Influence of three-magnon decays on electromotive force generation by magnetostatic surface waves in integral YIG – Pt structures

. The purpose of this work is to find out the influence of three-magnon decay processes on the electromotive force (EMF (U)) generated by propagating magnetostatic surface waves (MSSW) with the help of the inverse spin Hall effect in the “yttrium-iron garnet (YIG) – platinum (Pt)” structure. Methods. The experiments were carried out using the delay line structures based on YIG films with the thickness of 8.8 and 14.6 µ m, on the surface of which antennas were formed for MSSWs excitation and reception and a Pt film between antennas. Results. It was shown that the three-magnon parametric instability can significantly change the character of EMF dependences on frequency and on power of MSSW that resulted both from the effect of power limitation and from the participation of parametric spin waves (PSW) and secondary spin waves (SSW) in the processes of electron-magnon scattering on the YIG/Pt interface. Conclusion. It was demonstrated that the effect of amplification of EMF generation at the frequencies that are close to the long-wavelength limit of the MSSW spectrum is related with the PSW and SSW population of the region of anisotropic dipole-exchange spin waves spectrum, which is characterized by the presence of singularities in the magnon density of states (Van Hove singularities).


Introduction
Recently, there has been a surge of interest in studying the effect of EMF generation in structures based on films of yttrium iron garnet(YIG) and platinum (Pt), where due to exchange and spin-orbit interactions, the conduction electrons of the metal are sensitive to the state of magnetization ⃗ of the YIG film at the interface [1][2][3][4][5][6]. At the same time, a change in ⃗ , when coherent or incoherent (thermal) spin waves (SW) are excited due to the spin pumping mechanism, creates a spin current through the YIG/Pt interface, which leads to the generation of EMF at the ends of the electrically open Pt layer due to the inverse spin Hall effect (ISHE) [7]. Such effects are important for spintronics [8,9], because they open up the possibility of building not only SW detectors [10,11], but also spin logic devices [12,13], magnetic memory [14], magnonic transistors [15], as well as amplification and generation of SW [16][17][18].
In experiments on pumping by traveling magnetostatic waves (MSW), the density of the magnonic spin current through the cross-section of the YIG film can be associated with the microwave power flow of the spin waves ∼ ≈ | ⃗ | 2 , where | ⃗ | and , the amplitude and the group velocity are of the MSW. The spin current pumped through the interface ( < ) due to the ISHE, leads to the appearance of an electric current in the Pt film with For the practical application of spin current detectors, the proportionality of to the MSW power, characterized by the volt-watt sensitivity κ= / , is essential. Since an increase in the power of microwave pumping can lead to parametric instability in the SW system, much attention is paid to the study of the mechanisms of EMF generation in YIG/Pt structures under the conditions of the development of SW instability [11,18,[25][26][27][28][29][30][31][32][33][34][35][36]. Parametric instability occurs when the power of the MSW is above a certain threshold level th ( > th ) and when the conservation laws [37][38][39] are fulfilled where the frequencies ,1,2 and the wave vectors ⃗ ,1,2 correspond, respectively, to pumping and parametric spin waves (PSW), and the integer takes the values = 1, 2 and corresponds to processes of the first ( = 1, three-magnon (3M)) or the second ( = 2, four-magnon (4M)) orders of magnitude. Studies of the influence of processes (2) on the generation of spin current in YIG/Pt structures have shown the absence of dependence κ on the wave numbers PSW | ⃗ 1,2 | both in conditions of parallel [25,27,29,32] and perpendicular [27,29] pumping. At the same time, in the works [25,32], an increase in sensitivity κ was noted for such values of and the magnetic field , at which it is possible to generate secondary spin waves (SSW) as a result of non-threshold processes of PSW fusion: The possibility of detecting the microwave component of the spin current associated with the excitation of the PSW at the frequency 1,2 = /2 was considered in the work [11]. It was also demonstrated that the drop in microwave magnetic susceptibility at > th , as well as self-oscillations and bistability in the PSW system [37][38][39] the YIG/Pt structures lead to the nonlinearity of the dependence = ( ) [25][26][27][28][33][34][35][36], the occurrence of oscillations [30] and bistability of EMF [31] .
Special attention was paid to the study of the influence of processes (2) on the nature of frequency dependence κ( ) in tangentionally magnetized YIG/Pt structures under excitation of ferromagnetic resonance (FMR) at a frequency close to the frequency of the long-wave limit of the SW spectrum ( ∼ = 0 ) [26-28, 33, 34]. This interest was stimulated by the discovery of the effect of «enhancing the efficiency of spin current generation by 3M decay processes» [26,27], manifested in an increase in the values of κ with an increase in the pumping power , despite the limitation of the amplitude of the magnetization precession | ⃗ |. In the works [26][27][28], where structures based on YIG films with micron thicknesses were studied, the increase in the efficiency of detecting spin current at 3M decays was explained by the transmission of an impulse to the magnon system from the lattice, as well as the influence of PSW on the relaxation rate for magnetization in YIG [27,28]. However, in the works [33,34], where structures based on a 200 nm thick YIG film were studied and 3M decays were prohibited due to the exchange shift of the «bottom» of the SW spectrum [40], a maximum at frequencies ∼ = 0 ≈ 1 GHz was also observed in the κ( ) dependence that indicated the contribution of four-magnon and two-magnon scattering processes to the EMF maximum at frequencies ∼ = 0 ≈ 1 GHz. In this paper, we consider the effect of 3M decays on the generation of EMF by travelling MSSW in the YIG/Pt structures, based on the YIG films with a thickness = 8.8 and = 14.6 microns, where MSSW are predominantly dipole. Unlike previous works, where microstrip antennas with a width ≫ were used to create the spin pumping, we used antennas with < . This resulted in possibility to investigate the effect of 3M decays on EMF in the entire frequency band of the existence of MSSW [ 0 , ]. The paper also discusses a possible mechanism of the effect of «amplification of spin current emission at 3M decays» [26][27][28] in YIG/Pt structures based on YIG films of micron thicknesses associated with the population of the anisotropic dipole-exchange waves spectrum regions, which are characterized by singularities in the density of states, by SSW.
