (see Fig. 3a) as shown in Fig. 3b. Using this result, we plot the relationship between f_{c} and M_{s} in Fig. 3c, which is consistent with the above discussion—f_{c} increases with M_{s}. Thus, the observed change in f_{c} can be attributed to the change in the magnon dispersion, caused by the temperature variation of M_{s} in the Pt/YIG film.
The experimentally measured temperature dependences of f_{c} and M_{s} shown in Figs 2c and 3b illustrate the exact magnon dispersion of the Pt/YIG film at each temperature using equation (1). When the external magnetic field H is varied, the magnon dispersion is shifted in frequency space and the relationship between the frequencies f_{0}/2 and f_{min} also changes. Figure 4a shows a plot of the H dependences of f_{0}/2 and f_{min} at T=100 and 300 K, where f_{0}/2 was calculated using the Kittel formula with the experimentally obtained M_{s}. f_{min} was plotted by finding the minimum frequency of the magnon dispersion using equation(1). The crossing points of f_{0}/2 and f_{min} correspond to the cutoff frequency f_{c}, where f_{min} and the crossing point can be varied by changing the exchange interaction constant D; D for which the crossing point reproduces the measured f_{c} value at different temperatures is plotted in the inset to Fig. 4b. The experimentally obtained D is almost temperature independent, consistent with the literature^{25}. Using the values of M_{s} and D, we plot the lowest branch of the magnon dispersion at the cutoff frequency, f_{0}=f_{c}=2f_{min}, for different temperatures in Fig. 4b. Figure 4b shows that the shape of the lowest branch of the magnon dispersion changes slightly by changing temperature, which is the origin of the temperature dependence of the cutoff frequency.
Discussion
The observed enhancement of the spincurrent emission is enabled by spindamping tuning triggered by the redistribution of magnons due to the nonlinear magnon scattering. Here note that the steadystate spincurrent emission from the magnetic insulator is stabilized by balancing the creation and decay of the magnons; the steadystate angular momentum stored in the spin system is governed by the magnon damping. Assuming that the spin current emitted from the magnetic insulator is proportional to the total number of nonequilibrium magnons^{26,27}, and using the energy balance relation P_{abs}=∑_{k}ℏω_{k}η_{k}N_{k}, we find that the conversion efficiency κ=V_{ISHE}/P_{abs} is proportional to the spinlattice relaxation time τ of the pumped system: κ∝τ. The relaxation rate 1/τ is given by
where N_{k} and η_{k} are the number and the damping of nonequilibrium magnons with the wavenumber k, respectively, and N_{t}=∑_{k}N_{k} is the total number of nonequilibrium magnons in the system. This simple model provides a general picture of the nonlinear spincurrent emission in the presence of magnon scatterings; the above model indicates that the spincurrent emission can be enhanced through the magnon redistribution in a system with η_{0}>η_{q}, where η_{0} and η_{q} are the damping of the uniform and secondary magnons, respectively. In this system, the magnon redistribution triggered by magnon interactions destroys the magnons with the largedamping η_{0} and creates the magnons with the smalldamping η_{q}, changing the relative weight N_{k}/N_{t} in equation (2) and decreasing the dissipation rate 1/τ of the angular momentum from the spin system to the lattice. This spindamping tuning, triggered by the magnon redistribution, increases the steadystate angular momentum stored in the spin system, giving rise to the stabilized enhancement of the spincurrent emission.
The spin damping 1/τ can be measured directly by monitoring the temporal evolution of the spincurrent emission using the ISHE, which provides direct evidence that the spin damping plays a key role in the spincurrent enhancement. Figure 5a shows the temporal evolution of the electric voltage due to the ISHE in the absence of the threemagnon splitting: f_{0}=8 GHz>f_{c}. The temporal profile of the ISHE voltage, V_{R}(t), at the resonance field, H=H_{R}, was measured by applying a microwave pulse that was switched off at t=t_{MW}. Figure 5a shows that V_{R}(t) decays exponentially after switching off the microwaves; the spincurrent emission decreases with time due to the dissipation of the angular momentum from the spin system to the lattice in the YIG layer^{28}. The spincurrent decay was also observed at f_{0}=2.8 GHz<f_{c} as shown in Fig. 5b, where the dipoleexchange magnons are created by the threemagnon splitting. For these results, we extracted the relaxation time, τ, by fitting the voltage with V_{R}(t)/V_{R}(t=t_{MW}−500 ns)=exp(−2t/τ) for t>t_{MW}. The extracted relaxation time τ for different f_{0} at T=300 K is shown in Fig. 5c. Figure 5c shows that τ is almost constant for f_{0}>f_{c}. In contrast, τ increases with decreasing the excitation frequency for f_{0}<f_{c}, reminiscent of the f_{0} dependence of κ shown in Fig. 2a. These results, therefore, demonstrate the direct relationship between the spin damping and the conversion efficiency in the nonlinear spincurrent emission.