Note that the effects of EMF generation in the YIG/Pt structure under the conditions of MSW excitation by microstrip transducers were studied in the works [35,[41][42][43][44][45]. In the works [41][42][43][44][45], the propagation of MSSW in structures based on films of both micron [41][42][43][44] and nanometer [45] thicknesses was considered. At the same time, in the work [38], an increase in the detection efficiency κ was observed with an increase in the MSSW wave number . The influence of the non-reciprocity of the MSSW propagation on the value of EMF and the nature of the distribution of over the plane of the structure [36,37], the influence of the temperature gradient over the thickness of the structure due to microwave heating of the Pt film on the effect of EMF generation [37], and the effect of DC current in Pt on attenuation of MSSW in YIG/Pt structure [39] were also investigated. However, the influence of parametric instability of the MSSW on the EMF generation effect was not discussed in the works [41][42][43][44][45].
The influence of parametric processes on the dependence = ( ), generated in the YIG/Pt structure by travelling MSW, was studied, apparently, only in the work [36], where the case of propagation of a magnetostatic backward volume wave (MSBVW) in the direction of the ⃗ ℎ in terms of 4M processes. It was shown that at pumping supercritical levels = 10 log th > 15...20 dB the contribution of PSW to the generated EMF is comparable to the contribution of the MSBVW pumping. At the same time, the question of the influence of singularities in the SW spectrum on the value of the ISHE was not discussed.

The studied structures and methods of the experiment
The experiments were carried out with the delay line (DL) on the MSSW based on the integral YIG/Pt structures, the photo of which is shown in Fig. 1. For fabrication of the structures, YIG films with a thickness 1 ≈ 8.8 microns and 2 ≈ 14.6 microns grown by liquid-phase epitaxy on a substrate of gadolinium-gallium garnet of crystallographic orientation (111) were used. YIG films had a magnetization 4π ≈ 1750 Gs, relaxation parameter α ≈ 3 · 10 −4 (FMR linewidth ∆ ≈ 0.5 Oe) and a cubic anisotropy field ∼ = −40 . A Pt film with a thickness of ≈ .9 nm with a resistivity ρ ≈ 50 µ ℎ · was deposited on the surface of the YIG film by DC magnetron sputtering and rectangular elements with a width of 110 microns and a length of ≈ 430 microns were from this film by photolithography and ion etching. Copper microantennas (1,2) in the form of conductors with a width ≈ 4 microns and a length ≈ 110 microns with rectangular contact pads at the ends, as well as contacts and supply lines (3,4) to Pt elements were formed using magnetron sputtering, photolithography and ion etching. The distance between antennas 1 and 2 was ≈ 490 microns, and the distances ξ between the antennas and the edges of the Pt film were ξ ≈ 30 microns.The coordinate system was related to the film as shown in Fig. 1.
To measure the effect of Pt films on the dispersion and attenuation of MSW, the amplitudefrequency (frequency response) and phase-frequency (frequency response) characteristics of DL based on the YIG/Pt structures and YIG films without metallization were compared. For this purpose, simultaneously with the YIG/Pt structures, some DL were made without platinum films and copper contacts 3, 4.
The structure under study was placed between the poles of the electromagnet (see Fig. 1) in the magnetic field ⃗ ‖ ⃗ tangent to the surface of the film, which varied within −2473 < < 2473 Oe. The specified geometry corresponds to the excitation and propagation of the Damon-Eshbach MSSW [21] along ⃗ ( ⃗ ‖ ⃗). Using a vector network analyzer (Keysight M9374A) , the frequency dependences of the transmission coefficients 21 ( ) between antennas 1 and 2 and reflection 11 ( ) from the antenna 1 were measured at various levels of incident power in and values H . The foray of the MSSW phase Θ( ) in the structure was defined as Θ( ) = arctan Im [ 21 ( )]/ Re [ 21 ( )] and was used to calculate the MSSW wave number ( ) = Θ( )/ [46]. Electrical contact with antennas 1 and 2 was provided using microwave microprobes (Picoprobe Model 50A), see Fig. 1.