The observed spincurrent enhancement and spindamping tuning is realized by the small damping and longlived nature of the dipoleexchange secondary magnons created by the threemagnon splitting. When f_{0}<f_{c}, the threemagnon splitting creates the dipoleexchange secondary magnons with wavevectors of k=±q from the k=0 uniform magnons near the bottom of the magnon dispersion as shown in Fig. 1a. Although the damping of the uniform magnon is dominated by the elastic twomagnon scattering due to nonuniformities, where a magnon is destroyed and another magnon at the same frequency is created^{29}, the twomagnon scattering can be suppressed for the dipoleexchange secondary magnons created by the threemagnon splitting. For the secondary magnons, because of the nearly zero group velocity due to the competition of the magnetic dipole and quantum exchange interactions, the scattering events due to nonuniformities are suppressed, resulting in the long lifetime, or the small damping. In the presence of the threemagnon splitting, the dynamics of these uniform and secondary magnons are modelled as and , because short wavelength magnons do not couple to the microwave field. n_{k}(k=0, ±q) is the number of the magnons with the wavenumber k and the damping η_{k}. is the thermodynamic equilibrium value of n_{k}. P_{abs} is the absorbed microwave power and ω_{0}=2πf_{0}. Here for simplicity, we assume that the damping due to the threemagnon splitting as η_{3s}=Γ_{3s}(n_{q}+n_{−q}), where Γ_{3s} is the splitting strength.^{5,30} Notably, the splitting strength Γ_{3s} increases with the decreasing excitation frequency f_{0}, because the number of the degenerate states at f_{0}/2 available for the splitting increases by decreasing f_{0} (see also the schematic illustrations in Fig. 5c)^{30}. Thus, by decreasing f_{0} below the cutoff frequency, the relative number of the smalldamping dipoleexchange secondary magnons increases and that of the largedamping uniform magnons decreases (see also equation (2)), resulting in the long lifetime τ of the spin system as demonstrated in Fig. 5c. The effect of the creation of an additional magnon through the splitting on κ becomes evident in a strong excitation condition. For Γ_{3s}≠0, from the above rate equations, the number of the nonequilibrium uniform magnons can be expressed as , where n_{q,−q}≡n_{q}=n_{−q} and η_{q,−q}≡η_{q}=η_{−q}. In the limit of strong excitation, that is, , the number of the nonequilibrium uniform magnons is limited to N_{0}≃η_{q,−q}/2Γ_{3s}. Thus, in this condition, and N_{t}≃2N_{q,−q}, resulting in κ_{3s}=V_{ISHE}/P_{abs}∝2/η_{q,−q} because of the approximated energy balance relation . The factor 2 arises from the fact that the threemagnon splitting creates two magnons from one magnon. Although the threemagnon splitting triggers the angular momentum flow from the lattice, the creation of the secondary magnons also affects the inherent dissipation of the angular momentum from the magnons to the lattice; when the redistribution of magnons creates secondary magnons with exceptionally short lifetimes, this redistribution process enhances the inherent dissipation of the angular momentum of the spin system or decreases τ of the system, masking the angular momentum flow from the lattice triggered by the nonlinear magnon interaction. Thus, in the steady state, it is nontrivial whether the magnon splitting increases or decreases the nonequilibrium angular momentum stored in the spin system. Since in the absence of the splitting, Γ_{3s}=0, the conversion efficiency is expressed as κ_{linear}∝1/η_{0}, the spincurrent emission can be enhanced by the threemagnon splitting when the damping of the uniform and secondary magnons satisfies κ_{3s}/κ_{linear}=2η_{0}/η_{q,−q}>1; the creation of longlived magnons with a damping of η_{q,−q}(<2η_{0}) through the nonlinear magnon scattering is responsible for the enhancement of the spincurrent emission.