The measurement of the EMF ( ) generated at contacts (3,4) to the Pt film during the propagation of MSSW at a frequency was carried out using a selective voltmeter (SR830) in the mode of modulation of incident microwave power in by a meander signal with a frequency Ω ≈ 11.33 kHz, see fig. 1. This approach for measuring the ( ) signal allows to reduce the influence of noise and spurious signals on the measurement process, as well as to reduce the contribution from the thermal EMF caused by inhomogeneous heating of the structure by microwave power. At the same time, the contribution to EMF from electron-magnon scattering processes characterized by the times τ − ∼ 10 −12 s [47], tracks power modulation almost without inertia.
The influence of parametric processes on EMF generation was studied for the range of bias fields from = 200 Oe to = 1000 Oe. The threshold value of the power of the MSSW th , at which processes developed at the frequency (2), was determined by the standard method [48][49][50][51]: by decrease in the modulus of the transmission coefficient ( , ) = | 21 ( , )| when > th . At the same time, the changes in the dependencies ( ) and ( ) with the pump supercriticality > 0 were compared with the form of spectrum of the pump signal of the MSSW that passed through the DL, for which the signal from the output antenna was applied to the spectrum analyzer through the directional coupler, see Fig. 1.

The effect of conduction electrons on the dispersion and MSSW attenuation in the YIG/Pt structure
Since the metallization of the YIG film can significantly change the dispersion and attenuation of MSSW [37,38], it is necessary to discuss the effect of the conductivity of the Pt film on the propagation of MSSW in the studied structures. And let us take into account that two mechanisms are possible. The first, long-range mechanism, is associated with the induction of volumetric microwave currents in the metal by the MSSW field, which lead to ohmic losses and shielding of MSSW fields [37,38]. In this case, the nature of the effect of the Pt film on the dispersion and attenuation of the MSW is determined by the value of the spin-electron coupling parameter [52,53]: where sk is the skin depth. For Pt films with ρ ≈ 50 µ ℎ · the skin depth in the frequency range 1...5 GHz will be sk ≈ 7...10 microns. If we assume that the use of (4) for evaluation is valid in the case when at least one wavelength λ fits the length of the Pt film (λ < ) then from (4) , for our case, we get < · /(2π 2 sk ) ≈ 0.07 ≪ 1. Note that at ≪ 1, the metal film mainly affects the MSSW [52], which , for propagating waves, can be characterized by the spatial decrement ′′ = ′′ + ′′ , where ′′ = Im [ ], ′′ and ′′ are the components of spatial decrement due to magnetic and ohmic losses, respectively. At the same time, the dispersion dependence of is close to the case of free YIG films.
The second mechanism results from the exchange interaction on the interface. In relation to the travelling MSW in the structures of YIG /Pt, it was considered in the works [54,55]. effect of spin pumping on the propagation of MSSW was taken into account with the help of boundary conditions for the surface spins pinning at the YIG/Pt boundary and manifested itself in an increase in MSSW losses. It has been shown that in structures based on YIG films with magnetic surface anisotropy characterized by the constant , the relaxation parameter ∆α associated with spin pumping does not depend on the thickness d of the YIG film if the condition [55] is fulfilled: where = 3.85 · 10 −7 erg/cm -the exchange stiffness in YIG. In this case, for typical YIG/Pt structures, the values of ∆α turned out to be of the same order as the magnetic damping parameter ∆α ∼ α [53,54]. Note that with typical values for the YIG, = 0.02...0.05 erg/cm 2 [56, 57] condition (5) is satisfied for films with a thickness ⩾ 0.5 microns.
Let us now turn to Fig. 2, where the frequency dependences of the magnitude of the transmission coefficient ( ) = | 21 ( )| , the reflection coefficient 11 ( ), the conversion coefficient ( ) of the incident power in ( ) to the power of the MSSW ( ), as well as the dispersion curves = ( ), measured for H = 939 Oe and in ≈ −20 dBm are given for the structure of the YIG (14.6 µm)/Pt(9 nm) and the free YIG(14.6 m) film. In Fig. 2, a-c curves 1 and 2 show such dependencies in the YIG/Pt structures and the YIG film, respectively. Dependencies ( ), curves 1 and 2 in Fig. 2, b, reflect the results of the calculation of the MSSW losses using the expression where the values ′′ = ′′ + ′′ were calculated similarly to [53]. Curve 3 in Fig. 2, b corresponds to the calculation when both ohmic losses and losses due to spin pumping are taken into account.
In the spatial decrement ′′ = ′′ + ′′ , losses in the magnetic system ′′ were calculated taking into account the renormalization of the relaxation parameter of spin waves due to spin pumping α = α + ∆α and it was assumed that ∆α = α = 3 · 10 −4 . Curve 3 in Fig. 2, a corresponds to the dispersion ( ) calculated using the dispersion equation for MSSW [21]: where 2 0 = 2 + , = γ , = γ4π , γ =2.8 MHz/Oe -gyromagnetic ratio for YIG. Vertical dotted lines mark the long-wave ( → 0) 0 ≈ 4.43 GHz and short-wave ( → ∞) = + /2 ≈ 5.09 GHz limits of the MSSW spectrum. The dependence of 11 ( ) is shown by curve 3 in Fig. 2 for the field * = 2473 Oe, when there is no excitation of MSSW at frequencies < 7 GHz. Measured dependencies 11 ( , * ) were used to calculate the coefficient ( ) by the ratio: where it is assumed that the MSSW power ( ) is defined as the difference of the reflected powers ( , ) from the input transducer at the bias field corresponding to the excitation of the MSSW at a frequency , and the field * ≫ when there is no MSSW excitation at the frequency f.
From a comparison of the results shown in Fig. 2, a-c, it can be seen that the dependencies ( ), ( ), 11 ( ) and ( ) in the structure of the YIG(14.6 µm)/Pt(9 nm) are both qualitatively and quantitatively close to the case of the YIG film. At the same time, the comparison of curves 2 and 3 in Fig. 2, b shows that, under the assumption ∆α ∼ α, the contribution of the Pt film to the attenuation of MSSW is mainly due to ohmic losses of induction currents. However, from a comparison of the experimental dependencies ( ) shown by curves 1 and 2 in Fig. 2, a, it is not possible to state unequivocally that the deposition of a Pt film leads to additional losses of MSSW in comparison with the free YIG film . Indeed, from Fig. 2, a it can be seen that at some frequencies the amplitude of the output signal in the YIG/Pt structure exceeds the values for the free YIG film. This behavior of ( ) can be explained, on the one hand, by the small distance between the input and output antennas in the DL, which is why the electronic contribution to the measured values of ( ) does not exceed 2...3 dB in the long-wave region, see Fig. 2, b. On the other hand, these changes in the losses of the MSSW can be compensated at some frequencies by the difference in the electrodynamic properties of the structures and the inhomogeneity of the magnetic field in the probe station, see, for example, the dependencies 11 ( ) and ( ) in Fig. 2, c. Fig. 2, d shows the frequency dependence of the generated EMF ( ) in the YIG/Pt structure at in ≈ −5 dBm. It can be seen that near the long-wave 0 and short-wave limits of the MSSW spectrum, depending on ( ), two pronounced EMF peaks are formed, marked in Fig. 2, d as 1 and 2 . This behavior of ( ) correlates with the frequency dependence of the density of states function η( ) for the spectrum of dipole MSSW obtained in [21], the form of the dependence of which for =939 Oe is shown in Fig. 2, d. Thus, it can be argued that in the structures under consideration, the platinum film does not lead to noticeable differences in the MSSW spectrum in comparison with the isolated YIG films. Therefore, when analyzing the development of parametric processes and their influence on EMF, it was assumed that the spectrum of SW and the character of the density of states function η( ) for the YIG/Pt structure are identical to the case of free YIG films. Note that the volt-watt sensitivity of the structure, taking into account the bidirectionality of the MSSW excitation by the antenna and the calculated values of the coefficient ( ) (see Fig. 2, c), is κ ≈ 2 · 10 −4 V/W for the frequencies 0 and The YIG(8.8 µm)/Pt(9 nm) structure, the measured dependencies ( ), ( ), 11 ( ), ( ) and ( ) for the field =939 Oe had a character similar to that shown in Fig. 2. Values of the parameter κ for peaks 1 and 2 turned out to be an order of magnitude higher than in the structure based on a 14.6 micron thick YIG , and amounted, respectively, to κ ≈ 2.1 · 10 −3 V/W and κ ≈ 1.1 · 10 −3 V/W. > > (4/3)π at frequencies 0 < < 2γ , the behavior of MSSW at > 0 is determined by 4M processes, whereas 3M processes dominate at frequencies 2γ < < . Fig. 3 shows characteristic changes in the dependencies ( ), ( ), ( ) and 11 ( ) caused at > 0 by the development of parametric instability of the MSSW on the example of the YIG(14.6 µm)/Pt(9 nm) structure for the fields 1 = 428 E< (4/3)π , (4/3)π < 2 = 809 E< 2π , 3 = 939 E> 2π values of th for MSSW at the frequency of were determined by decrease in the amplitude of the ( ) of the output signal of the MSSW at ∼ = th (Fig. 3, d ).
Taking into account the values of the conversion factor ( ) shown in Fig. 3, b, and the  Fig. 3 d, we get that the values of th at which parametric instability develops are 4 th ≈100 MW in the case of 4M processes and 3 th ≈ 0.6...6 MW in the case of 3M processes. The threshold values of the magnetization amplitude th MSSW can be matched to the specified threshold powers using the ratio [50,51]: where the product of · determines the cross-sectional area of the film through which the power of the MSSW is transferred. Calculated using (9) values of th with parameters corresponding to Fig. 3, d, are 4 th ≈20 Gs in the case of 4M processes, and 3 th ≈ 0.7...4 Gs in the case of 3M processes. The obtained values are several times higher than the typical values of th for YIG films, and higher than the estimates performed according to Sull's theory for homogeneous pumping, which gives the values of 3 th ≈ α · /(4πγ) ≈0.03 Gs and 4 th ≈ √︀ ( · α · /(4πγ)) ≈ 2 Gs. This discrepancy can partly be attributed to the absorption of part of the incident power by the conduction electrons of the Pt film, which is separated from the input antenna by a distance of ξ ≈30 microns, as well as the diffraction divergence of the MSSW with wavelengths λ ⩾ , which may affect the estimate of th using (9). However, the method used to determine th by the drop in the amplitude of the output signal ( ) can also give a noticeable error in determining 3 ,4 th due to the smallness of the distance between the antennas. Indeed, with a small supercritical pumping ⩾ 0, the additional nonlinear attenuation of the ′′ introduced by parametric instability may be too small to cause noticeable changes in the behavior of the ( ) for DL at the length of . Note that shown in Fig. 3, a, b, d changes in the dependencies ( ), 11 ( ) and ( ) at > 0 are characteristic [38,[48][49][50][51] for the parametric instability of the MSSW and are associated with a drop in magnetic susceptibility χ( ) and the growth of nonlinear losses of MSSW.