The temperature dependence of the conversion efficiency κ shown in Fig. 2b further demonstrates that the spincurrent emission is governed by the spin damping of the system. Figure 5a shows that the decay time of V_{R}(t) is almost independent of temperature when f_{0}=8 GHz>f_{c}, that is, in the absence of the threemagnon splitting. In contrast, the voltage decay time varies by changing the temperature at f_{0}=2.8 GHz<f_{c}, as shown in Fig. 5b. The temperature dependence of the decay time, τ, is summarized in Fig. 5d. This result shows that the magnon relaxation time τ increases with decreasing temperature under the threemagnon splitting, consistent with the experimentally obtained temperature dependence of κ shown in Fig. 2b. At the fixed excitation frequency f_{0}, the number of the available states at f_{0}/2 for the splitting increases with decreasing temperature (see Fig. 5d). This indicates that the relative number of the smalldamping secondary magnons increases with decreasing temperature, which reduces the damping of the spin system through the splitting. The damping of the secondary magnon itself is also expected to be reduced with decreasing temperature, since the damping of the dipoleexchange magnon is dominated by the threeparticle processes, which is almost proportional to temperature^{29,31}. Therefore, by decreasing temperature, both the splitting strength and the damping of the secondary magnons tend to increase the relaxation time of the spin system, giving rise to the experimentally observed longer τ at low temperatures.
The direct measurements of the spin damping at different frequencies and temperatures shown in Fig. 5 provide evidence that the creation of the longlived secondary dipoleexchange magnons through the redistribution is essential for the enhancement of the spincurrent emission. This picture is further evidenced by microwave excitation power P_{in} dependence of the conversion efficiency κ(P_{in})=V_{ISHE}(P_{in})/P_{abs}(P_{in}) for the Pt/YIG film shown in Fig. 6a. The applied microwave excitation frequency was f_{0}=7.6 GHz>f_{c}, that is, the threemagnon splitting is prohibited in this measurement. For P_{in}<P_{th}, the conversion efficiency is constant, consistent with the spincurrent emission in the linear regime^{32}; the emitted spin current is proportional to P_{in}. However, κ(P_{in}) clearly increases by increasing P_{in} for P_{in}>P_{th}, demonstrating the enhancement of the spin conversion efficiency without the splitting of a pumped magnon, or without breaking the angular momentum conservation of the spin subsystem. Notably, κ(P_{in}) shows a clear threshold power P_{th}, indicating that the fourmagnon scattering^{33,34}, where two magnons are created with the annihilation of two other magnons, is responsible for the enhanced spincurrent emission.
The enhanced spincurrent emission induced by the scattering process that conserve the number of magnons, that is, the fourmagnon scattering, demonstrates the general picture of the role of the magnon interactions in the nonlinear spincurrent emission: the spindamping tuning triggered by the magnon redistribution. In the absence of the redistribution of magnons, the angular momentum relaxation rate from the spin system to the lattice is 1/τ≃η_{0} because of N_{t}≃N_{0}. Although the fourmagnon scattering creates dipoleexchange magnons at the excitation frequency f_{0} not at f_{0}/2, the lifetime of these magnons can be longer than that of the uniform magnon because of the small group velocity^{35}. Thus, even in the absence of the threemagnon splitting, the fourmagnon scattering can decrease the relaxation rate of the spin system, 1/τ, through the annihilation of the uniform magnons with large η_{0} and creation of dipoleexchange magnons with small η_{q}, resulting in the steadystate enhancement of the angular momentum stored in the spin system or the enhanced spincurrent emission shown in Fig. 6a. This is further supported by a rate equation approach for the fourmagnon scattering^{33,36,37}: dN_{0}/dt=−[η_{0}+η_{sp}f(P_{in})]N_{0}+P_{abs}/ℏω_{0} and d(N_{t}−N_{0})/dt=−η_{q}(N_{t}−N_{0})+η_{sp}f(P_{in})N_{0}, where . η_{sp} is the decay constant of the uniform precession to degenerate magnons at f_{0} due to scattering from sample inhomogeneities. To keep the discussion simple, we define η_{q} as the average decay rate to the thermodynamic equilibrium of the degenerate secondary magnons. P_{th} is the threshold power for the fourmagnon scattering and χ″ is the imaginary part of the susceptibility. P_{abs}=ω_{0}χ″h^{2} and h is the applied microwave field strength. The steadystate solution of the above rate equation gives