The influence of parametric instability on the EMF generation in the YIG/Pt structure by travelling MSSW is illustrated by the results of measuring the dependencies of ( ) obtained for the values of in =2 dBm, 8 dBm and 10 dBm, which in Fig. 3, c correspond to curves 1,2 and 3, respectively. At 3 = 939 Oe, in ( ) dependence , two EMF peaks are observed with maxima near the frequencies 0 ≈4.43 GHz and ≈5.09 GHz, which correlates with the frequency dependence of the density of states in the MSSW spectrum shown in Fig. 2, e. With increasing power, the position of the maxima 1 and 2 shifts «down» in frequency by the values ∆ 1 and ∆ 2 , respectively, which amounted to ∆ 1 ≈ −25 MHz and ∆ 2 ≈ −60 MHz, at the level of in ≈10 dBm see fig. 3, c. Note that with increasing power in ( ), the maximum of the coefficient ( ) at frequencies ≈ also shifts «down» in frequency and decreases its value by about 15%, see Fig. 3, b. The values of ( ) grow with increasing in and reach the values of 1 ≈ 1200 nV and 2 ≈ 500 nV at in ≈10 dBm. It should be noted that the dependence of 1 ( ) is close to linear, whereas 2 ( ) has a nonlinear character, see Fig. 4, a, where the dependences of the 1,2 peaks on the power of the MSSW for the selected values of magnetic fields are given. In addition, the top of the peak 2 is noticeably smoothed, compared to the case of < 4 th . The marked difference in the nature of dependencies 1 ( ) and 2 ( ) can be associated with two consequences. Firstly, at > 4 th , the nonlinear contribution to the MSSW decrement [50,51] leads to the limitation of the MSSW power. Secondly, the increase in the power of the MSSW leads to a shift in the frequency band of the MSSW due to a decrease in the projection of the film magnetization 4π ( ) on the direction of the magnetic field ⃗ due to the heating of the film with microwave power and the effect of dynamic demagnetization [58] where the dependence on the coordinate reflects the attenuation of the amplitude of the MSSW ( ) along the direction of propagation. In Fig. 2, e curves 1 and 2 show the densities of states η( ) in the MSSW spectrum, calculated by the formulas [21] for magnetization values 4π = 1750 Gs and 4π = 1740 Gs, respectively. In this case, it is assumed that the value 4π = 1740 Gs corresponds to the film section near the input antenna, where the amplitude of the MSSW is ( = 0) maximum. With the propagation of MSSW, attenuation leads to an increase of 4π ( ) and a shift of the MSW frequency band «up» in frequency in accordance with the expressions for the limits of the spectrum: It can be seen that near the long-wave limit, the MSSW, excited at the input, integrates all the singularities in the density of states η( 0 ( )), whereas near due to the shift of the MSSW frequency band, the contribution to electron-magnon scattering from the singularities η( ( )) falls when > 0 . At 2 = 809 Oe, the dependence ( ) also demonstrates the presence of peaks of 1 and smaller and shifted «down» in frequency relative to the short-wave limit of the MSSW spectrum . Its position coincides with the boundary frequency 3 th ≈ 2γ for 3M decays [48][49][50][51]. In fact, 3M processes «cut» the values of ( ), ( ) and ( ) for frequencies > 3 th . At the same time, the absence of a peak at the frequency is explained by its small value 2 ( ) = κ · ≈ 1.2 nV under conditions of limited MSSW power due to 3M decays = 3 th ≈ 6 µ W.The magnitude of the peaks 1,2 monotonically increases with the input power, see curves 2 and 4 in Fig. 4, a. There were no noticeable changes in the position of the peak 2 with increasing power, since its position is determined by the frequency 3 th , which in this case is determined by the bias field. With an increase in the power of the MSSW, the position of the maximum 1 near the frequency 0 ≈ 4.09 GHz shifts by ∆ 1 ≈ −20 MHz, see Fig. 3, c.
At 1 = 428 Oe, the frequency 0 ( 1 ) ≈2.7 GHz exceeds the boundary frequency for 3M processes 3 th = 2γ 1 ≈2.4 GHz ( 0 ( 1 ) > 3 th ) and 3M processes are allowed in the entire range of the existence of MSSW. At the same time, the threshold values of the power of 3M processes 3 th in this case are 3 th ⩽ −18 dBm, see Fig. 3, d. From Fig. 3, a-c it can be seen that at the supercriticities of the MSSW signal ≈20 dB, parametric processes significantly affect the excitation ( 11 ( ), ( )) and transmission ( ( )) MSSW in the structure. In the ( ) dependence, only the peak EMF 1 was observed at frequencies near 0 . The values of 1 monotonically grew from in , reaching the values 1 ≈1630 nV at the maximum available power in ≈ 10 dBm , see Curve 1 in Fig. 4, a. At the same time, the frequency corresponding to the maximum of 1 decreased by ∆ 1 ≈ −10 MHz with increasing power.
In Fig. 4, a the dependences of the peak values 1,2 on the power of the MSSW = in · Fig. 4. a -Dependencies of 1,2 peaks on MSSW power = in · for the YIG(14.6 µm)/Pt(9 nm) structure. Number (1) shows the 1 peak at ≈ 428 Oe, numbers (2, 4) show 1 and 2 peaks, respectively, at ≈ 809 Oe and numbers (3, 5) denote 1 and 2 peaks, respectively, at ≈ 939 Oe. Insets show the character of 1,2() dependencies at low levels of power. b and c -Frequency dependencies of 1 with the change of magnetic field from 214 till 668 Oe at constant level of input power in ≈ 2 dBm for the YIG(14.6 µm)/Pt(9 nm) and YIG(8.8 µm)/Pt(9 nm) structures, respectively. Vertical dash line denotes the boundary field and frequency for 3M decays in an isotropic YIG film. Asterisk shows the maximum EMF value as well as the peak at = 428 Oe discussed in Figures 3 and 4 for 1,2,3 are presented. The dependencies 2 ( ) are shown by curves 4 and 5 and demonstrate monotonous growth. However, at a power of ⩾ 200µ , the dependence of 2 ( ) deviates from linear, as shown in inset to Fig. 4, a. The volt-watt sensitivity of the detector at frequencies near the short-wave limit is κ ≈ 2 · 10 −4 V/W at a power of < 200 µ and decreases to the values κ ≈ 10 −4 V/W at ≈ 5 µ . The dependencies 1 ( ) for values > 3 th are close to linear, see curves 1-3. At the selected values 1,2,3 , the volt-watt sensitivity of the detector κ = / reaches the highest values at the frequency 0 and at 1 ≈ 428 Oe it is κ ≈ 1.6 · 10 −3 V/W. In the case of the structure based on the YIG film with the thickness 1 ≈8.8 microns, the behavior of the dependencies of 1,2 ( ) was similar to the one shown in Fig. 4, a with the only difference that the values of the parameter κ were an order of magnitude higher. Fig. 4, b and 4, c show the dependencies of the peak 1 on the frequency 0 (or the field ) at a fixed input power level in ≈ 2 dBm for the structures based on the YIG films with a thickness of 14.6 microns and 8.8 microns, respectively. From Fig. 4, b it can be seen that the EMF reaches a maximum of 1 ≈ 1200 nV ( ≈ 2 · 10 −3 V/W) at * 0 ≈ 2.9 GHz, which corresponds to the field * ≈480 Oe. Note that at * ≈ 480 Oe for the frequency * 0 ≈ 2.9 GHz, the condition * 0 > 2γ * ≈ 2.71 GHz is fulfilled at which 3M-decay processes limit the power of the MSSW in the entire excitation frequency band, similar to that shown in Fig. 3, a for the field 1 = 428 Oe. In this case, the frequency * 0 and the field * are less than the usual estimates of the boundary values 3 th ≈ 3.25 GHz and 3 = (4/3)π ≈ 583 Oe for 3M decays of dipole MSSW at the frequency 0 in YIG films [35] at values of approximately 350 MHz and 100 E.
From Fig. 4, c it can be seen that in the YIG(8.8 µm)/Pt(9 nm) structure maximum EMF 1 ≈ 7.3 µ ( ≈ 2 · 10 −2 V/W) is achieved at the frequency of * 0 ≈2.8 GHz and the field * ≈ 460 Oe. The difference in the values of * 0 and * for the structures shown in Fig. 4, b, c is highlighted with a gray fill and does not exceed 100 MHz and 20 Oe, respectively, can be explained by the difference in the anisotropy fields in the YIG films, as well as power oscillations in the microwave cables with the frequency. Note that the oscillating nature of the dependence 1 ( , ) in Fig. 4, b, c has nothing to do with the mechanism of ISHE, does not depend on the magnetic field and is mainly associated with the influence of re-reflections in the measuring microwave path. The amplitude of the oscillations increased significantly if a microwave isolator was excluded from the microwave path between the modulator and the microwave probe.
In general, the type of dependencies 1 ( 0 ( )) in fig. 4, b, c is similar in a form to the ( ) dependencies observed earlier in [26-28, 33, 34]. It can be seen that the EMF reaches maximum values at frequencies * 0 and fields * , at which the power of the MSSW is limited to 3M decays. Such an increase in EMF due to ISHE in the YIG/Pt structures under conditions of limiting the power of the MSSW was called «amplification of spin current generation due by 3M decays» [26].

Influence of secondary spin waves on the effect of «amplification of spin current generation by three-magnon decays»
The increase in the efficiency of detecting spin current (Marked in Fig. 4, b) under conditions of limiting the power of MSSW at 3M decays was explained in the works [26][27][28] by the transfer of a impulse to the magnon system from the lattice, as well as the influence of PSW on the rate of relaxation of magnetization in the YIG [27,28]. It was believed that the appearance of two PSW instead of one pump quantum leads to an increase in spin current emission, and the difference in the spin moments of the pump magnon and two PSW is compensated by the lattice of the YIG at times of the order of spin-lattice relaxation. Since the measurements of the value of the ISHE performed in the pulsed mode showed that the stationary amplification of the spin current emission is determined by the attenuation of spin waves, it was concluded that the main contribution to the generation of spin current is given by PSW with a long lifetime. The PSW filling the «bottom» of the spectrum of the dipole-exchange waves were considered as such «long-lived» PSW. Note that for tangentially magnetized films at the "bottom"frequency bot , the negative dipole dispersion is compensated by the positive exchange dispersion, and the condition ( bot ) → 0 is fulfilled. Consequently, the frequency bot corresponds to the condition of the appearance of van Hove singularities in the density of states of SW.