where κ(P_{in})∝N_{t}/P_{abs} and χ″=(2γM_{s})/(η_{0}+η_{sp}f(P_{in})). Figure 6b shows the conversion efficiency κ(P_{in}) calculated using equation (3) for η_{q}/η_{0}=0.75 (the red curve) and η_{q}/η_{0}=1.25 (the black curve) with the assumption of V_{ISHE}∝N_{t}. In a system with η_{q}/η_{0}=1.25, κ(P_{in}) decreases with P_{in} for P_{in}>P_{th}, since the scattering process increases the relative number of the largedamping magnons and increases the dissipation rate of the angular momentum stored in the spin system (see equation (2)). In contrast, in a system with η_{q}/η_{0}=0.75, the fourmagnon scattering increases the relative number of the smalldamping magnons and decreases the damping of the spin system, resulting in the enhanced spincurrent emission, which is consistent with the experimental observation shown in Fig. 6a.
The observed enhancement of the spincurrent emission demonstrates the crucial role of the spin damping affected by nonlinear magnon interactions in spintronic devices. The damping of the spin system can be directly quantified using the timeresolved measurement of the spincurrent emission under various conditions, providing a way for further development of the physics of nonlinear spin conversion. These findings shed new light on nonlinear spin dynamics, promising important advances in spintronics with interacting magnons.
Methods
Sample preparation
The sample used in this study is a Pt/YIG film. A singlecrystal YIG (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. A 10nmthick Pt layer was sputtered in an Ar atmosphere on top of the film and two electrodes were attached to the edges of the Pt layer for voltage measurements.
ISHE voltage and microwave absorption measurements
The Pt/YIG film was placed at the centre of a coplanar waveguide, where a microwave signal with a frequency of f_{0} was applied to the input of the signal line. The signal line of the coplanar waveguide is 500 μm wide and the gaps between the signal line and the ground lines are designed to match the characteristic impedance of 50 Ω. An inplane external magnetic field was applied parallel to the signal line, such that the static magnetic field is perpendicular to the dynamic magnetic field produced by the microwave current. The microwave power absorbed by the sample P_{abs} was measured by monitoring the transmitted output power.
Additional information
How to cite this article: Sakimura, H. et al. Nonlinear spincurrent enhancement enabled by spindamping tuning. Nat. Commun. 5:5730 doi: 10.1038/ncomms6730 (2014).
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Acknowledgements
We thank A.L. Chernyshev, H. Kurebayashi, Y. Kajiwara and M.B. Jungfleisch for valuable discussions. This work was supported by PRESTOJST ‘Innovative nanoelectronics through interdisciplinary collaboration among material, device and system layers,‘JSPS KAKENHI Grant Numbers 26220604, 26103004, 26600078, the Mitsubishi Foundation, the Asahi Glass Foundation, the Noguchi Institute, the Murata Science Foundation and the Cabinet Office, Government of Japan through its ‘Funding Programme for Next Generation WorldLeading Researchers’.
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K.A. designed the experiments, developed the explanation and wrote the manuscript. H.S. and T.T. collected and analyzed the data. All authors discussed the results and reviewed the manuscript.
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Sakimura, H., Tashiro, T. & Ando, K. Nonlinear spincurrent enhancement enabled by spindamping tuning. Nat Commun 5, 5730 (2014). https://doi.org/10.1038/ncomms6730
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Yttrium Iron Garnet Thin Films with Very Low Damping Obtained by Recrystallization of Amorphous Material
Scientific Reports (2016)

Spincurrent emission governed by nonlinear spin dynamics
Scientific Reports (2015)
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