However, for the experiments on the generation of EMF by travelling MSSW in the YIG/Pt sructure, considered here, it is not obligatory that population of the spectrum «bottom» by parametric magnons necessarily leads to an increase in the efficiency of spin current generation through the interface. As a proof, let us turn to Fig. 3, where the results of EMF measurement for the structure magnetized in the field = 809 Oe are given. From Fig. 3, c it can be seen that when the MSSW frequency gets into the frequency range [ 3 th , ] (4.53...4.72 GHz), the EMF signal level does not exceed the noise level. At the same time, a noise peak with a maximum near the frequency 3 th ≈4.53 is formed in the spectrum of the MSSW pumping with a frequency from the interval 4.53 ⩽ ⩽ 4.72 GHz at supercriticities of ⩾ 15 dB. As an example Fig. 5 shows the spectrum of the pump signal at the frequency ∼ = 4.6 GHz for the field = 809 Oe, where a noise peak is formed near the frequency 3 th ≈ 4.53 GHz. The mechanism of the appearance of such a noise peak is associated with the non-threshold processes of merging PSW (3) with frequencies 1,2 ≈γ · ∼ = bot and, therefore, reflects the process of filling the «bottom» of the SW spectrum for films with parametric magnons [49][50][51].
To understand the mechanism of the «amplification of spin current generation due to 3M decays» let's turn to Fig. 5, where for the case of in =10 dBm, the spectra of the pump signal of the MSSW measured on the output antenna at a frequency fp close to the long-wave limit of the MSSW spectrum ( ≈ 0 ) are shown. In Fig. 5, the spectra are for the values of the field from the range from 316 to 652 Oe, which includes the * fields corresponding to the EMF maximum in Fig. 4, b, c and covers the areas of both 3M and 4M decays. The horizontal dotted lines in Fig. 5 shows the amplitude of the EMF peak at the pump frequency.
It can be seen that at a given level of in , there is a noise signal in the output signal spectrum associated with the parametric instability of the MSSW [39,[48][49][50][51]. The noise intensity is maximal near the pumping frequency . In this case, we can distinguish a noise peak with a frequency band δ = − , where the frequencies and correspond to the lower and upper the boundaries of the noise peak, which can be defined as frequencies where the amplitude of the noise components of the peak reaches a «plateau» in the spectrum, see Fig. 5. The higher amplitude of the noise on the plateau at the upper frequencies is explained by falling of PSW into the MSSW spectrum. At frequencies > , the amplitude of the noise signal drops.
From Fig. 5 it can be seen that the noise peak near the frequency is observed both in the case of 3M and 4M processes under the condition ≈ 0 1 . The amplitude of the noise components in the spectrum depends nonmonotonically on the magnitude of . In the range 440 ⩽ ⩽ 500 Oe, the amplitude of the noise components is maximum and exceeds the level of −60 dBm. At the same time, with a decrease in from 652 Oe, to 316 Oe" the width of the noise peak δ increases from δ ≈ 100 MHz to δ ≈ 400 MHz, as can be seen from Fig. 5 and fig. 6, where the dotted curves 9 mark the boundaries of the noise peak and . By gray fill in Fig. 6 an area of fields is highlighted in which the amplitude of the noise peak exceeds −60 dBm.
The described dynamics of the noise peak in the MSSW spectrum correlates with the dynamics of the amplitude of the EMF peak 1 ( , ) when the field changes, shown 1 In the case of 3M decays and > 3 th , the noise peak is localized near the frequency 3 th , see   Fig. 4, b, c. Indeed, the EMF value reaches maximum values in the range of fields ≈ ≈ 460...480 Oe, at which the noise peak in the spectrum has the largest amplitude values and its width is δ ≈ 400 MHz, see Fig. 5. This allows us to conclude that the effect of «amplification of spin current generation due to 3M decays» reflects the process of population of the spectrum of the film by SSW generated as a result of non-threshold processes of PSW fusion (3). It should be noted that this conclusion contrasts with the one that was made earlier when discussing the dependence ( ) in Fig. 3, c at = 809 Oe and where appearance of the noise peak in the pump wave spectrum (see Fig. 5 for = 809 Oe) was not accompanied by an increase in EMF. This contradiction can be explained if we take into account that for the range of fields from 440 to 525 Oe, a noise peak is formed near the long-wave limit of the spectrum of dipole MSSW 0 , where the density of SW states is characterized by van Hove singularities. At the same time, to explain the EMF maximum at the field * = 480 Oe and the pumping frequency * ≈ 2.9 GHz, when the boundary of the noise spectrum is shifted relative to * at ∆ = * − ≈ 200 MHz, it is necessary to use the representation of the MSSW spectrum taking into account the influence of the fields of crystallographic anisotropy of the YIG [59,60].
Indeed, taking into account the anisotropy fields leads to two main changes in the SW spectrum relative to the case of an isotropic film. Firstly, anisotropy leads to a change in the values of the characteristic frequencies in the SW spectrum with respect to the magnetized film. In particular, for dipole MSW in a YIG film with a crystallographic orientation (111), the frequencies of the long-wave ( 0 ) and short-wave limits of the MSSW ( ) and MSBVW ( bot ) can be estimated using the expressions [59]: where = · ( + + ), = − , = γ · 2 / , = γ · 1 / , and 1uniaxial and cubic anisotropy constants, -angle between direction the in-plane magnetic field and the crystallographic direction [110] lying in the plane of the film with the crystallographic orientation (111). In Fig. 6 for a cubically anisotropic YIG(111) film characterized by a field = −40 Oe, magnetized along the crystallographic direction [110] ( = 0), the calculated dependences of 0 , bot , 2 bot and 2γ on magnetic field are shown. Vertical dotted lines mark the values of the magnetic fields 3 th ≈ 585 Oe and 3 th ≈ 545 Oe, below which 3M decay is possible at the frequency 0 ( ). Note that the field 3 th ≈ 545 Oe is close enough to the value = 550 Oe when the amplitude of the noise peak in the pumping MSSW spectrum increases, see Fig. 5.
Secondly, anisotropy leads to the appearance of frequency intervals in the spectrum of waves traveling perpendicular to the tangent field , in addition to the MSSW, where anisotropic magnetostatic forward (MSFVW) and backward (MSBVW) volume waves propagate. Such anisotropic MSFVW and MSBVW occupy, respectively, the frequency bands [ 0 , ] and [ , 0 ], where the frequency 0 is the long-wave limit of MSFVW and MSBVW, and the frequencies and -the short-wave boundaries of the MSFVW and MSBVW defined by expressions (15) and (16), respectively [59]: = ) and bottom ( ) boundaries of noise peak around the pumping frequency . Grey fill shows the field region where amplitude of the noise peak is above −60 dBm The dependencies of and on the field are shown in Fig. 6. It can be seen that the width of the noise peak correlates with the width of the frequency band occupied by anisotropic volumetric MSW. The discrepancy may be due to the fact that the calculations in Fig. 6 were performed neglecting the contribution of the uniaxial anisotropy field, as well as limiting the range of wave numbers ⩽ π/ ≈ 8 · 10 3 cm −1 , due to the finite width ≈4 microns of microantennas. In Fig. 7, a the result of micromagnetic modeling [61][62][63] of the spectrum of spin waves with wave numbers | | ⩽  The intensity of the gray color reflects the amplitude of the Fourier component for spin waves traveling in the direction , perpendicular to the magnetic field when the film is excited by the pulse of the magnetic field normal to the surface of the film ℎ = · sinc (2π max [ − 0 ]), where = 100 A/m -amplitude, max = 5 GHzmaximum frequency, -time, 0 = 50 ns -time shift. It can be seen that there is a frequency band near the long-wave limit, where the dispersion in the region of small wave numbers corresponds to the dispersion of the MSFVW and MSBVW. In Fig. 7, b, the dots highlight the areas of dispersion where → 0 and van Hove singularities are formed in the density of states. The ocupation of these branches by the SSW spectrum leads to an increase in the efficiency of spin current generation.
The reason why the EMF reaches a maximum at a certain field * is due to the fact that at the field * , the frequency band in which van Hove singularities exist in the spectrum of anisotropic MSFVW and MSBVW optimally overlaps with the frequency band where the SSW are formed. To clarify what has been said, let us turn to the position of the boundaries of the noise peak and relative to the limit frequencies 2 bot and 2γ in Fig. 6. It can be seen that for the interval of magnetic fields * , highlighted by a gray fill, the detuning of the frequency from the frequencies 2 bot and 2γ is minimal and increases with decreasing . Since SSW populate the spectrum of the film at frequencies > 2 bot , the number of SSW populating a section of the spectrum of anisotropic MSFVW and MSBVW at < * turns out to be less than at = * . At the fields > * , the frequency 2 bot lies above the frequency and the SSW fill only part of the frequency band ∆˜, which reduces the efficiency of spin current generation. At the fields < * , the frequency 2 bot lies below the frequency .

Conclusion
Thus, the influence of the processes of three-magnon decays on the EMF generated due to the inverse spin Hall effect in the structure of the YIG/Pt during the excitation of traveling MSSW, the dispersion of which is close to the dispersion of dipole MSSW in nonmetallized LCG films, has been studied. It is shown that the three-magnon parametric instability can significantly change the form of the dependencies ( ) and ( ). It was found that 3M processes significantly limit the EMF signal in the short-wave part of the spectrum of dipole MSSW, whereas at pumping frequencies close to the long-wave limit 0 of the MSSW spectrum ( ≈ 0 ), the EMF signal demonstrates a quasi-linear growth with an increase in the MSSW power . A mechanism is proposed to explain the increase in the efficiency of EMF generation under conditions of limiting the power of MSSW due to 3M decays. The mechanism is associated with the SSW population of the region of the spectrum of anisotropic dipole-exchange spin waves, characterized by the presence of singularities in the density of the magnon states (van Hove singularities).